Shadows of the circle : conic sections, optimal figures, and non-Euclidean geometry

Conic Sections, Optimal Figures and Non-Euclidean Geometry

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Non-Euclidean Geometry

Vagn Lundsgaard Hansen Department of Mathematics Technical University of Denmark

1 | | ; | World Scientific Singapore • New Jersey • London « Hon^ Kbngf

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SHADOWS OF THE CIRCLE Conic Sections, Optimal Figures and Non-Euclidean Geometry Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-3418-X

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Printed in Singapore by Uto-Print

Preface Geometrical concepts and considerations enter the description of the world around us on many levels. For this reason alone, it is important to pay attention to the teaching of geometry throughout the educational system. Geometrical shapes appeal to the imagination, and the study of geometry offers opportunities for developing the faculty of visualization. In a challenging way geometry combines the concrete with the abstract. Together with investigations of the properties of the integers, the study of geometrical figures and their properties lies at the roots of mathematics, and geometry plays an important role in the history of mathematics. Restrictions of a geometrical kind often occur in the applications of mathematics, in engineering as well as in mathematical models of real world phenomena, and knowledge of geometry is necessary both to formulate and to solve the problems. The aim of this book is to throw light on various facets of geometry through development of four geometical themes. The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola. The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations. Here, too, we meet all three types of conic sections briefly. The emergence of non-Euclidean geometries in the beginning of the nineteenth century is one of the dramatic episodes in the history of mathematics. In the last theme we develop the non-Euclidean geometry in the Poincare disc model of the hyperbolic plane. v

VI

Preface

It is my hope that these topics will be an inspiration in connection with teaching of geometry at various levels including upper secondary school and college education. The book is, however, also written for the benefit of the reader who has heard about some of these subjects before, but who would like to see them presented in a non-traditional way, either in connection with further education or simply out of an interest in mathematics. The first three chapters of the book is a translation of my book "Temaer fra Geometrien" published in 1992 by the Danish Association of High School Teachers in Mathematics. The English translation of these chapters has been prepared with the very efficient help of Bodil Kirketerp Nielsen. The figures have been drawn by Beth Beyerholm. The chapter on non-Euclidean geometry follows closely my article "The dawn of non-Euclidean geometry," Int. J. Math. Educ. Sci. Technol. Vol. 28, No. 1 (1997), 3-23. The figures in this chapter have been created using a program for geometrical constructions written for the purpose by my good colleague Jens Gravesen. My friend, Professor Robert Greene from UCLA, has read most of the text and has suggested numerous improvements in the language and in the presentation of the material. I am extremely grateful for his help. Lyngby, 1997

Vagn Lundsgaard Hansen

Contents Preface Chapter 1: An ellipse in the shadow

1

The ellipse as a plane section of a cylinder The equation of the ellipse A parametrization of the ellipse The ellipse as a locus Directrix for the ellipse Geometrical determination of foci and directrices for the ellipse The tangents of the ellipse An application to gear wheel movements Sources for Chapter 1

3 5 6 8 10 12 13 17 21

Chapter 2: W i t h conic sections in the light The ellipse as a plane section in a cone Geometric determination of foci and directrices for a conic section The parabola The hyperbola Hyperbolic navigational systems Conic sections as algebraic curves Epilogue Sources for Chapter 2

vn

23 23 27 29 31 35 38 42 44

Contents

vm

Chapter 3: Optimal plane

figures

Isosceles triangles Perron's paradox Some simple geometrical problems without solutions A fundamental property of the real numbers Maxima and minima of real-valued functions The equilateral triangle as optimal figure The square as optimal figure The regular polygons as optimal figures Some limit values for regular polygons The isoperimetric problem Epilogue: Elements of the history of the calculus of variations Sources for Chapter 3

Chapter 4: The Poincare disc model of non-Euclidean geometry Euclid's Elements The parallel axiom and non-Euclidean geometries Inversion in a circle Inversion as a mapping Orthogonal circles and Euclid's Postulate 1 in the hyperbolic plane The notion of distance in the hyperbolic plane and Euclid's Postulate 2 Isometries in the hyperbolic plane Hyperbolic triangles and n-gons The Poincare half-plane Elliptic geometries Sources for Chapter 4

Exercises Index

45 46 48 48 49 51 54 57 59 60 62 67 69

71 71 74 76 79 84 86 89 91 93 94 96

97 109

Chapter 1

A n ellipse in t h e shadow In the car on our way south during the summer holidays, I suddenly saw an ellipse on the front panel of the car. It came from a round suction disc on the windshield that cast a sharp shadow in the baking sun. After we arrived at the holiday resort, we later saw an ellipse again, as we were playing ball; the shadow of the ball was an oval in the sand. It then struck me that the fact that such a shadow is an ellipse is somehow obscured by the way ellipses are normally introduced at school. I decided that it could be interesting to turn things around and take this fact as the point of departure. It began with the Greeks. People presumably had always noticed that round objects, that is, spheres, cast shadows that are usually oval-shaped, rather than circular themselves. But the formal understanding of these oval shapes, now called ellipses, began only after the Greeks initiated the systematic study of geometry. The ancient Greeks admired geometry, that is to say the study of shapes and figures and their properties, and they gave it a prominent position in their intellectual life. They also came to know surprisingly much about it; Greek geometry had a level of sophistication that was not regained until the Renaissance. (A short introduction to the early history of geometry is given in Chapter 4, Section 4.1.) Around 300 B.C. Euclid collected the knowledge of his time in the famous Elements, and this work was used in the teaching of geometry well into this century. The brilliant Greek mathematician Apollonius, who lived circa 262190 B.C. and was called 'The Great Geometer' by his contemporaries, wrote the second major work from the classical Greek period of geometry. It contains a systematic treatment of the conic sections: ellipse, parabola and hyperbola. The Greeks considered the circle to be the most perfect figure of all. This idea of a 'perfect circle' persists in common language and continued in astro1

2

An ellipse in the

shadow

nomy for a surprisingly long time. It was supposed almost automatically t h a t astronomical motions must occur in circular orbits until the discoveries of Kepler in the early seventeenth century of the ellipse as the shape of orbit of planetary motions. T h e first two chapters of this book are devoted to a study of the conic sections. In our discussion of the ellipse and the other conic sections, we shall take a slightly different p a t h from Apollonius, who did, however, know most of the geometry here. He did not know analytic geometry (coordinate geometry); t h a t was not developed until the seventeenth century by Descartes and, independently, Fermat. Nor did he have trigonometric functions at his disposal in suitable form. T h e missing topics made his investigations more complicated t h a n they have to be if these aids are used, as they will be here.

T h e Surface of a Cylinder

T h e Surface of a Cone F i g u r e 1.1

F i g u r e 1.2: Plane section of a sphere

The ellipse as a plane section of a cylinder

3

1.1 The ellipse as a plane section of a cylinder Greek geometry in the plane put special emphasis on the straight line and the circle. The corresponding shapes in space are the plane and the sphere (a ball). There are also two surfaces that can be built from straight lines and circles, namely the cylinder (a tube) and the cone; cf. Figure 1.1. If you cut a plane section in the surface of a sphere, you get a circle; cf. Figure 1.2. What do you get, if you cut a plane section in the surface of a cylinder? This question has a direct connection to the shape of the shadow of a disc or a sphere on a plane, in particular the shadow of a ball on a beach. The rays of the sun are all parallel, so that the ball shades a circular column (a right circular cylinder), and the shadow in the sand is just the area of intersection between this cylinder and the plane of the sand; cf. Figure 1.3.

A Figure 1.3: Shadow of a ball In Figure 1.4 we have extracted from the shadow idea the parts that are interesting from the mathematical point of view. The figure represents a segment of the surface of a right circular cylinder that is bounded above by a plane section. Let the radius of the cylinder be b, and assume that the section plane forms the angle u with the axis of the cylinder, where 0 < u < | . Notice that the angle u corresponds exactly to the angular elevation of the sun in Figure 1.3. For u—f, the curve of intersection is a circle with radius b. When u approaches 0 through decreasing values, the curve of intersection gets more and more flattened in relation to the shape of a circle. The curve of intersection is called an ellipse. The real number e = cos(u), corresponding to the

4

An ellipse in the

shadow

angle of inclination u, is called the eccentricity of the ellipse. We notice t h a t the eccentricity of the ellipse is a real number in the interval 0 < e < 1. Any line segment t h a t connects two different points on the ellipse is called a chord of the ellipse. T h e largest chord is called the major axis of the ellipse, and the smallest chord through the axis of the cylinder, the minor axis of the ellipse. Clearly, the minor axis must be a diameter in the cylinder. Let a denote the length of the major semiaxis. Obviously, the length of the minor semiaxis is 6, the radius of the cylinder.

,b£2

ea B'

F i g u r e 1.5

F i g u r e 1.4

T h e plane section through the major axis perpendicular to the minor axis is shown in Figure 1.5. Since e = cos(u), the leg B'B in the right triangle OB'B has the length ea. Then, with the help of the P y t h a g o r e a n Theorem, we immediately get the following equation to determine the eccentricity of the ellipse from its semiaxes: b2 + e22a~ 2 - a 2 , or, equivalently,

1 - e2 =

^r.

The equation of the ellipse

5

1.2 The equation of the ellipse We now install a system of ^-coordinates in the plane of the section in such a way that the ar-axis lies along the major axis, and the y-axis along the minor axis of the ellipse; cf. Figure 1.4 and Figure 1.6.

Figure 1.6

Figure 1.7

Let P(x,y) be an arbitrary point on the ellipse. We are looking for an equation, that is to say a relation between the coordinates x and y that describes the points on the ellipse. With this in mind, we consider the plane that contains the minor axis of the ellipse and is perpendicular to the axis of the cylinder. In this plane we install a system of z'y-coordinates as shown in Figure 1.6. We observe, then, that the point P(x,y) is situated on the ellipse if and only if its right-angled projection onto the ar'y-plane, namely the point P'(x -sin(M), y)), is situated on that circle with radius b in which the x'y-plane intersects the surface of the cylinder; cf. Figure 1.6 and also the enlargement of the triangle CPP' shown in Figure 1.7. In other words, it holds that P(x, y) is on the ellipse if and only if (x • sin(w))2 + y2 = 62. By substitution of b2 sin 2 (u) = 1 — cos 2 (u) = 1 — e 2 = - j , into this equation we immediately get that

6

An ellipse in the shadow

b2 2

,

2

i2

cr

or, equivalently, that y2 • + — = 1

This is the equation of the ellipse that we are looking for. The quantity b is the radius of the cylinder. The quantity a, with b given, then determines the eccentricity e and thereby the angle u. So the two quantities a and b determine exactly which ellipse we are considering. Therefore it makes sense to say 'the ellipse with major semiaxis a and minor semiaxis b.' And so we now have the following theorem. T h e o r e m 1.1 In a standard system of rectangular xy-coordinates in the Euclidean plane, the ellipse with major semiaxis a and minor semiaxis b can be described by the equation

a?

b2

1.3 A parametrization of the ellipse Since the equation of the ellipse can be written in the form

(DM!) 2 -. it follows that the point (£, |-) is situated on the unit circle and, accordingly, that the ellipse can be described by the point set {(*, y) | ( ^ , | ) = (cos(t),sin(*)), 0 < t < 2TT} . From this we immediately find a parametrization of the ellipse, that is to say a description of the points on the curve of the ellipse by a single parameter, in this case the parameter t.

A parametrization

7

of the eUipse

T h e o r e m 1.2 In a standard system of rectangular xy-coordinates in the Euclidean plane, the ellipse with major semiaxis a and minor semiaxis b can be described by the parametrization (x,y)

= {acos(t),bsm(t)),

0 < t < 2TT.

(bcos(t),bs\n{t)) (acos(t),asin(t)) (acos(t),bsin(t))

/yS^~~b

ll

0

a\

x

F i g u r e 1.8 From the above parametrization of the ellipse one easily sees t h a t the ellipse can be constructed from the circle with the major axis as diameter by compressing in the ratio - around the major axis or, correspondingly, from the circle with the minor axis as diameter by enlarging in the ratio |- around the minor axis; cf. Figure 1.8. One can use this to prove t h a t the shadow of a circle (or a disc) cast on a plane surface by parallel light is always an ellipse, as asserted in the first p a r a g r a p h of the book. In fact, there is a diameter in the circle orthogonal to the light rays such t h a t the shadow of the circle is congruent to the ellipse obtained by compressing or enlarging the circle in a certain ratio around this diameter; the diameter is unique except when the light rays are orthogonal to b o t h the plane of the circle and the plane of the shadow. A degenerate situation occurs when the plane of the circle is parallel to the light rays; in this case the shadow is a line segment. See Exercise 7 for a special case. In the following we shall continue t o consider the ellipse and the system of ^^-coordinates as before. Moreover, we shall now consider the points E(—ea, 0) and F(ea,0) shown in Figure 1.9. These points are called the focal points, or the foci, of the ellipse. Furthermore, from an arbitrary point P on the ellipse we have drawn the line segments to the foci, the so-called focal radii from P.

8

An ellipse in the shadow

1.4 The ellipse as a locus In Theorem 1.1 and Theorem 1.2, we have described the ellipse by an equation and a parametrization, respectively. These are useful, but they are not very geometric descriptions. We might ask whether the ellipse as a plane figure can be characterized by a geometrical property. Or, as the mathematicians put this question: Can you describe the ellipse as the locus for the points in the plane that complies with a suitable geometrical property? This is possible: The ellipse is the locus of points having its distances to two fixed points (the foci of the ellipse) adding up to a constant. For example you can mark out an oval flower bed in your garden with an ellipse as boundary curve by sticking two pegs into the ground (the foci of the ellipse), connecting them by a suitably long string, and next moving a stick around along the string in such a way that it is kept tight all the time. This characterisation of the ellipse can be established by completely geometrical reasoning as we shall see in Section 1.6. But since the geometrical proof historically came much later than the theorem we shall first establish it analytically as follows. Theorem 1.3 Let a be an arbitrary positive real number, and let e be a real number in the interval 0 < e < 1. Let E and F be two points in the Euclidean plane with distance lea. Then any ellipse with major semiaxis a and eccentricity e is congruent to the locus of points P in the plane, for which the sum of the distances to the points E and F is constantly 2a. In other words, the ellipse with foci E and F and major semiaxis a is characterized by \PE\ + \PF\ = 2a.

Figure 1.9

The ellipse as a locus

9

Proof. We install a standard rectangular system of zy-coordinates in the plane so that the x-axis contains the points E and F and so that the y-axis is the perpendicular bisector of the line segment EF. An arbitrary ellipse with major semiaxis a and eccentricity e can then be placed as in Figure 1.9. First we consider a point P(x,y) on the ellipse. We then have to show that the point satisfies the desired relation between distances to the foci. This emerges from the calculations below where, in the first two calculations, we use the Euclidean distance formula, the equation of the ellipse, plus the formulas e V = a 2 - b2 and e2 = 1 - g . \PE\ = y/(x + ea)2 + y2 = \/x2 + 2eax + e2a2 + y2 = \/x2 + 2eax + a2 - b2 + b2

-x2 - J a2 + 2eax + ( 1

) x2

= v a2 + 2eax + e2x2 = \/{a + ea;)2 = a + ea

\PF\ = A/(Z - ea) 2 + y2 = \Jx2 - 2eax + e 2 a 2 + y2 = \jx2 — 2eax + a 2 — b2 + b2

-x2 = t a2 — 2eax + I 1

, ) x2

= v a 2 — 2eax + e2x2 = \/(a — ex)2 = a — ex. \PE\ + \PF\ = a + ex + a - ex = 2a. Conversely, assume now that P(x,y) is a point in the plane for which IP-E"! + \PF\ = 2a. We then have to show that the pair of coordinates (x,y) satisfies the equation of the ellipse. By applying the distance formula, we first get the equation y/(x + ea)2 + y2 + yj(x - ea)2 + y2 = 2a, which we rewrite as \J(x + ea)2 + y2 = 2a — \/(x — ea)2 + y2. By squaring, we next get the equation {x + ea)2 + y2 = 4a 2 + (a: - ea) 2 + y2 - Aa^/(x - ea)2 + y 2 ,

10

An ellipse in the shadow

which can be reduced to y/(x — ea)2 + y2 = a — ex. Squaring again, we get the equation (x - ea)2 +y2 = (a-

ex)2,

which can be reduced to

(l-eV+02 = (l-eVIf we now insert 1 — e 2 = ^ , this equation can finally be rewritten as X

-+y- 2 = l a2 b ' which is exactly the equation of the ellipse. This completes the proof of Theorem 1.3.

1.5 Directrix for the ellipse There is also another description of the ellipse as a locus. This description uses a single focus and a line called a directrix for the ellipse. Theorem 1.4 Let a be an arbitrary positive real number and let e be a real number in the interval 0 < e < 1. Let I be a line in the Euclidean plane and let E be a point in the plane with distance (- — e)a from I. Then any ellipse with major semiaxis a and eccentricity e is congruent to the locus of points P in the plane whose distances \PE\ from E and \Pl\ from I have the ratio e. In other words, the ellipse with focus E, directrix I, and eccentricity e is characterized by \PE\=e\Pl\. Proof. We use a standard system of ^-coordinates in the plane so that the z-axis contains the point E and is perpendicular to / and so that the point of intersection A between the z-axis and I has the coordinates A(—-,0), and E has the coordinates E{—ea,0). Notice that \El\ = \EA\ = -ea + - = ( - - e)a, e e

Dirextrix for the ellipse

11

as desired. An arbitrary ellipse with major semiaxis a and eccentricity e can thus be placed relative to the axes as shown in Figure 1.10. First we consider a point P(x,y) on the ellipse. We then have to show that the point satisfies the desired relation between the distances to / and E. From the calculations in connection with the proof of Theorem 1.3 we extract in particular that \PE\ = y/(x + ea) 2 + y2 = a + ex. Then we immediately get the desired relation between distances by the following rewriting: \PE\ = e(x + - ) = e\PL\ = e\Pl\.

,

i(-f,y)

A(-f,0)

—^,P(x,y) J

1

^S*

\E{-ea,0)

O

J ~x

I

Figure 1.10 Now assume conversely that P(x,y) is a point in the plane for which \PE\ = e|P/|, or, equivalently, \PE\ = e2\Pl\ . We then have to show that the pair of coordinates (x, y) satisfies the equation of the ellipse. By applying the Euclidean distance formula we first get the equation (x + ea)2 + y2 = e2(x +

-)2,

which by calculation makes „2„2

ar + 2eax + e ' V + y = e'xz + 2eax +
12

An ellipse in the shadow

( l - e V + y2 = ( l - e V If we now insert 1 — e 2 = |y this equation can finally be rewritten as x2 y2 \— = 1 a 2 + b2 ' which is exactly the equation of the ellipse. This completes the proof of Theorem 1.4.

1.6 Geometrical determination of foci and directrices for the ellipse In this section we shall briefly outline how one can determine geometrically the foci and directrices for an ellipse. The method is described by the Belgian mathematician G.P. Dandelin in a paper from 1822 and, no doubt, would have filled the ancient Greeks with enthusiasm if they had discovered it. Consider an ellipse produced as the curve of intersection in Euclidean threespace between a vertically oriented cylinder and a plane 0 that forms the angle u with the vertical direction. Place two spheres with the same radius as the cylinder inside the cylinder so that they both touch /?; one of the spheres lies below and one above /?; cf. Figure 1.11. We shall now show that the two points of tangency E and F for the spheres with /? are the foci of the ellipse and that the directrices of the ellipse are the two secants IE and lp between /? and the two horizontal planes that contain the circles of tangency between the cylinder and the two spheres. In order to show the above we choose an arbitrary point P on the ellipse. Let P\ be the point in which the generating line of the cylinder through P intersects the circle of tangency between the lower sphere and the cylinder and let L be the point in which the line IE intersects the line through P perpendicular to IE- Notice that the triangle PP\L has a right angle at P\. Since the line segments PE and PP\ are both tangent lines to the lower sphere we immediately get that \PE\ = | P P i | . From the right triangle PP\L we get that | P P i | = |PL|cos(«) = e | P i | , and thereby that \PE\ = e\PL\ = e\PlE\.

13

The tangents of the ellipse

According to Theorem 1.4 this shows exactly that E and IE is a corresponding pair consisting of a focus and a directrix for the ellipse. Correspondingly it can be shown that F and lp is a corresponding pair consisting of a focus and a directrix for the ellipse.

Figure 1.11 With reference to Figure 1.11, we can also give an alternative proof of Theorem 1.3. First notice that l-Pi?!, being equal to |-P-Pi|, is the distance of P to the lower horizontal plane. Similarly, \PF\ is the distance to the upper horizontal plane. For all points P on the ellipse it therefore holds that IPE'I + IP-F'I is constant equal to the distance between the two horizontal planes. For P at one of the end points of the major axis, it is clear that the sum of distances to the foci is the length la of the major axis. Hence \PE\+\PF\

= 2a,

for all points P on the ellipse as asserted in Theorem 1.3.

1.7 The tangents of the ellipse Consider the ellipse with foci E and F, eccentricity 0 < e < 1, and major semiaxis a. The minor semiaxis b is determined by the formula b2 = (1 — e 2 )a 2 .

14

An ellipse in the shadow

We place the ellipse in a system of ^-coordinates in the usual way; cf. Figure 1.12.

tangent at Po x (-a,0)

x = —a

x= a

Figure 1.12 Consider an arbitrary point PQ{XQ, 2/o) on the ellipse. We seek to determine an equation for the tangent of the ellipse at the point. At the point (a, 0), respectively (—a, 0), the tangent is vertical and is therefore given by the equation x = a, respectively x = — a. If yo 7^ 0 the ellipse can be described in the vicinity of XQ as the graph of a function y = f(x). By applying the equation of the ellipse we see that f(x) = ±yjb2 - £x2 with the sign determined by the sign of j/o- Then the tangent of the ellipse at -Po(a;o,yo) has the equation

V-yo =

f'{x0)(x-xo).

Since (x, f(x)) is a point on the ellipse it satisfies the equation

x2 . f{*Y _

1

a2 b2 If we differentiate this equation with respect to x we get

2x

2f{x)f'(x)

0, b2 and since f(xo) = yo it then follows immediately that

+

f'(xo)

b2x0 a2yo'

The tangents

of the

15

ellipse

Thereby we get the following equation for the tangent

y-

b2x0 .

2/o =

— X

2

.

a yo

which can be rewritten as x0(x - xQ) a2

+ 2 / o ( 26/2-

2/o)

and further as x0x a?

+ 2/02/ fc2 ~

1,

since [xo, 2/0) satisfies the equation of the ellipse. In this form the equation also includes the equations x = a and x = — a for the t a n g e n t s at ( a , 0 ) , respectively ( a , 0 ) . Thereby we have proven the following theorem. T h e o r e j n 1.5 For the ellipse with the

a2 the tangent

+

equation

62

at the point -Po(^o>J/o) has the xpx a2

+

' equation

y0y _ fc2 ~~

For the tangents of the ellipse the following theorem holds. T h e o r e m 1.6 The tangent at an arbitrary point af an ellipse bisects the exterior angle between the focal radii. Or equivalently: The normal to the ellipse at an arbitrary point bisects the angle between the focal radii. Proof. Consider an arbitrary point Po{xo, 2/0) ° n the ellipse. W i t h reference t o Figure 1.13 we shall prove t h a t the angles u and v are equal. For this, we consider the unit vector v # , respectively vp, on the focal radius from E, respectively F, towards P0, and a normal vector n to the ellipse at Po- T h e o r e m 1.5 shows t h a t the vector n below is a normal vector as required. From t h e calculations in connection with t h e proof of T h e o r e m 1.3 we extract in particular t h a t \EPQ\ = a + exo and \FPo\ = a — exo. Therefore, the vectors VE and vp can immediately be written in the form given below.

An ellipse in the

EP0

(xQ +

ea,y0)

\EP0\

a + exQ

\FPQ\

(x0 - ea,y0) a — exo

2i2/a;0

3/0 N

shadow

/,2

n = a Y - r z , —z ) = (¥x0, a b

'

2

i

a Z/o)-

normal at P0

Po{xo,yo)

F i g u r e 1.13 We shall compare the angles u and v by the help of their cosine values, and we therefore calculate the following scalar products making use of the fact t h a t (xo, Vo) satisfies the equation of the ellipse: n v

£

=

n • vp =

b2x2 + b2x0ea + a2y^

a2b2 + ex0ab2

a + exo

a + exo

b2x$ — b2x0ea + a2y^

a2b2 — exoab

a — exo

a — exo

= ab2

= ab2.

These expressions show t h a t n • vE = n -Vf, and hence t h a t cos(u) = cos(w). From this follows t h a t u — v and T h e o r e m 1.6 is proved.

An application to gear wheel movements

17

Theorem 1.6 shows that a ray of light or a sound wave emitted from a point source at one of the foci of an ellipse converges at the other focus after reflection in the ellipse. Since the paths that the various rays in the wave have traversed from one focus to the other all have the same length (the length of the major axis) the waves meet in the same phase and so intensify each other. This fact is made use of in kidney stone crushers. A kidney stone crusher has the form of an ellipsoid of revolution. When an ultrasonic impulse is emitted from one focus and the patient is placed in such a way that the kidney stone lies at the other focus, the energy of the wave that hits the kidney stone will be concentrated at the location of the stone, and it will be crushed. Since the energy is spread out as it passes through the surrounding tissues, these tissues are not damaged in the process. The treatment is known as Shockwave lithotripsy.

1.8 An application to gear wheel movements Let me end the first part of my discussion of the ellipse by mentioning an application of some geometrical results in engineering. The mechanical side of the matter I shall pass over lightly. It should be stressed, though, that a well-developed theory lies behind. The subject is the realization of a planar motion by gear wheel movements. In mechanics it is well known that if a solid plane figure moves in a fixed plane with a rotatory movement, the movement can be realized by gear wheels. In order to explain exactly what is meant by this we imagine that the figure is 'glued' to a moveable plane which, then, is taken along as any movement of the figure, and vice versa. In Figure 1.14 it is a triangle for blackboard drawing that is the moveable figure. If we let a curve in the moveable plane roll along a curve in the fixed plane the figure (which follows the moveable plane, of course) will perform a certain movement. The fundamental principle of gear wheel movement is this: Provided that a given motion of the moveable figure always involves some rotation (i.e. its angular velocity is never zero), then there is a curve in the fixed plane and another curve in the moveable plane such that the given motion of the moveable figure is the result of rolling the moveable-plane's curve along the fixed-plane's curve. The two curves are called the poJe curves of the movement, the fixed and the rolling pole curve respectively. They are uniquely determined as the curves that the instantaneous center of rotation of the motion describes in the fixed and the moveable plane, respectively, during the movement. The instantaneous centre of rotation at a given moment in the movement can be determined as the point of intersection for the normals (at

18

An ellipse in the shadow

the given moment) of the paths that the individual points in the solid figure follow in the fixed plane. This agrees with what happens if the total movement is a smooth rotation about a fixed point: the paths are circles and the normals to the circles all meet at the center of the rotation.

fixed plane Figure 1.14

JFigure 1.15 Let us now, for example, consider the movement of a bar PQ where the end points P and Q are connected, by bars of equal length that cross each other, to two fixed points A and B. Also the distance between A and B in the fixed plane is equal to the length \PQ\ of the bar PQ. In the little strip cartoon in Figure 1.15 we show the movement of the bar PQ. It is instructive to make a

19

An application to gear wheel movements

model for oneself from 4 bars and 4 bolts. How can the movement of the bar PQ be realized as a gear wheel movement? The considerations in the following apply to Figure 1.16. If we consider the point P alone it will, during the movement, move on a circle with its centre at A. The instantaneous centre of rotation M of the movement must be situated along the normal to the motion of P, namely it must be located on the line PA. In the same way you see, by considering the movement of the point Q, that the instantaneous centre of rotation M is situated along QB. Therefore, the instantaneous centre of rotation M must be the point of intersection between PA and QB. Now, it is exactly the curve that the instantaneous centre of rotation M describes in the fixed plane (respectively the moveable plane) that is the fixed (respectively the rolling) pole curve in the movement. And now we come to geometry. How do we characterize the position of the point M in the fixed (respectively the moveable) plane?

Figure 1.16

Figure 1.17

By plane geometry you see that the triangles AMB and QMP are congruent; cf. Figure 1.16. It follows, then, that the lengths \AM\ = \QM\ and \MB\ = \MP\ and consequently that

and

(1)

\AM\ + \MB\ = \QM\ + \MB\ = \QB\

(2)

\QM\ + \MP\ = \AM\ + \MP\ = \PA\.

According to (1) the sum of the distances from M to the points A and B in the fixed plane is constant. Therefore, according to Theorem 1.3, M moves on an ellipse in the fixed plane. In the moveable plane the points P and Q are our fixed reference points (the bar is 'glued' to the moveable plane). According to (2) the sum of the distances from M to P and Q is constant too. Therefore, also the moveable pole curve is an ellipse, which furthermore is congruent to the first one, since \PA\ = \QB\.

20

An ellipse in the

shadow

So, the movement of the bar can be realized by letting an ellipse roll along another ellipse congruent to it as shown in Figure 1.17. It would probably have been hard to guess this. But once you have looked at Figure 1.17 and thought about the characterization of the ellipse as the locus of points with a fixed distance sum to two given points, then you can imagine the rolling motion and the motion linkage altogether quite easily. T h e motion linkage in Figure 1.17 also shows the possibility of using elliptic wheels instead of circular wheels to exchange rotatory motions. If elliptic wheels corresponding to the ellipses in Figure 1.17 each rotates around an axis through one of the foci, say respectively A and P, then a constant speed input to the wheel rotating around A will cause a constantly varying speed of the wheel rotating around P depending on the varying relation between the radii of rotations \AM\ and | P M | . Such gear wheels are used in a number of machines where one needs a slow feed and a quick return mechanism. Also the question of the correct elaboration of the teeth of a gear wheel in order to avoid vibrations and to obtain maximal wearing qualities leads to geometry. It can be shown t h a t it works if the gear teeth are cut out as socalled involutes of a circle. An involute of a circle is the curve t h a t the end point T on a thread ST describes when it is unreeled from a circular spool while the thread is all the time held tight in the plane of the spool; cf. Figure 1.18. Investigations of involutes of circles and other s m o o t h curves belong to differential geometry which, as the n a m e suggests, exploits differential and integral calculus in the investigation of geometrical questions. In Figure 1.19 we show a gear wheel cut out along involutes of a circle.

F i g u r e 1.18

F i g u r e 1.19

Sources for Chapter 1

21

Sources for Chapter 1 B. Bolt: "Mathematics meets Technology," Cambridge University Press, 1991. R. Courant and H. Robbins: "What is Mathematics?," Oxford University Press, 1941. W. Fenchel, F. Handest, H. Meyer, P. Neerup: "Elementaer matematik," bind II og III, Munksgaard 1968. D. Hilbert und S. Cohn-Vossen: "Anschauliche Geometrie," Springer-Verlag, 1932. W. Wunderlich: "Ebene Kinematik," Bibliographisches Institut AG, Mannheim 1970.

Chapter 2

With conic sections in the light We have already described the ellipse as being a 'conic section.' In fact, the ellipse is one of the three possible types of curve that can be produced by cutting a circular cone by a a plane. The two additional 'conic sections' are the parabola and the hyperbola. You can see the three conic sections in light on a screen that is placed in a spotlight with a conical beam, by varying the angle of the screen to the light. If you have a conical glass, you can demonstrate the conic sections by filling the glass halfway to the brim with water and tilting the glass. When the inclination of the glass is changed from a vertical position, the boundary curve in the surface of the water is one of the three types of conic sections, but you will have to put a plate across the top of the glass to keep the water from running out if you tilt the glass far enough to see a parabola or a hyperbola. In this second part of the discussion about the ellipse we shall mainly talk about its relationship with the parabola and the hyperbola and about the properties all three types of conic sections have in common.

2.1 The ellipse as a plane section in a cone Consider a cone in three-dimensional Euclidean space with vertical axis and vertex T. The vertex T divides the cone into two parts called nappes of which we show the upper nappe in Figure 2.1a looking like an ice-cream cone. The 23

24

With conic sections in the light

cone can be visualized as produced by turning a nonvertical line through T around the vertical axis so that a surface in 3-space is swept out. The rotated lines through T that together constitute the cone are called the generators of the cone, and the angle v which the generators form with the axis is called the angular aperture. The angular aperture lies in the interval 0 < v < ^.

Figure 2.1b

Figure 2.1c We now drop a ball into the ice-cream cone. That is to say, mathematically

The ellipse as a plane section in a cone

25

we consider a sphere in the inside of the cone that touches the cone along a circle lying in a horizontal plane a, cf. Figure 2.1b. The radius of the sphere can be chosen arbitrarily, but is kept fixed in the following. Next, with a plane /? that touches the sphere and does not pass through T we cut off the cone as shown in Figure 2.1c. The boundary curve thereby produced is called a conic section. The shape of the conic section depends on the angle u that the section plane f3 forms with the axis of the cone, with u in the interval 0 < u < £. There will be three essentially different cases corresponding to u > v, u = v, and w < v. For u > v the shape of the curve is an ellipse, for u = v it is called a parabola, and for u < v a hyperbola. The three cases are shown in Figure 2.2. The ellipse and the parabola both have one piece, whereas the hyperbola has two separate pieces, usually called branches. Now we must prove that in the case of u > v we really get an ellipse, as we have previously defined this shape of curve. For this purpose we make use of the characterization of an ellipse by a focus and a directrix that we found in Theorem 1.4. However, we might as well treat all three cases together. Then we shall also get a characterization of the parabola and the hyperbola by a focus and a directrix. Referring to Figure 2.1c, the line I is the line of intersection between the plane a, which contains the circle of tangency between the sphere and the cone, and the section plane /?, which determines the conic section. Let F be the point at which the sphere touches /?, and let m be the line through F perpendicular to /. Then consider an arbitrary point P on the conic section and let Pi be the point in a in which the generator of the cone through P intersects the circle of tangency between the sphere and the cone. Furthermore, let L denote the point in which the line / intersects the line through P perpendicular to /. Since the line segments PF and PPi are both tangents from P to the sphere we get immediately that \PF\ = | P P i | . It is also obvious that the line segments PL and PP\ have the same right-angle projection onto the axis of the cone, since the points L and P\ both lie in the plane a, and this plane is perpendicular to the axis of the cone. It therefore holds that |PP|cos(v) = |PPi|cos(u) = |PI|cos(u) = |P/|cos(w). Now, from this follows immediately that

\PF\=C^\\Pl\.

cos(w)

The number cos(w) 6 —

——

cos(t;)

26

With conic sections in the light

is called the eccentricity of the conic section. Conversely, it is not difficult to show t h a t a point P in the section plane 0 lies on the conic section if the distances from P to the point F and the line / respectively have the ratio e, t h a t is to say t h a t \PF\ = e\Pl\. This ratio therefore determines a conic section as a plane figure. T h e point F is called a focal point, or a focus, and the line / a directrix for the conic section. In other words: T i e conic section with eccentricity e, focus F and directrix I is the locus of points P in the plane j3 for which \PF\ — e\Pl\. For y > u > v the real number e lies in the interval 0 < e < 1. Theorem 1.4 then shows t h a t this is an ellipse in the same sense as we defined it earlier. For u — y , t h a t is to say e = 0, we get a circle. In this case, the focus-directrix description does not apply as such. As you can see also in Theorem 1.4, this case corresponds to the directrix being 'infinitely far away'. In other words, an ellipse is a conic section with eccentricity e in the interval 0 < e < 1 (corresponding to u > v). W h e n the eccentricity e is 1 (corresponding to u = v) the conic section is a parabola, and when e > 1 (corresponding to u < v) the conic section is a hyperbola, corresponding to the definitions already m a d e .

Ellipse

Parabola

Figure 2.2

Hyperbola

Geometric determination of foci and directrices for a conic section

27

Now one can understand the Greek origin of the names of the conic sections. Ellipse originates from the Greek word 'elleipsis' which means 'deficiency', and in fact we see that the distance \PF\ falls short in being as large as the distance | P / | . Correspondingly, hyperbola originates from 'hyperbole' which means 'exaggeration' and parabola from 'parabole' which means 'compare'.

2.2 Geometric determination of foci and directrices for a conic section In this section we shall outline how you can determine foci and directrices geometrically for a conic section according to essentially the method already used for the ellipse (Dandelin's method). Previously we applied the method to plane sections of a cylinder. But the same kind of reasoning applies to sections of a cone. First we consider an ellipse produced as the curve of intersection in 3space between a cone with vertical axis and a plane /?. There are just two spheres inside the cone that touch the cone along a circle and also touch /?; one above and one below j3; cf. Figure 2.3 which shows a cross section in the cone. The two osculation points E and F for the spheres with /? are the foci of the ellipse. The two directrices for the ellipse, IE and lp, are the lines of intersection between /? and the two horizontal planes that contain the circles of tangency of the two spheres with the cone.

Figure 2.3

28

With conic sections in the light

For a parabola produced as the curve of intersection in 3-space between a cone with vertical axis and a plane /?, just one sphere can be placed in the cone that touches the cone along a circle and also touches /?; cf. Figure 2.4 which shows a cross section in the cone. The osculation point F for the sphere and /? is the (only) focus of the parabola, and the line of intersection / between /? and the horizontal plane that contains the circle of tangency of the sphere with the cone is the (only) directrix of the parabola. Finally we shall consider a hyperbola (both branches) produced as the curve of intersection in 3-space between a cone with vertical axis and a plane /?. There are just two spheres inside the cone that touch the cone along a circle and also touch /?; one in the upper and one in the lower piece of the cone; cf. Figure 2.5, which pictures a cross section of the cone. The hyperbola has two foci; these are the osculation points E and F for the two spheres with P. Correspondingly the hyperbola has two directrices, IE and lp, that are the lines of intersection between /? and the two horizontal planes that contain the circles of tangency of the two spheres with the cone.

Figure 2.4

Figure 2.5

In Figure 2.6 we have placed the hyperbola in a standard system of xycoordinates so that there is symmetry with respect to each of the two axes. The two foci and the corresponding directrices are also shown in the figure. Likewise, from an arbitrary point P on the hyperbola we have drawn the two line segments to the foci, called focal radii from P. We shall need also to consider the line through P perpendicular to the two directrices. Since e is the eccentricity of the hyperbola, we immediately get the following calculation \\PE\ - \PF\\ = e • \\PlE\ - \PlF\\ = e • \lElF\,

29

The parabola

which shows that the difference between the distances from an arbitrary point P on the hyperbola to the two foci E and F is constant. This provides a description of the hyperbola as a locus corresponding to the result for the ellipse in Theorem 1.3.

Figure 2.6

2.3 The parabola In agreement with the previous definition of a parabola as a conic section with eccentricity e = 1, we can also introduce the parabola by the following definition. Definition 2.1 In the Euclidean plane let there be given a line I (the directrix) and a point F (the focus) at the distance | from I where p > 0 is an arbitrary positive real number. A parabola with parameter p is by definition a plane figure that is congruent with the locus of points P in the plane that satisfy the equation \PF\ = \Pl\. In Figure 2.7 we have installed a standard system of ^-coordinates in the plane so that the y-axis passes through the point F and intersects the line I at a right angle at the point A and so that the i-axis is the perpendicular bisector of the line segment AF. Thereby the points mentioned have the coordinates

A(0,-f)andF(0,f).

30

With conic sections in the light

Now consider an arbitrary point P(x, y) on the parabola. The line through P perpendicular to I intersects this line in the point L{x, — | ) . We have also drawn the line segment from P to the focus F, called the focal radius from P.

P{x,y)

!(*,-§) \

A(0,-D

Figure 2.7 From the definition of the parabola follows that \PL\ — \PF\ or, equivalent!^ that \PL\2 = | P F | 2 . Then the distance formula gives the equation

(»*5)'"'+(»-!)"•

which by a simple calculation can be reduced to a;2 = py. Thereby we have proven the following theorem. Theorem 2.1 In a standard rectangular system of xy-coordinates in the Euclidean plane the parabola with parameter p can be described by the equation x

=py.

From the equation of the parabola we immediately find by differentiation that the inclination of the tangent at the point P(x, y) is given by dx

p

Therefore the tangent at a fixed point Po(xo,yo) has the equation

p(y-yo)

=

2x0(x-x0).

For the tangents to the parabola the following theorem applies:

The hyperbola

31

Theorem 2.2 The tangent at an arbitrary point of a parabola bisects the angle between the line to the focus and the line perpendicular to the directrix. An elementary proof can be given by noticing that the slope |a; of the tangent at the point P(x, y) is the negative reciprocal of the slope of the line from L(x, — | ) to F(0, | ) , so that the tangent at P is perpendicular to LF. Since \PL\ = \PF\ the tangent is therefore a bisector of the angle in triangle PLF. The result in Theorem 2.2 is utilized in an obvious way in the construction of parabolic antennas; cf. Exercise 17. When you throw a stone it follows a parabolic path. You can also see the parabola as the curve of a stream of water in the air, as when water comes out of a hose. The individual particles of water behave as small projectiles, like the stone, and each follows a parabolic path. Since they all have essentially the same initial velocity, they all follow the same parabola, and one sees the familiar stream of water, where the water moves but the parabolic shape of its path in the air stays the same.

2.4 The hyperbola Just as with the parabola we can also introduce the hyperbola by an alternative definition that agrees with its previous definition as a conic section with eccentricity e > 1. Definition 2.2 Let a be an arbitrary positive real number and let e be an arbitrary number > 1. Let I be an arbitrary line in the Euclidean plane and let F be a point in the plane with distance (e — -)a from I. Then, a hyperbola with transverse semiaxis a and eccentricity e is defined to be a plane figure that is congruent to the locus of points P in the plane whose distances \PF\ from F and \Pl\ from I have the ratio e. In other words, the hyperbola with focus F, directrix I, and eccentricity e is exactly characterized by the fact that \PF\ = e\Pl\. We would like to determine an equation for the hyperbola. For this purpose we install a standard system of xy -coordinates in the plane so that the xaxis contains the point F and is perpendicular to / and so that the point of intersection A between the x-axis and I has the coordinates A(^, 0), and F has the coordinates F(ea, 0); cf. Figure 2.8. Now consider an arbitrary point P(x, y) on the hyperbola. The line through P perpendicular to I intersects this line in the point L(-,y).

32

With conic sections in the light

From the definition of the hyperbola follows that \PF\ = e\PL\ or, equivalently, that \PF\ = e2\PL\ . The distance formula then gives the equation (x - ea)2 + y2 — e2 (x

J ,

which by a simple calculation can be rewritten as ( e 2 _ l ) a ; 2 _ y2

=

a2(fi2

_

1}

Figure 2.8 The transverse semiaxis of the hyperbola, namely the number a > 0, is part of the considerations already, and now, then, we define the conjugate semiaxis b > 0 by the formula b2 =

a2(e2-l).

Notice that the eccentricity e and the number b determine each other mutually, when a is prescribed (since we are taking 6 and e to be positive numbers). Also b > a for the hyperbola, when e > \ / 2 . The above equation for P(x,y) can now be reduced to the equation b2 " 2

2

i2

a1 which on the other hand immediately can be rewritten as

The hyperbola

33

This is the equation of the hyperbola that we were looking for. We state this as a theorem. T h e o r e m 2.3 In xy-coordinates in the Euclidean plane, the hyperbola with transverse semiaxis a and conjugate semiaxis b can be described by the equation i

i

a2

62 ~

In Figure 2.8 we have also marked what are called the asymptotes of the hyperbola. They are defined to be the two lines with the equations 6

y = ±-x.

a A point P on the hyperbola will approach one of these lines when P recedes to infinity along an arm of the hyperbola. In order to prove this, it is enough, for symmetry reasons, to consider the situation in the first quadrant, that is to consider points P(x, y) on the hyperbola with x > 0 and y > 0. We then have to show that the point P(x, y) approaches the line y = -x, when x increases to infinity. The distance d from P(x,y) to the line y = -x is given by _ bx — ay

~ Va2 + 62 ' We have to show that d —» 0 for x —>• oo, when the set of coordinates (x, y) satisfies the equation of the hyperbola. We can prove this by multiplying both numerator and denominator of the fraction giving d by bx + ay and then using the equation of the hyperbola. In detail: b2x2-a2y2 (bx + ay)y/a2 + b2

_

a2b2

a2b2

(bx + ay)\/a2 + b2

bx^fa2-^!}2

Since a,b are fixed, the last fraction clearly goes to 0, and hence d goes to 0, as x goes to infinity. The points (a,0) and (—a,0) are called the vertices of the hyperbola. The line segment between the vertices is just the transverse axis of the hyperbola. The line segment that the asymptotes cut off on the line through a vertex perpendicular to the z-axis has the length 26 and is, consequently, the conjugate axis of the hyperbola. The hypotenuse in the right triangle with the legs a and 6 shown in Figure 2.8 has the length ea so that the eccentricity can also be determined from the figure. We have previously found that the points P on the hyperbola are characterized by the equation

34

With conic sections in the light

\\PE\-\PF\\

=

e-\lElF\.

Since lp has the equation x = ^, and lE has the equation x = —-, we immediately get t h a t 11

i

e -\lElF\

i

i

a

a

= e •| - + e e

i

=2a.

Thereby we have the following characterization of the hyperbola corresponding to Theorem 1.3 for the ellipse. T h e o r e m 2.4 Let a be an arbitrary positive real number and let e be a real number e > 1. Let E and F be two points in the Euclidean plane with distance 2ea. Then any hyperbola with transverse semiaxis a and eccentricity e is congruent to the locus of points P in the plane for which the difference between the distances to the points E and F is la. In other words, the hyperbola with the two foci E and F and the transverse semiaxis a is characterized by the fact that \\PE\-\PF\\ = 2a. For the tangents of the hyperbola we can prove the following theorem by a m e t h o d quite similar to the proof of Theorem 1.5 for the tangents of the ellipse. T h e o r e m 2.5 For the hyperbola with the

a2 the tangent

62 ~

at the point Fo(^o,2/o) has the XQX

a2

equation

' equation

yoy _ , 62 ~

By an argument quite similar to the proof of Theorem 1.6 you can also prove the following theorem. T h e o r e m 2.6 The tangent at an arbitrary angle between the focal radii.

point

of a hyperbola

bisects

the

Hyperbolic

navigational

systems

35

2.5 Hyperbolic navigational systems In this part we shall discuss an interesting application of hyperbolas in navigation. T h i s is what is called the Decca Navigator System, invented by the American engineer J . W . O'Brien and further developed during World War II by the British firm Decca Navigator, which is a subsidiary company of the record and C D company Decca. It was used during the invasion on the coast of N o r m a n d y by the Allied Forces in 1944 and, later, began to be used all over the world. In Denmark, a Decca Navigator System was set u p in 1948. T h e system is mainly used by the shipping trade, but also to some degree by the airlines. In the following we shall consider navigation at sea. We want a method to determine the position of a ship at sea as precisely as possible. Navigation via the Decca Navigator System is based on emission of radio waves from two fixed stations. T h e waves used normally have a frequency around 100 kHz corresponding to each cycle in the sinusoidal curve of the wave having a length (the wavelength) around 3 kilometers (1.86 miles). T h e emission of radio waves from the stations is adjusted so t h a t a system of curves can be determined along which the radio waves from the stations are in phase. Along these curves the radio waves will intensify the effect of each other, while between the curves they have a certain difference of phase and therefore neutralize each other more or less. Since radio waves propagate with a constant speed (the speed of light), the curves along which the signals are in phase will correspond to points for which the difference between the distances to the two stations is constant. Now, according to Theorem 2.4, such curves are just hyperbolas with the two stations as foci. T h e hyperbolas can be drawn on a m a p in which the radio stations are marked in their geographically exact positions. Now, the principle in Decca is t h a t you measure the phase difference between the radio waves from the two stations with a special instrument, a phase meter, which is connected to a so-called deccometer. It is designed as a clock with a large and an small hand. T h e small h a n d shows the phase difference between the radio waves and thereby the position of the ship in relation to the two nearest hyperbolas. T h e space between the two hyperbolas is called a lane, and while the ship travels through such a lane the small hand of the deccometer t u r n s once. T h e hand passes the zero point each time one of the hyperbolas, which are also called lines of position, is passed. T h e large h a n d of the deccometer indicates how m a n y lines of position the ship has passed; it moves one graduation forward or backwards for every complete turn of the small h a n d . Since the deccometer can only show on which line of position the ship is situated, but not at which point of the line of position, it is necessary to have several pairs of stations and several deccometers, so t h a t the position of the

36

With conic sections in the light

ship can always be determined as the point of intersection between two lines of position.

1

/

1°

/

/

*

^

•*£-,

5 ^ */

1 \ 1

v ^ ©1

V ft " ^ ^ ^

-TC

\

A .^\\i o I —\

\ 't

SftOUNIE R S S K < R I H G S P O filer CR DCN NVJAGTIGL PLADS

'

C

7 "T p^£

r-

/ ^

/—fo

/ /?

/ ,'p ^ J

J

0

/ /

z_yj^

_/

Jd*T n

v*

t >Cx^^g K

'

<

JSJ

/

/ ^ C ^

^Master)

\ s

o«

//ZDJ*TO?\-4X\

iff-,, J^Vo v> rss/

/ '

/^-^^^//£^^ / /

Dtcci-koortl'inai A/J Rid 0 S«0 X 1

V V H 50

btcokoorOmja VWolel 1 70.M

Figure 2.9: The Decca Navigator System of Denmark

Hyperbolic navigational

systems

37

In practice three pairs of stations are used in order to obtain approximately the same precision at the positioning within a large area. (If only two pairs of stations are used, the inexactitude gets large in positions where the angle at the point of intersection between the tangent lines to the lines of position is small). Hence every position can be referred to three systems of hyperbolas. The system of station pairs is called a Decca Navigator System and consists of a shared main station, the master station., and three side stations, the slave stations, that are placed in a triangle around the master station, which controls the emissions of radio waves. Figure 2.9 shows a map of the Danish Decca Navigator System in which some of the belonging lines of position are marked. The Danish Decca Navigator System has its master station at Sams0 and its three slave stations at M0n, in H0jer, and in Hj0rring, respectively. The slave stations are designated red, green, and violet slave, since the corresponding lines of position on the charts that are used during the navigation are printed in red, green, and violet colours, respectively. The scales of the deccometers, of which two are shown in the picture in Figure 2.9, have the same colours, and in order to avoid any mistakes the red lines of position are numbered from 0 to 23, the green ones from 30 to 47, and the violet ones from 50 to 79. The number of lines of position between two connected stations depends on the wave length and the distances between the stations. In the Danish Decca Navigator System the distance between stations is approximately 150 kilometers, and there are several hundreds lines of position. Therefore, the charts are further divided into zones with 24 red, 18 green, or 30 violet lines of position in each. The zones are denoted by letters from A to J, and the deccometers have a window in which a new zone letter appears every time the large hand has moved round once. On the map in Figure 2.9 only zone lines of position are marked, but in real charts there are lines of position corresponding to all integer Decca Navigator System graduations that determine the so-called Decca Navigator System coordinates. In the map it is shown how the position of a ship in the Kattegat is determined by the red and the violet Decca net. On the deccometers the red Decca system coordinate is read as D 9.60 and the violet Decca system coordinate as I 70.30. Actually, the functioning of the system is somewhat more complicated than indicated here, but the main idea is as described. This kind of navigation is also referred to as hyperbolic navigation. The method permits a determination of position within 25 meters and can be applied at distances of up to 500 kilometers from the transmitters.

38

With conic sections in the light

2.6 Conic sections as algebraic curves In the Theorems 1.1, 2.1, and 2.3 we have found quadratic equations in two variables x and y that describe the ellipse, the parabola, and the hyperbola, respectively, in appropriately placed systems of coordinates in the Euclidean plane. How do the equations look in other coordinate systems? This will appear indirectly from the following in which we shall examine the connection between conic sections and sets of solutions to quadratic equations. The general quadratic polynomial in two variables x and y has the form p(x, y) = Ax2 + Bxy + Cy2 + Dx + Ey + F, where A, B, C, D, E, and F are arbitrary constants with the only condition, though, that A, B, and C are not simultaneously 0. If the set of solutions £ = {{x,y)\ p{x,y) = 0}, to the corresponding equation p(x, y) = 0 is not empty, it is called an algebraic curve of degree two. We shall show below that the algebraic curves of degree two are exactly the curves of the conic sections except for special cases. Equation without mixed term. First we investigate the case B — 0, where no mixed term appears in the equation. In other words, we consider an equation of degree two of the form Ax2 + Cy2 + Dx + Ey + F = 0, for which the constants A and C are not both 0. 1. If A 7^ 0 and C ^ 0 , we transform the equation into „,

D-.2

„ ,

E .2

D2

E2

„

la. For A • C > 0 and ^ - (JJ + §c ~ ^) > ® ^ n e se * °^ s o m t i o n s i s an ellipse with its center at the point (—^j, ~^)\ °f- Exercise 25. lb. For A • C > 0 and JJ + f^ = F the set of solutions consists of exactly the

point ( - £ , - £ ) . lc. For A • C > 0 and A • (£j- + | ^ — F) < 0 the set of solutions is empty.

Conic sections as algebraic curves

39

Id. For A • C < 0 and n + ^ - f / 0 the set of solutions is a hyperbola with its center at the point (—^, — ^ ) ; cf. Exercise 29. le. For A • C < 0 and JJ + jc — F the set of solutions is the pair of lines E , [A 2C=±y-C^+2C^

,

D .

y+

2. If A 7^ 0 and C = 0, we transform the equation into

2a. For E ^ 0 we make the further transformation

that gives a parabola with vertex (—^j, - • § + 3375 )> directrix parallel to the x-axis and parameter | ^ | ; cf. Exercise 26. 2b. For E = 0 and A • (j^ — F) > 0 the set of solutions is the lines _ _D_, ~ 2A

X

D2 V4A»

F A'

2c. For E = 0 and £^ = F the set of solutions is the line D X =

-2A-

2d. For E — 0 and A • ( ^ — F) < 0 the set of solutions is empty. 3. If A = 0 and C ^ 0, we proceed as in case 2. The general equation Next we consider the general equation of degree two, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where there is no restriction on B. If we change system of coordinates from the xy-system to a new iy-system, the equation changes into

40

With conic sections in the light

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. In Figure 2.10 the xy-system is constructed by a rotation of the zy-system through the angle t. We shall show that it is possible to choose the angle t so that B = 0 in the new system of coordinates.

• • x

Figure 2.10 Referring to Figure 2.10 we see that the connection between the coordinates in the icy-system and in the ai^-system of an arbitrary point P in the plane is described by the linear substitution x = cos(t)x — sin(t)y y — sm(t)x + cos(t)y. First, we concentrate on the quadratic term q(x, y) = Ax2 + Bxy + Cy2. If we introduce the linear substitution we get the following expression after a small calculation that makes use of the trigonometric formulas cos(2f) = cos 2 (i) — sin 2 (i) and sin(2f) = 2sin(i) cos(i): q{x, y) - [A:os2(<) + B cos(i) sin(i) +

Csin2(t)]x2

+[(C - A) sin(2*) + B cos(2t)]xy + [,4sin 2 (*)-5 sin(i) cos(i)+Ccos 2 (*)]y 2 .

41

Conic sections as algebraic curves

It emerges from this that the mixed term disappears when the angle t is chosen so that (C - A) sin(2t) + B cos(2t) = 0. For 5 = 0 the mixed term is gone already, and nothing has to be done. I.e., the angle t = 0 works. For A = C and B ^ 0 the angle t = \ works, that is to say a rotation of the system of coordinates by \. For A^C and 5 ^ 0 , the angle t must be chosen so that

tan(2t) = -£-^. All in all it appears that there is an angle to in the interval — \ < t 0 < ^ so that a rotation of the system of coordinates by to removes the mixed term. After a rotation of the xy-system by the angle to to the new ary-system, the set of solutions K, in question therefore has the equation Ax2 + Cf

+ Dx + Ey + F = 0,

and thereby we are back to the case first treated. In order to get the most compact formulation of the final result about algebraic curves of degree two we introduce the idea of degenerate conic sections. Geometrically the degenerate conic sections arise by intersecting a cone with a plane that goes through its vertex or by intersecting a right circular cylinder with a plane parallel to the generators of the cylinder. In the degenerate cases of planes intersecting a cone, corresponding to the ellipse, the parabola, and the hyperbola we get a point, a line, and two lines intersecting with one another, respectively. In the degenerate cases for sections of a cylinder we get either two parallel lines, a single line, or the empty set. The above investigations can now be summed up in the following theorem. Theorem 2.7 If the set of solutions to the equation of degree two Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is not empty, then it is a conic section, possibly a degenerate conic section. Example 2.1 Consider the equation xy = 1. According to Theorem 2.7 the set of solutions is a conic section. To determine the type of conic section we perform the linear substitution

42

With conic sections in the light

y/2_

x = -~—x 2

y/2_

2

yy

V2_ , V2_ which corresponds to a rotation of the system of coordinates by the angle \. By this change of coordinates the equation xy = 1 transforms into the equation

According to Theorem 2.3 this equation represents a hyperbola. Consequently, the set of solutions to the equation in question is the hyperbola with the a;-axis and the y-axis as its asymptotes shown in Figure 2.11.

Figure 2.11

2.7 Epilogue The conic sections have fascinated mathematicians for more that two thousand years, and, our discussion has, one hopes, cast new light on these fundamental geometrical objects that appear in numerous applications of mathematics. In particular, the reflection properties of ellipses, parabolas, and hyperbolas (Theorems 1.6, 2.2, and 2.6) have many practical applications. From the viewpoint of the history of science, it should be mentioned that the ellipse appears

Epilogue

43

in the first of Kepler's three laws of planetary motions, published by Johannes Kepler (1571-1630) in 1609; the publication of Kepler's laws can justifiably be regarded as the point of transition from medieval thought to modern science. It has been maintained that if the shape of ellipse had not already been known, the observations made by Tycho Brahe (1546-1601) were not precise enough to have forced his collaborator Kepler to choose this shape of curve for his description of the motions of the planets. Sir Isaac Newton (1642-1727) in his celebrated work Philosophiae Naturalis Principia Mathematica published in 1687, deduced the ellipse as the shape of curves for the motions of the planets from theoretical considerations, supposing that the force of gravity is directed towards the Sun and follows an inverse square law. This deduction established once and for all the supremacy of mathematical methods in the description and explanation of the physical world.

44

With conic sections in the light

Sources for Chapter 2 R. Courant and H. Robbins: "What is Mathematics?," Oxford University Press, 1941. W. Fenchel, F. Handest, H. Meyer, P. Neerup: "Elementaer matematik," Volume II and III, Munksgaard, Copenhagen 1968. V. L. Hansen: "Geometry in Nature," A K Peters, Ltd., Wellesley, MA, 1993. D. Hilbert und S. Cohn-Vossen: "Anschauliche Geometrie," Springer-Verlag, 1932. M. Kline: "Mathematical Thought from Ancient to Modern Times," Oxford University Press, 1972. H.G. Zeuthen: "Keglesnitslaeren i Oldtiden," Videnskabernes Selskab, Copenhagen, 1885. The presentation of the application of hyperbolas in the Decca Navigator System is based on various encyclopaediae, in particular "Mentor" (an encyclopaedia in one volume) published by Grafisk Forlag, Copenhagen, 2nd revised edition 1958. References to the reflection properties of the conic sections can be found via the article: D. Drucker: "Reflection properties of Curves and Surfaces," Mathematics Magazine Vol. 65, No. 3, The Mathematical Association of America, 1992, pages 147-157.

Chapter 3

Optimal plane figures A Greek legend tells that around 800 years B.C. the Phoenician princess Dido fled from Phoenicia with some of her subjects. Her cruel twin brother Pygmalion, who owned the city-state Tyros, had killed her husband in order to appropriate his many riches. But Dido brought along her numerous treasures. After having crossed the Mediterranean, she landed on the coast of North Africa close to present-day Tunis. Here she entered into an agreement with the local inhabitants that she could buy as much land as could be held in an ox-hide. Dido was cunning as well as skilled in geometrical figures, so she cut the ox-hide into narrow strips and tied these into a long string which she placed in a semicircle with endpoints on the shore of the Mediterranean. The ox-hide string together with the sea marked out the land that Dido purchased. Here, according to the legend, Dido founded the city of Carthage that played such an important role in Roman history and literature. In the Aeneid, written in the first century B.C. and considered to be one of the masterpieces of world literature, the Roman poet Vergil lets the Trojan hero Aeneas land on the coast of present-day Tunisia, where he meets Dido, who falls in love with him. But Aeneas who - again according to legend - later founded the city of Rome, leaves her, and Dido commits suicide. Tales with a similar hidden geometrical content are also found in other places. Saxo Grammaticus, an ancient Danish historian, in his 'Danmarkskr0nike' from the end of the twelfth century tells that Regner Lodbrog's son Ivar fools the Saxon king Ella by performing a similar cutting up of a horsehide in so narrow strips that by this "he enclosed a piece of land so large that a whole city could easily stand there." And in Saxo's days cities were round. The background for this was that King Ella had let Regner Lodbrog meet his death in a snake pit, and in order to buy his peace with the sons, he had 45

46

Optimal plane figures

promised Ivar as much land, as could be held in a horsehide. This turned out to be quite a lot! It appears from the above that the Greeks guessed that the circular disc is the plane figure that contains the largest possible area in relation to its perimeter. The Greek mathematician Zenodor occupied himself with the maximal area property of the circle in a paper from circa 150 B.C., and also Archimedes (287-212 B.C.) is said to have worked on this problem. As will appear, it is not obvious that the underlying problem has a solution at all. The problem, which is known by the name the isoperimetric problem, is a problem in what is now known as the calculus of variations, which is an important field of applied mathematics. The problem can be formulated like this: Among all closed curves in the plane without self-intersections and with a fixed length find the one(s) that enclose the maximum area. The depth of the isoperimetric problem was hardly formally recognized by the Greeks. And to prove rigorously that the problem has a solution, or even to formulate it precisely, was not possible at the time. We have to go right up to the end of the nineteenth century until the problem was finally completely clarified. In contrast to the isoperimetric problem, there were other famous problems in mathematics that the Greeks formally recognized as difficult problems: The doubling of the cube, the squaring of the circle, and the trisection of the angle. We know now that these problems cannot be solved, when you allow yourself to use ruler and compass in the specified ways only. Again, this was not finally proved until the nineteenth century. The impossibility of squaring the circle, for instance, with the restrictions prescribed for the construction is a consequence of the fact that the real number n is not a root of any polynomial with integer coefficients (TT is transcendental), which was proved by the German mathematician Lindemann in 1882. This chapter deals with some questions about optimal geometic figures in connection with the isoperimetric problem. We shall return to the application aspect of such problems at the end of the chapter. Along the way, we also illustrate the importance of what are known as existence proofs in mathematics, i.e. proofs that solutions to problems exist.

3.1 Isosceles triangles Consider a triangle ABC, as in Figure 3.1, in which we keep fixed the length g of the base AC. We now ask how this triangle shall be designed so that it has the largest area possible, when its perimeter has fixed length L. For the

47

Isosceles triangles sum of the length of the sides AB and BC, it must hold that \AB\ + \BC\ =

L-g.

Points B with this property lie on the ellipse with the points A and C as foci shown in Figure 3.2. Let the altitude of the triangle have the length h. Then the area of the triangle is given by |/i • g. It is now obvious that the altitude h in the triangle is largest, whereby the area is largest, when B falls in the vertex for the ellipse. Hence the triangle with largest area is isosceles as in Figure 3.2, where the two sides AB and BC each have the length |(Z, — g). Thereby we have proven the following theorem. T h e o r e m 3.1 Among all triangles with a given base and a given perimeter the isosceles triangle has the largest area.

B

A

9

B

C

Figure 3.1

Figure 3.2

B L / V A

\ k V 1

'C

3

Figure 3.3 Stimulated by this first succes in determining an optimal geometric figure, we try our hand with a triangle ABC, in which we only keep fixed the length

48

Optimal plane figures

L of the perimeter of the triangle. How shall the triangle be designed, then, to obtain the largest possible area? We now argue in the following way. Assume that the triangle ABC solves the problem. According to what has just been proven, the sides AB and BC must have the same length, since otherwise we could at once produce a triangle with the same perimeter, but with a larger area. In the same way you see that the sides AC and BC have the same length. Therefore, the triangle ABC that solves our problem is the equilateral triangle with side length j^ shown in Figure 3.3. But there was something sneaky about this proof, when you think about it carefully. Who says that a triangle that solves the problem exists at all? And that is what we assumed at the beginning of the argument, as you will remember. And it might not have been true. (Is there a triangle of minimum area with a fixed perimeter?)

3.2 Perron's paradox Perron illustrated the gap in an argument of the above type by the following example. We known that the set of natural numbers does not contain a maximal element. But let us, for the fun of it, for a moment assume that TV > 1 is the greatest natural number. If TV > 1, then N • N > N, according to the rules of arithmetic, and that would contradict N being the greatest natural number. So, it must hold that N = 1, which is obviously absurd. What went wrong? We assumed that there exists a greatest natural number! There are also other simple situations in which a maximal or a minimal value does not exist. For example, the set of fractions {|, | , ^ , . . . , i , . . . } does not contain a smallest number, and { 2 , 3 , 4 , . . - , '"~ , • • •} does not contain a greatest number.

3.3 Some simple geometrical problems without solutions Examples with sets of numbers are one thing, geometrical figures another. Before the great German mathematician Karl Weierstrass (1815-1897) pointed out the deficiency of the reasoning, it was often accepted without proof that questions about the existence of a maximal or a minimal value in problems of a physical or geometrical nature always have a solution although people did

A fundamental property of the real numbers

49

realize that the solution might 'degenerate' in some sense. As an example, the minimum area for a triangle with a given perimeter is not obtained for exactly a triangle but rather for a straight line segment that one thinks of as a triangle with zero altitude. Weierstrass pointed out the shaky foundations for such considerations by giving several simple examples of geometrical problems without a 'minimal' solution. In Figures 3.4 and 3.5 we give two such examples. In Figure 3.4 we consider two points A and B o n a line and ask for the shortest polygonal path in the Euclidean plane that starts in A orthogonally to the line and ends in B. Obviously, there is no such shortest path. In Figure 3.5 we consider two points on a line through the origin in the plane. If we remove the origin from the plane, there is no longer a shortest path connecting the two points.

A

B

F i g u r e 3.4

A

~~

B

F i g u r e 3.5

Even famous mathematicians have overlooked the real problems that may lie where least expected, namely in just the question of the existence of the object searched for. Others did worry, though, but did not pinpoint the needs for existence proofs in mathematics as clearly as Weierstrass.

3.4 A fundamental property of the real numbers From the natural numbers one first gets the integers {..., —2, —1,0,1,2,...} and next the set of fractions, the so-called rational numbers. If you mark all the rational numbers on an oriented axis, you realize that there are points on the axis that are not included: There are 'holes' in the axis. For example, the Greeks discovered, to their great annoyance, that the diagonal of the unit square is a quantity that cannot be represented by a rational number. If we lay down this length from 0 we arrive at at new point, namely Vz. Before long we realize that there are many more 'holes' in the axis than points corresponding to rational numbers. To a great extent, these so-called irrational numbers caused the Greeks to concentrate their mathematical efforts on geometry, because the

Optimal

50

plane

figures

irrational numbers, as numbers rather t h a n geometric quantities, did not seem to t h e m to fit into a logical theory. T h e set of all points on the chosen axis represents the real numbers. If we want to describe the real numbers completely from the rational numbers, it can, for our purpose, be done most expediently by the following procedure. We simply imagine t h a t we catch the real numbers in so-called nested intervals. By a nested interval sequence we understand a decreasing sequence of closed intervals [ai, 61] D [02,62] D • • • 3 [a n , b„] D ..., in which the length of the interval [ a n , 6 n ] approaches 0 for increasing n. We can now introduce the real numbers as such 'limit points' for nested interval sequences, in which we use only rational numbers as end points of the intervals. After this has been done, every such nested interval sequence catches exactly one real number, namely the only point in common for all the intervals. It is i m p o r t a n t to use closed intervals in the sequences; consider for example the decreasing sequence ]0,1[3]0, | [ D . . .]0, ^[D . . . of open intervals. This sequence of intervals has no common point.

ai = ai 1 1

as 1

&2 = 63 1—I

I V2 §

61 1

•

2

F i g u r e 3.6 For example you can find y/2 in this way ; cf. Figure 3.6. Choose rational numbers a\ and 61 so t h a t a i < \ / 2 < 61, e.g. a\ — 1 and 61 = 2. Divide the interval [ai,6i] at the middle. Thereby we get two new intervals with rational numbers as end points, y/2 lies in exactly one of these intervals; call it [02, 62]. We actually know t h a t 02 = 1 and 62 = § in this case, but the m e t h o d is quite general. Now divide the interval [02,62] a t the middle. Again we get two new intervals with rational numbers as end points, and again y/2 lies in exactly one of these intervals; call it [03, 63]. In this case we actually know t h a t 03 = I and 63 = | . We go on like this, and clearly a nested interval sequence [ai, 61] D [02, 62] 2 ' ' ' 3 [ a m 6„] D . . . is thereby constructed, in which all the intervals have rational end points, and in which A/2 is the only c o m m o n point. Numerical analysts call this m e t h o d bisection, for obvious reasons. One of the i m p o r t a n t discoveries in the foundations of m a t h e m a t i c s in the nineteenth century was exactly t h a t a tenable basis can be found for the number system by this or some similar m e t h o d . You can take, as the basic property t h a t distinguishes the real numbers from the rational numbers, the following principle.

Maxima and minima of real-valued functions

51

The principle of nested intervals: Any decreasing sequence of closed intervals [ai,6i] D [02,^2] 2 • • • 2 lan,bn] 2 •••> f° r which the length of the interval [a n ,^n] approaches 0 for increasing n, has exactly one real number as common point. Not surprisingly the principle of nested intervals is of major importance in investigations of the kind we are occupied with here, for, ultimately, this gives exactly the existence of the quantities we are looking for. The system of real numbers had to be built on a logical basis before any systematic theory of existence of maxima and minima could be developed, even for simple questions. For instance, the function x2 — 2 does not attain a minimum when x ranges over the rational numbers only. Thus the Greeks could not deal even with simple minimum problems in full generality. The definitive construction of the real numbers by a purely logical arithmetical construction without appeal to intuition was not given until the end of the nineteenth century, when Karl Weierstrass (1815-1897), Hugues Charles Robert Meray (1835-1911), Richard Dedekind (1831-1916) and Georg Cantor (1845-1918), almost simultaneously and independently, each presented such a construction.

3.5 Maxima and minima of real-valued functions In this section we shall briefly explain an important consequence of the principle of nested intervals. We consider a function / : [a,b] —> M. defined on a closed and bounded interval [a, b] and with values in the real numbers TSL. In the usual way we may view the function by considering its graph as in Figure 3.7. We say that the function / is continuous at the point xo £ [a,b], if, for any open interval Ie no matter how small with f(xo) £ h, there exists an open interval Is around Xo, so that f(x) 6 Ic, when x £ [a, b] f]Is- If / is continuous at all points of the interval [a,b], the function / is said to be continuous in [a,b]. Continuity can be viewed in terms of the graph of / ; it is related to the fact that the curve of the graph is connected, in the informal sense that you can draw it without lifting the pen from the paper. All functions that you normally meet, defined by formulas, are continuous. For continuous functions, the following fundamental result holds. Theorem 3.2 Any continuous real-valued function f : [a, b] —> M defined in a closed and bounded interval [a,b] assumes a maximum value and a minimum value in the interval.

52

Optimal

plane

figures

Proof. First we prove t h a t / is bounded, t h a t is t o say t h a t there is a real number k, so t h a t \f(x)\ < k for all x £ [a, 6]. This is proved indirectly. Assume for t h a t purpose t h a t / is not bounded. As we shall now show this assumption leads to a contradiction. For notational convenience put [a, 6] = [ai,&i]. Divide the interval [cti,&i] at the middle. Thereby we get two equally large subintervals. In at least one of these subintervals / is not bounded. By continuing the process of bisection successively we construct a nested interval sequence [ai,6i] D [02,^2] 3 ' ' ' =? [a„, bn] D . . . , in which it holds, in every interval, t h a t / is not bounded. T h e nested interval sequence determines a real number XQ £ [a,b] — [a\,b{\. Since / is continuous at XQ , / is obviously bounded in an interval neighbourhood of XQ and hence on [a n ,6„] for all n large enough. Thereby we have obtained a contradiction, and it has been proved t h a t / is bounded in [a,b].

{x0,f(xo))

Figure 3.7 Next we shall prove t h a t / assumes a m a x i m u m value in the interval [a, b]. Since / is bounded in [a, b], we can choose real numbers c and d, so t h a t c < f(x) < d for all x £ [a,b]. Now we carry out a process of bisection on the interval [c, d\ on the y-axis, in which we at every bisection choose the upper subinterval, if there are function values for / in it, and otherwise the lower subinterval. Thereby we get a nested interval sequence t h a t determines a (uniquely defined) number ko £ [c,d], so t h a t f(x) < ko for all x £ [a,b] and so t h a t ko is the smallest number with this property. Now, the crux of the m a t t e r is t h a t there exists a point xo £ [a, b] for which / ( z o ) = &o- This we prove as follows. Choose a sequence of numbers {xi, X2, • •., x„,...} in the interval [a, 6] = [ai,&i] so t h a t f(xn) > ko — —. T h i s we can do, since ko — ^ is not an upper bound for / , since ko, as you know, is the smallest such bound. If the

Maxima

and minima

of real-valued

functions

53

sequence {x\,X2, • • •, x„,... } consists of finitely m a n y distinct numbers only, you easily see t h a t one of these numbers can be used as the XQ searched for, since f(xn) with increasing n gets arbitrarily close to &0. Now assume therefore t h a t the sequence {x\,X2, . . . , £ „ , . . . } consists of infinitely many m u t u ally distinct points. Then divide the interval [ai,&i] into two equally large subintervals. At least one of these subintervals, let us say [02,62], contains infinitely m a n y mutually distinct points from the sequence {x\, X2, •. •, xn,...}. By continuing successively the process of bisection, we get a nested interval sequence [01,61] D [02,62] 2 ' ' ' -5 [ a n , 6 n ] D . . . , in which any of the intervals [a„, 6 n ] contains infinitely m a n y mutually distinct points from the sequence {x\, X2, • • •, xn,...}. This interval sequence determines a real number XQ £ [0,6]. Since f(xn) with increasing n move towards ko, and there are points xn with arbitrarily high index n as close t o XQ as we want, it must hold t h a t f(xo) = ko, since / is continuous at x0. Thereby it has been proved t h a t / assumes a m a x i m u m value in the interval [a, b]. In a similar way it can be proved t h a t / assumes a m i n i m u m value in the interval [a, 6]. This completes the proof of Theorem 3.2. A corresponding theorem holds for continuous functions / with real values f(x,y) t h a t depend on two real variables x,y. (You m a y for example think of the pressure f(x, y) of a gas as a function of the volume x of the gas and the t e m p e r a t u r e y of the gas). Here the proofs can be carried through by considering decreasing sequences of closed rectangles, in which a rectangle is viewed as the product of two intervals. Even more generally, you m a y consider continuous real-valued functions f(x, y, z,.,.) t h a t depend on a finite number of real variables x, y, z,.... In this general case, the proofs are carried through by considering decreasing sequences of 'higher dimensional intervals' t h a t arise as products of a number of ordinary intervals corresponding t o the number of variables in the function. In order to get the existence of a m a x i m u m value and a m i n i m u m value, the point set in a higher dimensional number space, from which the variables x,y,z,... are taken, must be bounded and closed with respect t o limit processes, such as generalized nested interval sequences. Such point sets are said to be compact. T h e compact point sets in a higher dimensional n u m b e r space are exactly the point sets t h a t are bounded and contain all their b o u n d a r y points. By way of example, any point set in the twodimensional number space (can be identified with the plane) t h a t is bounded by a finite n u m b e r of closed curves without self-intersections is compact, if you regard the b o u n d a r y curves as part of the set.

54

Optimal plane figures

3.6 The equilateral triangle as optimal figure Consider a triangle ABC the perimeter of which is a fixed length L. The object is to choose the triangle so that its area is as large as possible. From the previous treatment of isosceles triangles, we know that we can assume from the start without loss of generality that the sides AB and BC are equally long, for otherwise we can, according to Theorem 3.1, produce a triangle with the same perimeter as ABC, but with a larger area. We therefore consider an isosceles triangle ABC that we place in a rectangular coordinate system, as shown in Figure 3.8. For the triangle ABC we assume in other words that \AB\ = \BC\, and that \AB\ + \BC\ + \CA\ = L.

i

y

B>

/

A

V

0

\Jx2 + y2 X

Q

*X

Figure 3.8 Now put \OC\ = x and \OB\ = y. Then \BC\ = ^Jx2 + y2 according to the Pythagorean Theorem. Notice that the pair of real numbers (x, y) determines the triangle ABC completely. We seek to determine the pair of real numbers (x, y) in such a way that the corresponding triangle ABC has maximum area, when (x, y) is subject to the side condition corresponding to the restriction on the perimeter of the triangle, namely that 2x + 2y/x2 +y2 = L,

0<x,

0
After division by 2, transfer of x to the right hand side of the equation, followed by a squaring, this side condition can first be rewritten as

55

The equilateral triangle as optimal figure

x2 + y2 = C; and next as y2 =

^-(^-2x

Hence the possible pairs of numbers (a;, y) which describe one of the relevant triangles ABC lie on a segment of a parabola as shown in Figure 3.9.

O Figure 3.10

Figure 3.9 The area A(x, y) of triangle ABC is given by A(x,y) = x y.

We have to maximize A(x, y) under the side condition 2/2 = | - ( | - 2 z ) ,

0<x,

0
For any fixed positive real number k > 0, the equation A(x,y) = x -y = k describes a hyperbola of which we show the relevant branch in Figure 3.10. For k small the branch of the hyperbola in Figure 3.10 intersects the segment of the parabola in Figure 3.9 in two points, and for k large it does not intersect the parabola at all. For exactly one value ko of k the branch of the hyperbola touches the parabola at exactly one point (#o,2/o); cf. Figure 3.11. For the triangle (#0,2/0) that corresponds to the pair of real numbers (xo,2/o),

56

Optimal plane figures

we evidently obtain the maximum area among all triangles with the given perimeter L. We can also argue using the main Theorem 3.2 about continuous functions. This is done as follows. It is an equivalent problem to maximize the square of the area A(x,y)2 under the given side condition. Using the side condition we perform the rewriting A{x,y)2

= x2 -y2 =2'X

-(2 L L-x2-{--x)

•2x)

in which x shall be considered in the interval 0 < x < -r only.

Figure 3.11

Figure 3.12

The function f(x) = L-x2-(j — x) is a continuous function on the closed and bounded interval [0, j] and therefore assumes a maximum value in the interval. In Figure 3.12 we show the graph of the function. Since /(0) = f(j) = 0, the maximum value is assumed at a point xo with 0 < XQ < -^. The corresponding value t/o is given by yQ = J^ • (^ - 2x0). Again the pair of real numbers (x0, j/o) corresponds exactly to the triangle that has the maximum area among all triangles with the given perimeter L. Hence there does exist a triangle ABC with maximum area among all triangles with a given perimeter. But then we can argue as already done earlier on, and we find that the triangle ABC must be equilateral. Thereby we have conclusively proved the following result.

The square as optimal figure

57

Theorem 3.3 Among all triangles with a prescribed perimeter the equilateral triangle has the largest area. If we make use of differential calculus in the considerations, the point xo, in which the function f(x) = L • x2 • ( \ — x) assumes its maximum value in the interval [0,^-], can be determined by finding the zeros of the derivative f'(x) of f(x). We find f'(x)

= =

2L-x-(--x)-L-x2 --x-(L-6x).

It follows that f'(x) = 0 for x = 0 and x — jr. By considering the sign variation of f'(x) we conclude that the function f(x) in the interval [0, ^] assumes its maximum value at the point a?o = j • Then the corresponding t/o is j/o = r ^ s , and we find that all the sides in the corresponding triangle ABC has the length |f-. This provides an alternative proof of the above theorem.

3.7 The square as optimal figure Consider a quadrilateral ABCD for which the perimeter has a fixed prescribed length L. How shall the quadrilateral be designed, then, in order to have the largest possible area? It turns out that this problem is somewhat easier than the corresponding problem for the triangle that we have just treated.

Figure 3.13

Figure 3.14

58

Optimal plane figures

First we notice that without loss of generality we can assume that the quadrilateral ABCD is convex, that is to say that the diagonals AC and BD fall completely within the quadrilateral as in Figure 3.13. For if there is a re-entrant angle in the quadrilateral as in Figure 3.14, then by reflection in the line segment BD we can produce a quadrilateral with the same perimeter, but with a larger area.

Figure 3.15

Figure 3.16

By dividing the quadrilateral ABCD in Figure 3.13 into two triangles along the diagonal BD we see that unless \AB\ — \AD\ and | C J 3 | = \CD\ , we can, according to Theorem 3.1, produce a new quadrilateral A\BC\D shown in Figure 3.15, such that \AXB\ = \AiD\ and |CiJB| = \CiD\, and such that A\BCiD has the same perimeter as ABCD, but has a larger area. Now we divide the quadrilateral A\BC\D into two triangles along the diagonal A\C\. Since A\BC\D is symmetrical with respect to A\C\, we can, by a quite similar argument as before, produce a new quadrilateral A\B\C\D\ shown in Figure 3.16, in which all sides have the length ^ , and in which A\B\C\D\ has a larger area than A\BC\D, unless the sides oi A\BCiD already have the same length.

Figure 3.17

The regular polygons as optimal figures

59

Therefore the quadrilateral AiBidDi is a rhombus with side length \ . If the altitude of the rhombus has the size h, then its area is given by j • h. Clearly the altitude is largest, whereby the area is largest, when all the angles in the rhombus A\B\CiDi are right angles, in other words, when it is a square; cf. Figure 3.17. Thereby we have proved the following theorem. Theorem 3.4 Among all quadrilaterals with a prescribed perimeter, the square has the largest area.

3.8 The regular polygons as optimal figures A regular n-gon for n = 3,4, 5 , . . . is a closed, convex polygon with n equally long sides, in which all angles in the corners of the polygon have the same measure. A regular n-gon is constructed by dividing a circle in n equally large segments and drawing the inscribed chords. In Figure 3.18 and Figure 3.19 we show the regular pentagon and the regular hexagon, respectively.

0

Figure 3.18

Figure 3.19

In Figure 3.20 we consider an arbitrary closed polygon (without self-intersections) with n sides and the corners {Ai,A2,--.,An} cyclically ordered in an orientation of the polygon. Such a polygon can be divided into triangles and thereby assigned an area. We ask again how the polygon shall be chosen so that it has maximum area, when the perimeter has fixed length L. That there exists a closed polygon with n sides and perimeter L of the type considered that has maximum area among all such polygons can be seen as follows. First we notice that if we install a rectangular system of coordinates

60

Optimal

plane

figures

in the Euclidean plane, then the polygon can be described by n pairs of real numbers, namely the coordinates for the n corners {Ai,A2,...,An} in the polygon. Since the polygon must have the fixed perimeter L, the 2n real numbers we need to describe it can be reduced to vary within a compact set of points in the number space of 2n real variables. Since the area function depends continuously on the coordinates of the corners, it follows from the previously mentioned general result about continuous functions on compact point sets t h a t the area function assumes a m a x i m u m value. In other words, there does exist a closed polygon of the type considered with m a x i m u m area.

^2(^2,^2), Ai(xi,yi)

A„(xn,t/„)' An-l(xn-l,yn-l) Figure 3.20 By now it should come as no surprise t h a t the following theorem holds. T h e o r e m 3.5 Among all closed polygons with n sides and with a prescribed perimeter, the regular n-gon has the largest area. In connection with the t r e a t m e n t of the isoperimetric problem in Section 3.10, we shall in particular indicate a proof of Theorem 3.5.

3.9 Some limit values for regular polygons In this section we shall investigate how the radius of the circumscribed circle and the area of the regular n-gon with a prescribed perimeter L changes with increasing n. In this connection it is appropriate to make use of the trigonometric functions.

Some limit values for regular

61

polygons

As it is well known, for angles x (measured in radians) in the interval 0 < x < j , the values of the functions sine, cosine, and tangent of x, denoted respectively sin(a;), COS(K), and t a n ( « ) , can be defined as the lengths of the line segments shown in Figure 3.21 t h a t are connected to a segment of the circle with centre O and radius 1. W i t h reference to Figure 3.21 it holds in other words t h a t \0P\=

\OA\=l,

and t h a t sin(«) = \PQ\,

cos(a;) = \OQ\,

Notice, furthermore, t h a t tan(ai) =

sln

tan(a-) =

\AB\.

> (.

,_

n

tan(x)

0

cos(x)

Q A

Figure 3.21

Figure 3.22

Since angles are measured in radians, the length of the arc PA is exactly x. T h e n it appears directly from Figure 3.21, by a consideration of areas, t h a t smla;) < x < t a n ( z ) =

—• cos (a;)

.

From these inequalities you derive by division by tan(a;) and sin(.r) t h a t cos(a;) <

tan(a;)

< 1<

1 sin(a;) < cos(a;)

Since cos(a;) approaches 1 t h r o u g h increasing values, when x approaches 0 through decreasing values, it follows immediately from the above inequalities t h a t b o t h sin(rr) and tan(a;) approach 1, when we let x approach 0 t h r o u g h

62

Optimal

plane

figures

decreasing values. [By interpreting 5i2i£2j respectively t a n ^ ' , as the slope of the line segment from (0, 0) to the point (x, sin(a;)), respectively (x, tan(a:)), you can actually d e m o n s t r a t e t h a t ai5x\ approaches 1 through decreasing values and t h a t t a ^/ \ approaches 1 through increasing values, when x approaches 0 through decreasing values.] Now back to the regular n-gon with the prescribed perimeter L and thereby the edge length ~. W i t h r„ we denote the radius in the circumscribed circle for the regular n-gon, and with An its area. Since the altitude hn and the angle n = ^ are the magnitudes shown in Figure 3.22, we get, by utilizing t h a t the two triangles in Figure 3.21 are similar to the triangle in Figure 3.22 for x = <j>n, the following expressions for hn, rn, and An '•

h — JL 2n

1 tan(^n)

_ L In

_ _^_

1 sin((/>„)

An=2n--hn2 "

2ir _

L 2-7T

n tan(0n) (f>n sin((^ n )

K — = — 2n 47T tan(<^„)

W i t h a view towards these formulas, note t h a t the circle with perimeter L T

T

2

has the radius -k- and the area %—. 27T

47T

Since „ = - approaches 0 decreasingly for increasing n, we get as a result of the above calculations the following theorem. T h e o r e m 3.6 For increasing n, the radius rn in the circumscribed circle for the regular n-gon with perimeter L approaches through decreasing values the radius •£- in the circle with perimeter L. At the same time, the area An of the polygon approaches through increasing values the area j ^ of this circle.

3.10 The isoperimetric problem T h e isoperimetric problem, as already discussed, can be stated in outline like this: A m o n g all closed curves in the plane without self-intersections and with a fixed length find the one(s) t h a t enclose the m a x i m u m area. Of course, to make this precise, certain things have to be clarified. First of all, what is m e a n t by a curve? T h e n a t u r a l answer from the viewpoint of modern m a t h e m a t i c s is t h a t a plane closed curve without self-intersections is the image in the Euclidean plane of a continuous function defined in the unit

The isoperimetric problem

63

interval [0,1] so that 0 and 1, and only these points, are mapped into the same point of the curve. It seems to be assumed in the statement of the isoperimetric problem that such a closed curve divides the plane into two regions, a bounded region (the 'inside'), and an unbounded region (the 'outside'). This happens to be true, but it is far from obvious; indeed its precise proof is a famous accomplishment of rather modern mathematics, known as the Jordan curve theorem after the French mathematician Camille Jordan (1838-1922) who stated the theorem in his Cours d'analyse from 1887. The first rigorous proof of the Jordan curve theorem was given by the American mathematician Oswald Veblen (18801960) in 1905. Closed curves in the plane without self-intersections as defined above are nowadays called Jordan curves. There is also a problem with assigning length to Jordan curves in general. Continuity alone is not enough to ensure that a Jordan curve has a well defined finite length. For instance, the so-called fractal Jordan curves eat up perimeter indefinitely and hence cannot be assigned a length. In the following we shall only consider Jordan curves in the plane that have finite length. We shall also restrict attention to curves that can be approximated by polygonal Jordan curves as closely as we wish simultaneously with respect to length and enclosed area. (This is in fact true for all Jordan curves with finite length but we shall not attempt to prove this here.) For brevity, we call such curves rectifiable Jordan curves. Examples of such curves include Jordan curves that have continuously varying tangent lines, possibly except at finitely many points. As mentioned in the introduction to this chapter, the ancient Greeks took it for granted that the solution to the isoperimetric problem is the most 'perfect' of all closed curves, namely the circle. By now it should be evident, though, that the problem is not so simple after all. The German mathematician Jacob Steiner (1796-1863) suggested several brilliant proofs, of which we shall present one from around 1840 below. It should be remarked that Steiner, too, overlooked the existence problem. One can illustrate by the following experiment that the circle is the solution to the isoperimetric problem. Suspend a closed curve in a small frame. The curve and the threads by which it is attached can for example be sewing thread. Now dip the frame in a not too thin soap solution. Thereby the closed curve gets to lie on a soap film. Prick a hole in the film inside the closed curve. Immediately the closed curve will be reshaped into a perfect circle; cf. Figure 3.23. From this follows that the circle must be the solution to tl 3 isoperimetric problem, since the potential energy in the soap film outside the curve is proportional to the area, so that the soap film will seek to minimize the area, which means, conversely, that the area inside the closed curve must be maximized.

64

Optimal plane figures

Of course, this physical experiment is not a conclusive mathematical proof of the existence of a solution to the isoperimetric problem. Such a proof was not given until Weierstrass did so in famous lectures at the university of Berlin in the 1870s.

—{BSli»

Figure 3.23 However, let us for a moment assume that a solution does exist. So, we consider a closed plane curve C without self-intersections and of a fixed prescribed length L with the property that among all such curves it encloses the maximum area. Following Steiner, we shall now argue mathematically that C is a circle. By considering Figure 3.24 it is obvious that C must be convex, since otherwise we could create a closed curve of the same length L as C, but enclosing a larger area, by reflecting an inbuckling to an outbuckling.

Figure 3.24

Figure 3.25

Now choose two points A and B on C that divides the curve into two equally long arcs of the lengths \ . The chord AB divides the enclosed figure into two pieces that must each have the same area, since otherwise we could construct a figure of larger area by replacing the smaller piece with the reflection in AB of the larger piece; cf. Figure 3.25. Also, each of the two pieces has the property that it has the largest possible area among figures formed by an arc of length j and a straight line segment of arbitrary length (Dido's problem). For if we could find a larger 'half-figure', we could fit it and its reflection in the line together to enclose a larger area with perimeter L.

The isoperimetric problem

65

Now consider the 'half-figure' bounded by one of the arcs from A to B of length j and the line segment AB. Let P be an arbitrary point on the arc, and consider the triangle APB. Imagining that the arc is made of steel and that we have placed a hinge at P, we can bend the triangle at the vertex P without changing the length of the arc. Since the shaded areas do no change by the bending, we see that the 'half-figure' has maximum area among those obtained by bending at the hinge precisely, when the triangle has a right angle at P; for then the altitude h in the triangle is largest among those with fixed length sides emanating from P; cf. Figure 3.26. Through the three points A, B, and P passes exactly one circle, the centre of which is the point of intersection between the perpendicular bisectors of AB and PB. Since /.APB = 90°, this point of intersection must be the midpoint of AB. As P was chosen arbitrarily, we conclude that the arc from A to B must be a semicircle and the original closed curve C therefore a circle with AB as diameter.

Figure 3.26 We now return to the question of the existence of a solution to the isoperimetric problem. This can be proved in a mathematically correct way by approximating the closed curve by closed polygons. So, we remind the reader that we are in fact only considering rectifiable Jordan curves in the sense we defined earlier. As it turns out we prove at the same time that the circle is the solution to the isoperimetric problem. First note that any rectifiable Jordan curve can be approximated by a closed polygon with an even number 2m of edges and corners, which approximates it as closely as we wish simultaneously with respect to length and enclosed area. We can even adapt the approximating polygon so that it gets the same perimeter L as the closed curve under consideration, by slightly magnifying or slightly shrinking an arbitrary approximating polygonal curve. Let us now look at such an approximating polygon. We shall prove (Theorem 3.5) that its area is less than or equal to the area of the regular 2m-gon of perimeter L and hence, by Theorem 3.6, less than or equal to the area ~of a circle with circumference L. This will enable us to complete the proof

Optimal plane figures

66

momentarily, by a limiting argument, that the original curve bounds an area of no more than 4 - . In Section 3.8 we proved that there exists a 2ra-gon that has maximum area among all 2m-gons of perimeter L. Such a 'maximum' polygon has to be convex, by the argument used in our discussion of maximum-area quadrilaterals in Section 3.7. Moreover, in such a 'maximum'polygon with cyclically ordered corners {Ai, A2,..., ^bm}, any of the triangles determined by a corner and its two neighbouring corners must be isosceles according to Theorem 3.1. From this follows that all the edges of the 'maximum' polygon have the same length.

Figure 3.27 Now divide the 'maximum' polygon by the diagonal AiAm+i. We can then argue exactly as above in Steiner's argument and prove that all the corners must be situated on a circle with a radius r2m determined by m and L. In other words, the 'maximum' polygon is the regular 2m-gon with perimeter L; cf. Figure 3.27. According to Theorem 3.6, the area of this polygon is less than that of the circle with circumference L. Hence the area of any 2m-gon of perimeter L is less than the area of the circle of circumference L. Returning now to our original rectifiable Jordan curve of length L, we recall that its area is a limit of areas bounded by polygons of perimeter L with even numbers of sides. Since each of these polygons bounds an area less than the area of a circle with circumference L, it follows that the original curve bounds an area less than or equal to the area of a circle with circumference L, as we wanted to prove. Thereby we have reached the goal aimed at, namely to prove the following theorem. Theorem 3.7 Among all closed curves in the plane without self-intersections and with a prescribed length (rectifiable Jordan curves), the circle encloses the largest area.

Epilogue: Elements of the history of the calculus of variations

67

We notice that we, as part of the proof of Theorem 3.7, in particular have proved Theorem 3.5 for closed polygons with an even number of edges. From this, one can deduce Theorem 18 for polygons with an odd number of sides, as well. For if we connect only every other corner of a regular 2m-gon, we get a regular m-gon. And if this regular m-gon does not have maximum area in relation to its perimeter, it can be replaced by the 'maximum' m-gon corresponding to this perimeter. (Here we utilize that all edges of the 'maximum' m-gon must necessarily have the same length.) This is a contradiction because we would thereby produce a new 2m-gon with the same perimeter as the original regular 2m-gon, but with a larger area.

3.11 Epilogue: Elements of the history of the calculus of variations As mentioned at the beginning of this chapter, the isoperimetric problem is one of the classical problems of the early history of the calculus of variations, which is an extremely important and central field in applied mathematics. The development in this field has taken place especially after Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) got the calculus with infinitesimal quantities (differential and integral calculus) systematized at the end of the seventeenth century. Among the pioneers of the calculus of variations must be singled out the brothers Jakob Bernoulli (1655-1705) and Johann Bernoulli (1667-1748) plus, not least, the extremely productive mathematician Leonhard Euler (1707-1783). Even though the strong tools from infinitesimal calculus are decisive for the calculus of variations, it has always been considered to be especially attractive to be able to carry through purely geometical arguments. It is rare, however, to succeed so well in this as with the isoperimetric problem. Among the mathematicians from the nineteenth century that have contributed significantly to the calculus of variations, we have already mentioned Weierstrass, who got the very basis of the theory set right in the 1870s. In the twentieth century, calculus of variations has developed tremendously, and we shall content outselves with singling out the American mathematician Marston Morse (1892-1977), who has lent his name to a fruitful topological theory within the field. As an example of geometric problems in the calculus of variations, we mention, among other subjects, the study of properties of curves in curved surfaces that locally minimize distances (geodesic curves). A particularly fascinating subject is the study of the so-called minimal surfaces, i.e. (curved) surfaces that locally minimize area. Such surfaces you meet, among other places, as

68

Optimal

plane

figures

m e m b r a n e s in n a t u r e or as the soap films t h a t are formed, when a closed wire curve of arbitrary shape is dipped in a suitably strong soap solution. Since most physical theories are based on a variational principle, the calculus of variations becomes one of the i m p o r t a n t subjects of m a t h e m a t i c a l physics. As a good example we can mention F e r m a t ' s principle for light propagation. According to this principle, light always follows the fastest p a t h . In a homogenous m e d i u m for light propagation this exactly corresponds t o the shortest p a t h , while in an inhomogeneous m e d i u m the 'least t i m e ' p a t h is usually not a straight line. This difference explains what is called refraction of light in inhomogeneous media. T h e above presented method of solving the isoperimetric problem is a good example of a type of method in the calculus of variations t h a t is known as a direct method: you demonstrate, in a problem (here the isoperimetric problem), the existence of a solution with a required extremal property and, at the same time, lay down the solution (here the circle) by constructing an appropriate approximating sequence of objects (here regular polygons). T h e isoperimetric problem can be generalized in m a n y ways, and the t e r m an isoperimetric problem is now being used about a whole series of problems, in which a quantity (usually an integral of some parameters) must be maximized while preserving another quantity. T h e i m m e d i a t e generalization to higher dimensional number spaces is as expected. For example, the sphere is t h a t figure in three dimensional space t h a t encloses the largest volume with a prescribed surface area. This was first proved by H.A. Schwarz (1843-1921) in an article from 1884. This problem is more difficult to deal with t h a n the isoperimetric problem in the plane. Partly, this is because there are no obvious candidates for polyhedra with large numbers of faces t h a t enclose maximal volume with a given area. Only five regular solids exist, and there is no analogue in three dimensional space of the infinite sequence of regular m-gons, m = 3 , 4 , 5 , . . . in the plane.

Sources for Chapter 3

69

Sources for Chapter 3 W. Blaschke: "Kreis und Kugel," Verlag Von Veit & Comp., Leipzig, 1916. R. Courant und D. Hilbert: "Methoden der Mathematischen Physik I," Springer-Verlag, 3. Auflage 1968. R. Courant and H. Robbins: "What is Mathematics?," Oxford University Press, 1941. H.G. Eggleston: "The Isoperimetric Problem," Chapter 7 in the book "Exploring University Mathematics 1" (editor N.J. Hardiman), Pergamon Press, 1967. V. L. Hansen: "Geometry in Nature," A K Peters, Ltd., Wellesley, MA, 1993. V. L. Hansen: "The Magic World of Geometry - I. The Isoperimetric Problem," Elemente der Mathematik Vol. 49, No. 2, Birkhaiiser, 1994, pages 61 65. D. Hilbert und S. Cohn-Vossen: "Anschauliche Geometrie," Springer-Verlag, 1932. J. Liitzen: "Cirklens kvadratur, Vinklens tredeling, Terningens fordobling," Forlaget Systime, 1985.

Supplementary literature S. Hildebrandt: "The calculus of variations today," The Mathematical Intelligencer Vol. 11, No. 4, Springer-Verlag, 1989, pages 50-60. S. Hildebrandt and A. Tromba: "Mathematics and Optimal Form," Scientific Amer. Library, W.H. Freeman and Co., 1985. (Beautiful and exciting book about forms in nature as solutions to optimization problems.) M. Kline: "Mathematic Thought from Ancient to Modern Times," Oxford University Press, 1972. (An excellent book on the history of the developments of mathematics.) R. Osserman: "Bonnesen-style isoperimetric inequalities," The Amer. Math. Monthly Vol. 86, No. 1, The Mathematical Association of America, 1979, pages 1-29. G. Polya and G. Szego: "Isoperimetric inequalities in mathematical physics," Princeton University Press, 1951. (The classic on this subject.)

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Optimal plane figures

B. White: "Some recent developments in differential geometry," The Mathematical Intelligencer Vol. 11, No. 4, Springer-Verlag, 1989, pages 41-47.

Two survey papers for specialists R. Osserman: "The isoperimetric inequality," Bulletin of the AMS Vol. 84,American Mathematical Society 1978, pages 1182-1238. L.E. Payne: "Isoperimetric inequalities and their applications," SIAM Review Vol. 9, Society for Industrial and Applied Mathematics 1967, pages 453-488.

Chapter 4

T h e Poincare disc model of non-Euclidean geometry The emergence of non-Euclidean geometries in the beginning of the nineteenth century is one of the dramatic episodes in the history of mathematics. When it finally became clear after more than 2000 years of speculations that the parallel postulate in Euclidean geometry is independent of the other postulates in Euclid's Elements, the famous Gauss - fearing to be ridiculed - did not dare to reveal his findings until Lobachevsky and, independently, Bolyai published their results in this respect around 1830. Non-Euclidean geometries are one of the most fascinating and surprising intellectual constructs of mankind and even today they attract considerable attention. In this chapter we provide a short, but relatively complete, exposition of the geometrical constructions from Euclidean geometry needed to develop non-Euclidean geometry in the Poincare disc model of the hyperbolic plane. The chapter is based on my article "The dawn of non-Euclidean geometry" mentioned among the sources. We start out with a brief introduction to the early history of geometry with emphasis on Euclid's Elements, in particular his postulates.

4.1 Euclid's Elements Geometry derives from the greek word geometria, which means measurement of land. The word was used by the greek historian Herodotus in the fifth century B.C. in his great epic on the Persian wars in which he writes that 71

72

The Poincare disc model of non-Euclidean

geometry

'geometria' was used in old Egypt to find the right distribution of land after the floods of the Nile. As a framework for the description and measurement of figures, geometry was developed empirically in m a n y cultures several thousand years ago. Geometry as pure m a t h e m a t i c s , which encompasses a collection of abstract statements about ideal forms and proofs of these statements, was founded around 600 B.C. in the Greek culture by Thales, who according to legend proved several theorems in geometry. Also the famous school of Pythagoras in the sixth century B.C. deserves mentioning. From the early period one should, however, in particular single out Eudoxus (around 391-338 B.C.), who is known for a theory of proportions and the so-called m e t h o d of exhaustion making rigorous determinations of areas and volumes possible. Classical Greek geometry has first of all survived through the famous 13 books written by Euclid around 300 B.C. known as Euclid's Elements. In these books the m a t h e m a t i c a l , in particular the geometrical, knowledge possessed by the Greeks at the t i m e of Euclid is summarized and systematized in such a way t h a t the exposition has p u t a s t a m p on m a t h e m a t i c a l writings ever since. T h e geometrical content is now known as Euclidean geometry. T h e second peak in classical Greek geometry was reached with the major work on conic sections by Apollonius from around 200 B.C., which we have treated in Chapters 1 and 2. T h e methods which were developed to handle geometrical questions by Euclid, Apollonius and their successors u p to the development of analytic geometry (coordinate geometry) in the seventeenth century are now known under the n a m e synthetic geometry. Euclid's Elements contains definitions, postulates and theorems. T h e basic definitions are not particularly precise. For example a point is defined as an object without parts, and a line as an object without width. A line corresponds to what we nowadays call a segment of a curve. A straight line is defined as a line which runs directly along its points, and hence corresponds to what we nowadays call a line segment. Since one cannot define a notion by listing all the properties it does not possess, these definitions leave much to be desired and should be taken as a point of departure only. In other definitions, Euclid defines notions such as angle and parallelism and geometrical figures such as the circle and the different types of triangles and quadrilaterals. Here we shall only state Euclid's definition of a right angle: "When a straight line is erected on another line, so t h a t the angles next to each others have the same measures, then any of the angles having the same measures is right; and the erected line is said to be vertical to the other line." In a certain sense postulates and theorems have the same s t a t u s in Euclid, but with the difference t h a t postulates are facts directly accepted as true without proofs, whereas theorems have to be proved by rational arguments from definitions and postulates.

Euclid's

73

Elements

Constructions of geometrical figures play an i m p o r t a n t role in Greek geometry, providing the necessary existence proofs. In particular such constructions as can be performed with ruler and compass were considered i m p o r t a n t . Which rules shall be satisfied in this connection? Which constructions are admissible? In the first three of his postulates, Euclid therefore writes down some basic admissible constructions with ruler and compass. In the English translations of Euclid's Elements, the postulates are formulated in the following manner. Let there be

postulated:

1. That one can draw a straight 2. That one can extend

line from any point to any other

a Unite straight

3. That one can construct

line continuously

in a straight

a circle with any centre and any

4. That all right angles are equal to one

point. line.

radius.

another.

5. That, if a straight line intersects two straight lines and make the interior angles on the same side less than two right angles, then the two lines, if they are extended indefinitely, meet on that side on which are the angles less than two right angles. Difficulties are hidden behind the above system of postulates, and it is not sufficient to characterize Euclidean geometry. Only more t h a n two thousand years later did the G e r m a n m a t h e m a t i c i a n David Hilbert (1862-1943) succeed in formulating a complete set of axioms for Euclidean geometry in his book "Grundlagen der Geometrie" from 1899. Hilbert's system, building on previous works of Moritz Pasch (1843-1930) and Guiseppe Peano (1858-1932) among others, is, however, so complicated t h a t usually it is not presented in its entirety in the teaching of Euclidean geometry. W i t h o u t explicitly mentioning it, Euclid assumes t h a t there is an underlying measure of distance between points, a measure of area of figures and a measure of size of angles. In Postulate 2 it is expressed t h a t a line segment can be arbitrarily extended into other line segments, or with the interpretation of a line given nowadays, t h a t a straight line has unbounded length. Taking the set of real numbers as point of departure, the latter is most easily m a d e precise by postulating t h a t for every line there is a bijective correspondance with the set of real numbers, so t h a t the distance between any two points on the line is the absolute value of the difference between the two real numbers to which the points correspond.

74

The Poincare disc model of non-Euclidean

geometry

4.2 The parallel axiom and non-Euclidean geometries It was, however, Euclid's 5th postulate t h a t attracted most attention. Right from the beginning, mathematicians were concerned whether this postulate is necessary, or whether it is a consequence of the four other postulates. Down through the centuries, vain a t t e m p t s were m a d e to deduce Euclid's 5th postulate from the other postulates. During these efforts m a n y equivalent formulations of the postulate were found. T h e most famous equivalent formulation is due to Playfair 1795 and is known as Playfair's axiom, or T h e p a r a l l e l a x i o m : For any given line in the plane and any given point outside the line, there passes exactly one line through the given point t h a t does not intersect the given line. As a consequence of his defintions and the five postulates Euclid shows in the Elements t h a t the following theorem holds. T h e s u m o f t h e a n g l e s i n a t r i a n g l e : T h e sum of the angles in a triangle is equal to the sum of two right angles (180°). T h e most famous attack on the problem in connection with Euclid's 5th postulate is due to the Italian m a t h e m a t i c i a n Girolamo Saccheri (1667-1733), who tried to prove the theorem about the sum of the angles in a triangle only using the first four of Euclid's postulates. Saccheri tried to obtain a contradiction by assuming the existence of certain special quadrilaterals (now known as Saccheri quadrilaterals) corresponding to the sum of the angles in a triangle being greater t h a n , respectively less t h a n , the sum of two right angles. It can be proved t h a t the theorem about the sum of the angles in a triangle is in fact equivalent to Euclid's 5th postulate. Saccheri did not succeed in giving a correct proof of the theorem as desired. As it turned out later, one can construct geometries corresponding to all the three cases for the sum of the angles in a triangle t h a t Saccheri had considered in his work. Geometries, where the sum of the angles in a triangle is respectively greater t h a n , equal to and less t h a n 180°, are now called respectively elliptic, parabolic and hyperbolic with a terminology introduced by the G e r m a n m a t h e m a t i c i a n Felix Klein (1849-1925). T h e most c o m m o n n a m e for parabolic geometries is of course Euclidean geometries. Around 1830 came the breakthrough, when the Russian m a t h e m a t i c i a n Nicolai Ivanovich Lobachevsky (1793-1856) in 1829 and the Hungarian m a t h ematician Janos Bolyai (1802-1860) in 1832 independently publicized t h a t they could construct geometries satisfying all the properties known from Euclidean geometry except the parallel axiom. As it turned out, Euclid had m a d e a clever decision when he formulated the 5th postulate in the description of his geometry. T h e G e r m a n m a t h e m a t i c i a n Karl Friedrich Gauss (1777-1855) h a d

The parallel axiom and non-Euclidean

geometries

75

in fact obtained similar results already back in 1816 but had kept his findings to himself since they deviated so strongly from accepted philosophical thinking of the time. The German philosopher Immanuel Kant (1724-1804) even took the point of view that the human brain "thinks Euclidean and can only conceive of space in Euclidean terms." In a paper in 1887 the French mathematician Henri Poincare (1854-1912) described a particularly well known model of a non-Euclidean geometry: the hyperbolic plane. The points in Poincare's model of the hyperbolic plane are the points within the boundary of a Euclidean disc $ , and the hyperbolic lines are the Euclidean circular arcs in $ that intersect the boundary circle of $ at right angles; cf. Figure 4.1. As hyperbolic lines we also include all Euclidean diametres in $; one can think of these hyperbolic lines as Euclidean circles with 'infinitely large' radius. An angle between two intersecting curves in $ is measured by the Euclidean angle between the tangents to the curves at the point of intersection. In the following we shall prove that Poincare's model satisfies Euclid's Postulate 1, and that $ can be equipped with a notion of distance that satisfies the interpretation of Euclid's Postulate 2 given nowadays.

Figure 4.1: Poincare's model of the hyperbolic plane In this century Poincare's model has been used by the Dutch artist M.C. Escher in a series of pictures. In Escher's picture 'Circle limit III' the fishes swim along circular arcs similar to hyperbolic lines, cf. Figure 4.2. The circular arcs in 'Circle limit III' do in fact not all intersect the boundary circle at right angles and hence are not perfect hyperbolic lines; Escher has been more concerned with the artistic aspects in the picture. As a matter of fact, they are so-called equidistant curves, as pointed out by H.S.M. Coxeter in the article "The Trigonometry of Escher's Woodcut Circle Limit III," The Mathematical Intelligencer Vol. 18, No. 4 (1996), 42-46. See also Excersise 61.

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T i e Poincare disc model of non-Euclidean geometry

F i g u r e 4.2: M.C. Escher: Circle limit III, 1959. ©1996 M.C. Escher/Cordon Art-Baarn-Holland. All rights reserved. Due to the fine agreement between theory and observations in nature people had in the course of centuries become accustomed to thinking of Euclid's postulates as selfevident truths. The construction of non-Euclidean geometries raised the question which kind of geometry describes the physical world in the best possible way. Thereby began one of the golden periods in the interaction between mathematics and physics, which in the beginning of this century led Einstein to develop his theory of relativity. This is described in some detail in my book "Geometry in Nature."

4.3 Inversion In a circle In order to prove that the geometry in Palmare's model of the hyperbolic plane satisfies Euclid's Postulate 1, i.e., through two arbitrary points in $ passes exactly one hyperbolic line, we need a geometrical construction, known as inversion in a circle.

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Inversion in a circle

Consider a circle £ in the Euclidean plane with centre in the point O and radius r. Let P be an arbitrary point in the plane different from O. Then there is a unique point P* in the plane lying on the half-line from O through P so that the Euclidean distances \OP\ and |0.P*| satisfy \OP\-\OP*\

= r2.

The point P* is called the inversion of P in the circle £. Reciprocally, it is clear that P is the inversion of P*. A geometrical construction of P* for a point P in the interior of £ is indicated in Figure 4.3. Consider the perpendicular to the half-line from O through P at the point P and choose one of the two points C in which the perpendicular intersects £. Then P* is the point of intersection between the tangent to £ at the point C and the half-line from O through P. This follows using that the triangles AOPC and AOCP* are similar whereby

\OP\=\OC\_ \oc\ \OP*\' Since |OP|-|OP*| = \OC\2 = r2 we conclude that P* is indeed the inversion of the point P in the circle £. For a point P outside £ one finds P* as follows. First construct the circle with diameter OP. This circle intersects £ in two points. Next drop the perpendicular onto the half-line from O through P from one of these points C. Where the perpendicular intersects the half-line one finds P*. Finally, we note that a point P on £ is mapped into itself under inversion in £.

Figure 4.3: Inversion in the circle £

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The Poincare disc model of non-Euclidean

geometry

For a point P in the interior of £ as in Figure 4.3, the real numbers a = r-\OP\

and

a* =

\OP*\-r

describe the distances from P, respectively P*, to £. By an easy geometrical argument it follows t h a t a* for fixed a decreases with increasing radius r. By definition of inversion in £ we get: (r - a) • (r + a*) = \OP\ • \OP*\ = r2, which after a small computation gives „ aa* a —a = . r W h e n the radius r in £ grows without bound we see from this t h a t a* approaches a, since a* for fixed a is bounded under the limit procedure. It follows t h a t inversion in £ transforms into s t a n d a r d reflection in a line (circle with 'infinitely large' radius) under the limit procedure r —>• oo. By passing t o Euclidean 3-space one can give an alternative construction of inversion in a circle, which we shall now describe.

F i g u r e 4.4: Inversion in the circle £ T h i n k of the Euclidean plane and the circle £ respectively as the equatorial plane and the equatorial circle for the sphere with centre O, radius r, north pole N and south pole S; cf. Figure 4.4. Consider an arbitrary point P in the Euclidean plane different from O. T h e line from N t h r o u g h P intersects the sphere in the point T. Where the line from S through T intersects the equatorial plane we find the point P*.

Inversion

as a

79

mapping

T h a t the point P* is indeed the inversion of the point P in the circle £ can be proved by the following considerations. First remark t h a t the whole construction takes place in the plane containing P and the diameter SN in the sphere. By considering the triangles AOPN and ATPP* it is easily seen t h a t £{ONP) = Z(OP*S). T h e n it follows t h a t the triangles AONP and AOP*S are similar, and therefore t h a t \OP\_

\ON\

\os\ ~ \op*y whereby \OP\ • \OP*\ = \OS\ • \ON\ = r 2 , which completes the proof.

4.4 Inversion as a mapping Inversion in a circle defines a bijective mapping of the Euclidean plane minus the centre of t h e circle onto itself, which interchanges t h e interior and t h e exterior for the circle. We need some results on this mapping. T h e m a i n result is t h a t straight lines and circles are m a p p e d into straight lines and circles by inversion in a circle. Precisely the following theorem is true. T h e o r e m 4 . 1 By inversion

in a circle £ with centre O

(1)

a line through O is mapped into itself,

(2)

a line which does not pass through O is mapped into a circle through O,

(3)

a circle through O is mapped into a line which does not pass through O,

(4)

a circle which does not pass through O is mapped into a circle which does not pass through O.

Proof: (1) is an immediate consequence of the definition of inversion in a circle. (2) W i t h reference to Figure 4.5 consider a line £ , which does not pass through O. T h e perpendicular from O onto £ intersects C in the point M. Let M* be the inversion of M in £ and construct the circle C with diameter OM*. Consider an arbitrary half-line from O t h a t intersects £ in the point P and C in the point P'. Since t h e triangles AOMP and AOP'M* are similar, it follows t h a t \OM\_ \OP'\ ~

\OP\ \OM*\'

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The Poincare disc model of non-Euclidean

geometry

and therefore that \OP\ • \OP'\ = \OM\ • \OM*\ = r 2 , where r is the radius in £. Consequently, P' is the inversion of the point P in £, and we conclude that the line £ is mapped into the circle C under inversion in £.

Figure 4.5 (3) Follows by considerations analogous to (2) with point of departure in Figure 4.6, if C does not intersect £.

Figure 4.6 Let M* be the point on the circle C so that OM* is a diameter in C. Next determine the point M as the inversion of M* in £. Now the image of the circle

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Inversion as a mapping

C by inversion in £ is the line C through M orthogonal to the line through O and M*. (4) Consider a circle C, which does not pass through 0. For an arbitrary point P on C we consider the inversion P* of P in 8. Then \OP\-\OP*\

= r2,

where r is the radius in £. Let Q be the second point (possibly coinciding with P) on C in which the half-line from O through P intersects C; cf. Figure 4.7. Consider the half-line from O through the centre of C and its points of intersection A and B with C. Since Z(QPA) and Z(QBA) are inscribed angles that span the same arc segment of C and therefore have equal measures, it follows easily that the triangles AOPA and AOBQ are similar. Hence \OP\ \OB\

=

\OA\ \0Q\'

which immediately leads to \OP\-\OQ\

= \OA\-\OB\

= k,

where A; is a constant independent of P. It follows that \OP*\ _ rf_ ~\OQ\~T'

Figure 4.7 From this we infer that P* is obtained from Q by multiplication from 2

the point O by the factor ^-. When the point P traverses C, the point Q also

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The Poincare disc model of non-Euclidean

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traverses C. The image of C by inversion in £ is therefore the same as the image of C by multiplication from the point 0 by the factor ^-. Since multiplication from a point maps a circle into a circle, the proof of (4) is completed. Inversion in a circle preserves angles, i.e. an angle is mapped into an angle of equal measure. We note this result as a theorem. Theorem 4.2 Inversion in a circle preserves angles. Proof: We consider the circle £ with centre O as in the proof of Theorem 4.1. Since the angle between two intersecting curves is the ordinary Euclidean angle between the tangents to the curves in the point of intersection, and since angles are added in the usual fashion, it is sufficient to consider the angle between a half-line from O and a circle C. (i) If C passes through O we get a situation as in Figure 4.6. Referring to this figure we see that the angle in the point P between the half-line from O through P and the circle C is a chord-tangent-angle that spans the same arc as the inscribed angle Z.{OM*P). These angles therefore have equal measures. Since the triangles AOPM* and AOMP* are similar, it further follows that Z(OM*P) and /.{OP*M) have equal measures thereby proving the result in this situation. (ii) If C does not pass through O we get a situation as in Figure 4.7. Since the two angles at the points P and Q in which the half-line from O intersects the circle C have equal measures, and since the image circle of C by inversion in £ is obtained by multiplication from O with a factor, the image circle will intersect the half-line at the same angle in the point P* as C intersects the half-line in Q. This proves the result in this situation. Altogether this completes the proof of Theorem 4.2. We also need to know how Euclidean distances transform under inversion in a circle. This information is contained in the following lemma. Lemma 4.1 Consider a circle C in the Euclidean plane with centre in the point M and radius s. Let P and Q be arbitrary points different from M in the plane, and lad P* and Q* be the points arising by inversion in C. Then the following formula holds for the Euclidean distances:

Proof: The proof is tied to Figure 4.8. Since the radius in the circle C is s, \MP*\

= 7T7W\ \MP\

and

M '\ Q*\ ^ ' = \MQ[

Inversion as a mapping

83

Let v be the angle between the half-lines from M through P and from M through Q. From the cosine relation for AMPQ follows: |PQ| 2 = | M P | 2 + \MQ\2 - 2\MP\ • \MQ\ • cos(u). (Note that the formula is true also for v = 0° and v = 180°.) Correspondingly, we get from the cosine relation for AMP*Q*

that

|P*Q*| 2 = \MP*\2 + \MQ*\2 - 2\MP*\ • \MQ*\ • cos(u). By inserting the values for |MP*| and |MQ*| we get: *n* |2 \P*Q

\MP\2

+ \MQ\2

\MP\2-\MQ\2 \MP\2-\MQ\2

\MP\-\MQ\

s(«)

•(|MQ| 2 +|MP| 2 -2|MF|-|MQ|-cos(v))

'|PQ' "

From this the formula in question follows.

Figure 4.8

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The Poincare disc model of non-Euclidean

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4.5 Orthogonal circles and Euclid's Postulate 1 in the hyperbolic plane In the following we shall consider a disc $ in the Euclidean plane with centre O and radius r, for which the set of points inside the boundary circle is taken as the set of points in our model of the hyperbolic plane. For the construction of hyperbolic lines we need the following lemma. Lemma 4.2 An arbitrary Euclidean circle C intersects the boundary circle for $ at right angles if and only if C contains a pair of mutually different points P and P* that corresponds to one another under inversion in the boundary circle of $ . This being the case, the circle C together with every point P on C will also contain the inversion P* of P in the boundary circle o / $ .

Figure 4.9 Proof: (i) First assume that C intersects the boundary circle of <J> at right angles and let P and Pi be the two points of intersection between C and a half-line from O; cf. Figure 4.9. We shall prove that the points P and Pi are mapped into one another under inversion in the boundary circle of $ . Let C be one of the points of intersection between C and the boundary circle of $ . Since the two circles intersect at right angles, the angle Z(OCP) is a chordtangent-angle in C, which spans the same arc as the inscribed angle Z.(OP\C) in C, so that these angles are equal. It now follows easily that the triangles AOCP and A O P i C are similar. Therefore

\oc\ \OPI\

\OP\

\ocy

which immediately leads to |OP|-|OPi| = \OC\2 = r 2 , showing what is desired.

Orthogonal circles and Euclid's Postulate 1 in the hyperbolic plane

85

(ii) Next assume that C together with one of its points P also contains the inversion P* of P in the boundary circle of $ and that P ^ P*. Then £ must intersect the boundary circle of $ in two points, and since these points of intersection are fixed under inversion, it now follows from Theorem 4.1 that C is mapped onto itself under inversion in the boundary circle of $. Elementary geometric considerations show that this is possible only if C intersects the boundary circle of $ at right angles. This completes the proof of Lemma 4.2. Now we can prove that the geometry in our model <J> of the hyperbolic plane satisfies Euclid's Postulate 1. This is exactly the content of the following theorem. Theorem 4.3 In our model <3> of the hyperbolic plane there passes through every pair of mutually different points P and Q exactly one hyperbolic line in the model.

Figure 4.10 Proof: Without loss of generality we can assume that P ^ O. Together with P we consider the inversion P* of P in the boundary circle of $ . Then there exists exactly one circle C (possibly a line through O), which passes through the three mutually different points P, P* and Q (could possibly be colinear). The construction of C in the circle case is shown in Figure 4.10. From Lemma 4.2 follows that C intersects the boundary circle of $ at right angles. The arc segment of C in the interior of $ is therefore a hyperbolic line, and by construction it contains the points P and Q. This completes the proof of Theorem 4.3.

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4.6 The notion of distance in the hyperbolic plane and Euclid's Postulate 2 Consider two different points P and Q in our model $ of the hyperbolic plane. According to Theorem 4.3 there is exactly one hyperbolic line £ in the model, which passes through these points. The hyperbolic line is an arc segment of a circle C (possibly a diameter in $ ) , which intersects the boundary circle of <J> at right angles in the points U and V, named so that the points in question follow the succession U, P, Q, V in relation to £; cf. Figure 4.11.

Figure 4.11 The hyperbolic distance from P to Q, denoted by d(P, Q), is now defined by the formula

«™=-(!M) where In denotes the natural logarithm function. For P = Q,we put d(P, Q) = 0. For P ^ Q we get immediately that

«*<»=H$!--PISince \QU\

KM

\PU\ \PV\'

it follows that d(P,Q) > 0 for P ^ Q, as one would expect from a notion of distance.

The notion of distance

in the hyperbolic

plane and Euclid's

Postulate

2

87

W h e n the hyperbolic distance d(Q, P) shall be computed, we have to consider the points in the succession V, Q, P, U. T h e n it is easily shown t h a t d(Q, P) = d(P, Q), so t h a t we can speak about the distance between the points P and Q. T h e following theorem shows t h a t the notion of hyperbolic distance satisfies the requirement corresponding to the interpretation of Euclid's Postulate 2 given nowadays. T h e o r e m 4.4 Consider a hyperbolic line C, which intersects circle of <& at the points U and V. Then

/(P) = l n j ^ j

for

defines a bijective mapping f : £ —> M, for d(P,Q)

=

the

boundary

PeC, which

\f(Q)-f(P)\

for all points P,Q 6 £ . Proof: W h e n the succession of the points on £ is U, P, Q, V as in Figure 4.11, we say t h a t the point P precedes the point Q. Thereby an ordering of the points on £ is defined. T h e above rewriting of d(P, Q) then shows t h a t d(P, Q) = f(Q) — f{P)i when P precedes Q. Numerical value takes into account t h a t the succession of the points can be U,Q, P,V, i.e. t h a t Q precedes P. It also follows from the above rewriting of d(P, Q) t h a t / : £ —» R is a strictly increasing m a p p i n g of £ onto the set of real numbers M, such t h a t f(P) tends to oo, when P approaches V, and / ( F ) tends to —oo, when P approaches U. This proves Theorem 4.4. Consider a hyperbolic line £ , which is an arc segment of the circle C with centre in M, and which intersects the boundary circle of $ in the points U and V. We are heading towards finding an appropriate m e t h o d for computing the function / = f(P) : £ —>• M, which determines the hyperbolic distance along £. First remark t h a t £ up to a rotation with centre O (which clearly does not change distances) is completely determined by the central angle Z(UOV) in $ ; cf. Figure 4.12. If we put this angle equal t o 2v, then the angle v lies in the interval 0 < v < £ . An arbitrary point P on £ is determined by the angle t = Z(PUO) in the interval 0 < t < | - v, since Z.(VUO) = f - v. Let N be the foot of the perpendicular from V t o the line through U and P. It t u r n s out t h a t Z(NVP) is constant a n d equal t o v. This follows since

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The Poincare disc model of non-Euclidean

the supplementary angle Z(VPU) in the circle C, whereby

= § + Z(NVP)

Z.{VPU) = |(27T - Z{VMU))

geometry

is also an inscribed angle

= |(2TT - (TT - 2v)) = § + v.

In the above computation we have used that £ is a hyperbolic line, such that the quadrilateral with corners O, U, M, V has right angles in U and V.

M

F i g u r e 4.12 By considering the orthogonal projections of the line segments | P ^ | and \UV\ onto respectively the line segments \UN\ and |JVV|, we get the following equations: (1) |PJ7| + \PV\ • sin(w) = \UV\ • cos(f -v-t) (2) \PV\ • cos(w) = \UV\ • s i n ( | -v-t)

= \UV\ • sin(t/ + *),

= \UV\ • cos{v + t).

From this we deduce the following equation by first forming the equation (l)cos(v) - (2)sin(w) and then using a well known formula from trigonometry: (3) \PU\-cos{v) = |[/V|-(sin(v+0-cos(i;)-cos(v+*)-sin(v)) =

\UV\-sm{t).

Dividing equation (3) by equation (2) we finally get the desired formula:

« | , ) = «') =

ta

5^0

fo

'

»<«!-•

Isometries in the hyperbolic plane

89

4.7 Isometries in the hyperbolic plane By an isometry of a geometrical object equipped with a notion of distance and a measurement of angles, we mean a bijective mapping of the object onto itself, which preserves distances and angles. In the following we shall describe the isometries in the hyperbolic plane. Consider a hyperbolic line C in our model $ of the hyperbolic plane, which is an arc segment of a circle C that intersects the boundary circle of $ at right angles; cf. Figure 4.13. According to Theorem 4.1 the boundary circle of $ is mapped into a circle by inversion in C, and since the boundary circle of $ intersects C at right angles, <3> must be mapped onto itself. This mapping of $ onto itself is called the hyperbolic reflection of $ in the hyperbolic line C. If the hyperbolic line £ is a diameter in $, the ordinary Euclidean reflection in C defines the associated hyperbolic reflection. Note that this is in agreement with the limit description of inversion in a circle, when we let the radius in the circle increase without bound. From Theorem 4.2 we know already that a hyperbolic reflection preserves angles. Now we prove that it also preserves distances. T h e o r e m 4.5 Every hyperbolic reflection preserves distances in the hyperbolic plane and thus defines an isometry of <E> onto itself.

'•C

M

Figure 4.13 Proof: (i) First consider a hyperbolic line £., which is an arc segment of the circle C with centre M and radius s. The computation below, which is tied to Figure 4.13 and employs Lemma 4.1, shows that the hyperbolic reflection in £ preserves distances in the hyperbolic plane.

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The Poincare disc model of non-Euclidean

d(P*,Q*)=ln

geometry

\Q*ir\\

\P*V*\ \P*U*\ '

\Q*V*\)

\MP\-\MV\ \PV\ P

\MP\\MU\\ U\

\MQ\\MU\\QU\ \MQ\\MV\ \QV\

- G S T O - ™ >• (ii) Next consider a hyperbolic line, which is a diameter in $ . In this case hyperbolic reflection is the same as ordinary Euclidean reflection, which clearly preserves distances in the hyperbolic plane. This completes the proof of Theorem 4.5. There are also other isometries of the hyperbolic plane $ onto itself. For example, a Euclidean rotation with centre in O through a given angle will also define a hyperbolic rotation in <£ through the same angle. However, there are hyperbolic rotations with centre in an arbitrary point Po in $ . To definethese we need the following lemma. L e m m a 4 . 3 An arbitrary point Po in $ different from O can by a defined hyperbolic reflection be mapped into O.

uniquely

Proof: Raise the perpendicular to the half-line from O through Po at Po. Next construct the tangent to the boundary circle of $ in one of the points C, in which the perpendicular intersects the boundary circle of 3>. T h e hyperbolic line, in which to reflect in order to m a p Po into O, then has its centre in the point, where this tangent intersects the half-line from O through Po, and it intersects the boundary circle of $ in C. This proves L e m m a 4.3. Now consider an arbitrary point Po in $ different from O together with the unique hyperbolic reflection t h a t m a p s Po into O. If we follow this hyperbolic reflection first by a rotation with centre O through an angle v and then again by the hyperbolic reflection, we have defined the uniquely determined hyperbolic rotation with centre Po through the angle v. T h e collection of all hyperbolic reflections and hyperbolic rotations generates the isometries in the hyperbolic plane <$. Precisely the following theorem holds. T h e o r e m 4 . 6 Every isometry in the hyperbolic plane can be written as a composition of at most two hyperbolic reflections and a hyperbolic rotation. Proof: Consider an arbitrary isometry in $ . Possibly first following this isometry by a hyperbolic reflection we can obtain an isometry t h a t fixes O. If necessary by composing with one further reflection in a diameter in $ , we can

Hyperbolic

triangles and

n-gons

91

obtain an isometry t h a t preserves a prescribed orientation on a small Euclidean circle with centre O. Such an isometry must be a hyperbolic rotation with centre in 0, possibly through an angle of 0°. If we compose this rotation in reverse order with the hyperbolic reflections we have applied in the process, we get back to the original isometry. This proves Theorem 4.6. Consider two different points P and Q o n a hyperbolic line C. T h e n it is not difficult to see t h a t there exist two uniquely defined hyperbolic reflections t h a t m a p the hyperbolic line £ onto itself, such t h a t the isometry composed of these hyperbolic reflections m a p s P into Q. [Proof: Consider the uniquely defined points R and 5 on £ between P and Q so t h a t the points follow in the succession P, R, S, Q and so t h a t each of the hyperbolic distances d(P, R) and d(S,Q) is ~d(P,Q). Then the hyperbolic reflections searched for are exactly the reflections in the two uniquely defined hyperbolic lines through R and S orthogonal t o £.] Such an isometry of the hyperbolic plane onto itself is called a hyperbolic parallel translation with axis C. A hyperbolic circle in $ with hyperbolic centre M and hyperbolic radius s - i.e. the pointset {P\d(M, P) = s} in <£ - can by a hyperbolic reflection in a suitable hyperbolic line be m a p p e d into a hyperbolic circle with centre O and hyperbolic radius s. It is easy to see t h a t such a hyperbolic circle is also a Euclidean circle with centre O, but with Euclidean radius r ^ s. Since a Euclidean circle in $ according to Theorem 4.1 is m a p p e d into a Euclidean circle by a hyperbolic reflection, it follows t h a t the original circle is not only a hyperbolic circle, but also a Euclidean circle; if M ^ O the hyperbolic centre M will be different from the Euclidean centre.

4.8 Hyperbolic triangles and n-gons A hyperbolic triangle is a figure in the hyperbolic plane bounded by three hyperbolic line segments. Again we model the hyperbolic plane by the Poincare disc <J>. For hyperbolic triangles we have the following fundamental result. T h e o r e m 4 . 7 The sum of the angles in a hyperbolic triangle is less than 180°. Proof: Preserving the angles, an arbitrary hyperbolic triangle in $ can, by a hyperbolic reflection, be m a p p e d onto a hyperbolic triangle with O as one of its corners. In such a hyperbolic triangle the edges meeting at the corner O are segments of diameters in $ ; cf. Figure 4.14. Since the s u m of the angles in the hyperbolic triangle with the corners O, P and Q is less t h a n the s u m of the angles in the Euclidean triangle with

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The Poincare disc model of non-Euclidean

geometry

the same corners, we conclude t h a t also the sum of the angles in the original hyperbolic triangle is less than 180°. This completes the proof of Theorem 4.7.

F i g u r e 4.14 By a hyperbolic n-gon, n > 3, we understand a figure in the hyperbolic plane bounded by n hyperbolic line segments. If all the edges have the same hyperbolic length and all angles between edges are equal, then the hyperbolic n-gon is said to be reguiar.

F i g u r e 4 . 1 5 : Tiling with regular hyperbolic 7-gons For every integer n > 5 one can construct a regular hyperbolic n-gon with the sum of the angles 360° and which is symmetric with respect to the centre

The Poincare half-plane

93

O in $ . In short the argument goes as follows. On the one hand a very small regular hyperbolic n-gon with centre O has a sum of angles, close to, but slightly less than, the sum of the angles in the corresponding regular Euclidean n-gon, which is (n — 2)-180°. On the other hand a hyperbolic n-gon with centre O and its corners close to the boundary circle of <J> has a sum of angles close to 0°. Since the sum of the angles depends continuously on the Euclidean distance of the corners from the centre O, there will be, up to rotation about O, exactly one regular hyperbolic n-gon with its centre in O for which the sum of the angles is 360°, when n > 5. For every n > 3 one can construct in a similar manner a regular hyperbolic n-gon with an arbitrarily prescribed sum of angles less than (n — 2) • 180°. In the hyperbolic plane, congruent regular hyperbolic n-gons in which the degree of the angle divides 360°, can be placed in a closed circuit around every corner. By moving such a regular hyperbolic n-gon around in our model $ of the hyperbolic plane by hyperbolic parallel translations and rotations one can therefore mark out a tiling of the hyperbolic plane. Thereby we obtain the following surprising theorem. Theorem 4.8 For every integer n > 3 the hyperbolic plane can be tiled with congruent regular hyperbolic n-gons. By a closer examination one finds that the angle in the regular hyperbolic n-gon can be chosen, so that p of these meet in each corner of the tiling of <2>, if p > 7 for n = 3, p > 5 for n = 4, p > 4 for n = 5, 6 and p > 3 for n > 7. Theorem 4.8 is in strong contrast with the situation in the Euclidean plane, which can be tiled with congruent regular Euclidean n-gons only for n = 3,4 and 6, and only such that respectively 6, 4 and 3 of these meet in every corner of the tiling. (This follows by elementary considerations of angles in regular Euclidean n-gons.)

4.9 The Poincare half-plane There is also another useful model of the hyperbolic plane known as t i e Poincare half-plane. A good description of this model and its properties can be found in the book by S. Stahl: "The Poincare Half-Plane." We can get to the Poincare half-plane H from the Poincare disc $ in the following way. Choose a circle £ that encloses $ and has its centre on the boundary circle of $ . According to Theorem 4.1 the boundary circle of $ is mapped into a line C by inversion in £. The interior of $ is mapped into one of the two open half-planes H determined by the line C. The hyperbolic lines

TAe Poincare disc model of non-Euclidean

94

geometry

in $ are transferred into semi-circles in H that intersect the boundary line £ at right angles, or into half-lines in H orthogonal to £; cf. Theorem 4.1 and Theorem 4.2. It follows from the above remarks that we get a model of the hyperbolic plane with the points in an open half-plane H in the Euclidean plane determined by a line £ as points, and the semi-circles in H with centres on £ and the half-lines orthogonal to £ as hyperbolic lines; cf. Figure 4.16. Note that in the Poincare half-plane it is very easy to construct the unique hyperbolic line through any two points. The hyperbolic distance between the points P and Q in H is also in this case defined by the formula d(P,Q)_ln^—•—J, where the meaning of U and V is evident from Figure 4.16. When P and Q lie on a line orthogonal to £ we have to put \ny\ = 1-

H Q

Q

P

P/ -i

V

u

u

Figure 4.16

4.10 Elliptic geometries There are also other geometries, which although not Euclidean, share many of the properties known from Euclidean geometry. Consider an arbitrary sphere in Euclidean 3-space. The obvious connection between two points on the sphere is the shortest of the great circle arcs con-

Elliptic geometries

95

necting the points; cf. Figure 4.17. Between two diametrically opposite points on the sphere there are infinitely many great circle arcs of the same length. In order to obtain a geometry that satisfies Euclid's Postulate 1, we therefore have to identify all pairs of diametrically opposite points on the sphere. The surface, which thereby arises, is called the projective plane. In small pieces, it looks like the sphere. A closer description of the projective plane can be found in my book "Geometry in Nature," Chapter 2. If as points we take the points in the projective plane and as lines the great circles, then we get a geometry in which Euclid's Postulate 1 is satisfied. All the lines in the geometry have finite length, namely the length of half a great circle, and hence the geometry does not satisfy the interpretation of Euclid's Postulate 2 given nowadays. Since any two great circles on the sphere intersect each other, there are no parallel lines in this geometry, and hence it does not satisfy Euclid's Postulate 5 either. It can be shown that the sum of the angles in a triangle in this geometry, a so-called spherical triangle, is > 180°. As mentioned earlier, a geometry with this property is said to be elliptic with a terminology introduced by Felix Klein.

Figure 4.17

96

The Poincare disc model of non-Euclidean

geometry

Sources for Chapter 4 R. Courant and H. Robbins:"What is Mathematics," Oxford University Press, New York, 1941. H.S.M. Coxeter: "Introduction to Geometry," Wiley, New York, 1961, 2nd ed. 1969 (reprinted 1989). V. L. Hansen: "Geometry in Nature," A K Peters, Ltd., Wellesley, MA, 1993. V. L. Hansen: "The dawn of non-Euclidean geometry," Int. J. M a t h . Educ. Sci. Technol., Vol. 28, No.l, 1997, pages 3-23. T.L. Heath: "The Thirteen Books of Euclid's Elements," 3 volumes, Dover (reprint), 1956. P. Kelly and G. Matthews: "The Non-Euclidean, Hyperbolic Plane," Universitext, Springer-Verlag, 1981. M. Kline: "Mathematical Thought from Ancient to Modern Times," Oxford University Press, 1972. H. Lenz: "Nichteuklidische Geometrie," Hochschultaschenbiicher - Verlag, Bibliographisches Institut, Mannheim, 1967. S. Stahl: "The Poincare Half-Plane," Jones and Bartlett Publishers, Boston, MA, US, 1993. J. Stillwell: "Mathematics and Its History," Springer-Verlag, 1989.

Exercises Chapter 1 Exercise 1 The shadow of a ball has eccentricity 0.5. How high in the sky is the sun? Exercise 2 The altitude of the sun is 30°. A ball with a diameter of 20 centimetres casts a shadow on the pavement. How long is the shadow? Exercise 3 Seen from the Earth the sun moves in the sky on a circular arc. In a given period of 24 hours we assume that the angle u of the altitude of the sun relative to the horizon can be described approximately by the formula • / \ r^ • i x \ V2 sin(u) = V2 • sm(— • TT) - — , in which x is measured in hours from midnight to midnight. In this formula the angle u can be negative. 1. The altitude of the sun is maximal at 12 o'clock. What is the maximum altitude of the sun in the 24 hours in question? 2. At what time does the sun rise, and at what time does it set in the 24 hours in question? 3. In the afternoon you play ball on the beach. The ball casts an elliptic shadow on the sand with eccentricity e = 0, 86. What time is it? [You are allowed to replace e = 0.86 by e = v A/3 — 1 and to use that in this case Vl - e2 = &(y/%- 1).] Exercise 4 For a given ellipse the major axis is 10 centimetres and the minor axis 6 centimetres long. How large is the eccentricity of the ellipse? Exercise 5 A standard rectangular system of ^-coordinates has been installed in the Euclidean plane. Find the lengths of the axes, the eccentricity, the coordinates of the foci, and the equations of the directrices for each of the ellipses given by the equations (i) %- + ^ = 1, and (ii) 4a;2 + 25j/2 = 100. 97

98

Exercises

Exercise 6 A standard rectangular system of aiy-coordinates has been installed in the Euclidean plane. For an arbitrary real number s > —1 we consider the ellipse given by the equation

4+s

1+s

Show that this ellipse always has the same foci independent of the value of s > — 1. (Such a family of ellipses is said to be a confocal family of ellipses.) Make a sketch of the ellipse for various values of s > — 1. Exercise 7 A sheet making an the angle v to a table contains a circular hole. A lamp over the table sends a vertical pencil of rays through the hole and produces a spot of light on the table. Explain that the spot of light is elliptical and determine the eccentricity of the ellipse. Exercise 8 Construct by ruler and compass the directrices of an ellipse for which you know the length of the major axis and the position of the foci. Exercise 9 In a standard rectangular system of ^-coordinates in the Euclidean plane an ellipse is given by the equation

— + J!L - 1 52 (|)2 ' Show that there are two tangents to the ellipse that pass through the point (—5,5), and determine the equations for these tangents. Exercise 10 Consider an ellipse with the major semiaxis a and the minor semiaxis b. Let P be an arbitrary point on the ellipse at which the tangent is not parallel to either of the axes. Then the tangent and the normal to the ellipse in P both intersect the line through the foci, say in the points T and N, respectively. When O denotes the centre of the ellipse (the point of intersection between the axes), then \OT\ • \ON\ = a2 - b2. Prove this. Exercise 11 For an ellipse, you know one of the foci and the line through the foci. Furthermore, you are given a point on the ellipse outside this line plus the tangent to the ellipse at that point. Construct by ruler and compass the other focus of the ellipse. Exercise 12 Show that the triangles A 0 B and Q 0 P in Figure 15 are congruent. You may for example first prove that the triangles BAP and PQB are congruent. By a chord for a plane curve we understand a line segment that connects two different points on the curve.

Exercises

99

E x e r c i s e 1 3 A straight line through the centre of an ellipse is called a diameter of the ellipse. Two diameters of an ellipse are said to be conjugate, if the midpoints of the chords in the ellipse t h a t are parallel to one of the diameters, lie on the other diameter. As an example, the lines for the major axis and the minor axis are a pair of conjugate diameters. 1. Let m i be an arbitrary diameter of an ellipse. First show t h a t all midpoints for chords k\ of the ellipse t h a t are parallel to m\ are situated on a diameter m2 of the ellipse. Next show t h a t m i bisects all chords hi in the ellipse t h a t are parallel to 7712. In other words, the diagonals m i and m2 are a pair of conjugate diameters. Hint: You may for example use t h a t an ellipse can be produced as a plane section in a cylinder. 2. Let m i be an arbitrary diameter in the ellipse. Prove t h a t there exists exactly one other diameter m2 of the ellipse which is conjugate to m i . 3. Install in the usual way a rectangular system of ^ - c o o r d i n a t e s in the plane of the ellipse. Prove t h a t if a diameter m i of the ellipse has the equation px+qy = 0, then the conjugate diameter mi has the equation —b2qx+a2py = 0. 4. Let m i and 7712 be an arbitrary pair of conjugate diameters of an ellipse. Install again, in the usual way, a rectangular system of xy-coordinates in the plane of the ellipse. Let (pi,P2), respectively (91,92), be one of the points of intersection between the ellipse and m i , respectively m,2 • Prove t h a t the ellipse permits a parametrization of the form

{x,y)

= (picos(t)

+ qisin(t),p2cos(t)

+ 9 2 sin(<)),

0 < t < 2w.

5. Let m i and m-i be a pair of conjugate diameters in the ellipse, and let &i be a chord in the ellipse parallel to, b u t different from, rti\. Prove t h a t the tangents at the end points of k\ intersect in a point on mi. E x e r c i s e 1 4 Consider an ellipse produced as the curve of intersection between a vertical circular cylinder and a plane /?. Place two spheres with the same radius as the cylinder inside the cylinder so t h a t they b o t h touch (3 and so t h a t exactly one of the spheres lies above a n d one below (3. (Dandelin's construction.) Show t h a t the length of the major axis of the ellipse is just the distance between the circles of tangency of the two spheres with the cylinder.

Chapter 2 E x e r c i s e 15 Consider a fixed line / in the plane and a fixed point F not on the line. Construct by ruler and compass a number of points on the parabola with directrix / and focus F.

100

Exercises

Exercise 16 Consider a fixed line / in the plane and a fixed point F not on the line. At your disposal you have a triangle for drawing whose longer leg is longer than the distance from F to I and a piece of string of the same length as the longer leg of the triangle. Now attach one end of the string to the vertex opposite to the shorter leg of the triangle and the other end of the string to F. Explain then how you can draw a segment of the parabola with directrix / and focus F by letting the shorter leg of the triangle slide along / holding the piece of string towards the point on the triangle to which it is fixed tight to its longer leg. Exercise 17 Consider a parabolic bicycle lamp, i.e. a bicycle lamp in which the reflector is shaped as the surface that emerges by revolving a parabola around its symmetry axis (a so-called paraboloid of revolution). Show that if the bulb is placed in the focus of the generating parabola, the light will be reflected in parallel rays. Exercise 18 Show that the parameter p > 0 for a parabola can be determined as the length of the chord in the parabola that passes through the focus and is parallel to the directrix. Exercise 19 Consider an ellipse with major semiaxis a, minor semiaxis b, and eccentricity e. The length of a chord of the ellipse through one of the foci and perpendicular to the major axis is denoted by p and is called the parameter of the ellipse. Show that | = a(l - e 2 ) = ^ . Exercise 20 Consider a hyperbola with transverse semiaxis a, conjugate semiaxis b, and eccentricity e. The length of a chord in the hyperbola through one of the foci and perpendicular to the transverse axis is denoted by p and is called the parameter of the hyperbola. Show that | = a(e 2 — 1) = —. Exercise 21 A standard rectangular system of xy-coordinates has been installed in the Euclidean plane. Determine equations for all tangents to the parabola y = x2 through the point (2,1). Exercise 22 A standard rectangular system of zy-coordinates has been installed in the Euclidean plane. Determine equations for all normal lines to the parabola y — x1 through the point (—1,2). Exercise 23 Define the hyperbolic functions cosine hyperbolic and sine hyperbolic oit, which we denote respectively cosh(t) and sinh(t), by the formulas ,,, exp(i) + exp(—t) cosh(<) = FV ; ^y—'-

, and

. , ... exp(t) — exp(—t) sinh(<) = ' —-,

Exercises

101

in which exp denotes the exponential function and t is an arbitrary real number. 1. Make a sketch of the graphs of the functions cosh(i) and sinh(tf). 2. Prove that cosh 2 (t) - sinh2(<) = 1. 3. Find a parametrization for the hyperbola with the transverse semiaxis a and the conjugate semiaxis b. Exercise 24 In the Euclidean plane let there be given two fixed points E and F. At your disposal you have a bar whose length exceeds the distance between E and F and a piece of string that is shorter than the bar. Attach one end of the string to one end point of the bar and the other end of the string to F. The other end of the bar is bolted to E so that it can revolve around this point. Now explain how you can draw a segment of a hyperbola that has E and F as foci by holding the piece of string towards the fixing point at the bar tight to the bar. Exercise 25 A standard rectangular system of ary-coordinates has been installed in the Euclidean plane. An ellipse has been placed in the system of coordinates so that the major axis is parallel to the ar-axis and the minor axis is parallel to the y-axis. The axes intersect in the point that has the coordinates (xo,yo) (the centre of the ellipse). The major semiaxis has the length a and the minor semiaxis the length b. Explain that the ellipse has the equation

(x-x0)2

jy-yo)2

a2

b2

Exercise 26 A standard rectangular system of xy-coordinates has been installed in the Euclidean plane. A parabola has been placed in the system of coordinates in such a way that its directrix is parallel to the x-axis and its focus lies below the directrix. Assume that the point of intersection between the parabola and the line through the focus perpendicular to the directrix (the vertex of the parabola) has the coordinates (xo,yo), and that the parameter is p > 0. Explain that the ellipse has the equation (x -x0)2

=

-p{y-y0).

Exercise 27 A standard system of xy-coordinates has been installed in the Euclidean plane. Determine the equation for the parabola that has horizontal directrix and passes through the points (2,3), (4,3) and ( 6 , - 5 ) . Exercise 28 A comet follows a parabolic path with the sun in the focus. When the comet is 100 mio. kilometres from the sun, the line from the sun to the comet is perpendicular to the symmetry axis of the parabola (the line through the focus perpendicular to the directrix). What is the minimum distance from the comet to the sun?

102

Exercises

Exercise 29 A standard rectangular system of ajy-coordinates has been installed in the Euclidean plane. A hyperbola has been placed in the coordinate system so that the transverse axis is parallel to the a;-axis and the conjugate axis is parallel to the y-a,xis. The axes intersect in the point with the coordinates (2:0,2/0) (the centre of the hyperbola). The transverse semiaxis has the length a and the conjugate semiaxis the length 6. Explain that the hyperbola has the equation (z-*o)2 a2

(y-yo)2 b2

_,

Exercise 30 A standard rectangular system of zy-coordinates has been installed in the Eucli3ean plane. Determine the equation for the hyperbola that has the foci (±10,0) and the asymptotes y = ± | a ; . Exercise 31 On a calm day a gun is fired at each of the points E and F with the mutual distance of 680 meters, on level ground and at the same time. A person at a distance of 34 meters from the line through E and F observes that where he is standing the bang from E arrives ^ seconds later than the bang from F. The velocity of sound is considered to be 340 meters/second. How far away is the person from the point F? Exercise 32 Determine which kind of conic section is defined by the equation 2x2 - 8a; - y + 5 = 0. Exercise 33 For every value of the real constant c, decribe the (perhaps degenerate) conic section defined by the equation of degree two x2 + cy2 + 2x - y = 0. Exercise 34 Determine which kind of conic section is defined by the equation 5x 2 + 6xy + by2 - 4\/2a; - 12-y/2y - 1 6 = 0. Exercise 35 Consider a circle C in the plane with centre E and radius r > 0. Let F be a point that is not on C. Determine the locus for the points P in the plane which are the centres of circles that each touches C in exactly one point and passes through F. Exercise 36 Consider a line m and a point F in the plane. Determine the locus for the points P in the plane which are the centres of circles that each touches m at exactly one point and passes through F.

Exercises

103

Exercise 37 Consider two fixed points E\ and E^ in the plane and let rj and r2 be real numbers so that r\ > r-i > 0 and r\ + T2 < \E\Ei\. Let C\ be the circle with centre E\ and radius r\ and C2 the circle with centre £2 and radius r2. Determine the locus for the points P in the plane which are the centres of circles that each touches both of the circles C\ and C2 at exactly one point. Exercise 38 The product of the distances from the foci in an ellipse, or a hyperbola, to an arbitrary tangent of the ellipse, respectively the hyperbola, equals the square of the minor axis in the ellipse, respectively the conjugate axis of the hyperbola. Prove this. Exercise 39 Consider an ellipse with the major semiaxis a and the minor semiaxis b that has its centre in the point O. Then show that the point of intersection between two tangents to the ellipse perpendicular to each other lies on the circle with centre in O and radius \Ja2 + b2.

Chapter 3 Exercise 40 Triangle ABC has fixed side lengths AB and AC. angle in the corner A does the triangle have the maximum area?

For which

Exercise 41 In the isosceles triangle ABC the two equally long legs AB and BC both have the fixed length s. Furthermore, the triangle has the largest area among all isosceles triangles with this length of legs. How long is the base line AC? Exercise 42 Two disjoint isosceles triangles have the same total perimeter L. How shall the triangles be chosen so that the total area gets as small as possible? Exercise 43 Two disjoint triangles (of which one possibly can degenerate to a point) have the total perimeter L. How shall the triangles be chosen so that the total area gets as large as possible? Exercise 44 Consider a triangle ABC with the base line AC kept fixed and a fixed prescribed area A. Show that the triangle has minimum perimeter when the sides AB and BC are equally long, i.e. when the triangle is isosceles. Exercise 45 How shall a triangle ABC with a fixed prescribed area A be chosen so that its perimeter gets as small as possible? Exercise 46 Consider a parallelogram with fixed edge lengths. How shall the parallelogram be chosen so that it gets maximum area?

104

Exercises

Exercise 47 Show that sin^ > approaches 1 through decreasing values and that tai^/ x approaches 1 through increasing values, when x approaches 0 through decreasing values. Hint: You may for example consider the position of the graphs for the functions sine and tangent in relation to the line y = x, and use that SiSi£2 and tan\x) Can be interpreted as the slope of the line segment from (0, 0) to the point (a?,sin(a;)), respectively to the point (x,tan(a;)). Alternatively, you may show that for small positive values of x, the differential coefficient of . xi s is sin i *u \

negative, and the differential coefficient of ta^,x\ is positive. Exercise 48 By the use of a pocket calculator or a computer, make a table of the magnitudes hn,r„, and An for the regular n-gon with perimeter 27T, when n increases from 10 to 100 in steps of 10. Also find these magnitudes for n = 500,1000,3000,5000. Exercise 49 Show that there passes exactly one circle through any three different points that are not on a line. Next show that if the triangle APB has a right angle at P, the point P lies on the circle with AB as diameter. Exercise 50 Consider a fixed line segment AB in the plane and a curve segment of a fixed length that connects A with B and only meets AB at the end points. How shall the curve segment be designed so that it together with AB encloses the maximum area? (You may assume without proof that the problem has a solution. If possible, demonstrate the solution by an experiment.) Exercise 51 Consider two line segments AB and AC that make an acute angle in A. Connect B to C by a curve segment of a fixed length. How shall the curve segment be designed so that it bounds a set with maximum area in the angle? (You may assume without proof that the problem has a solution. If possible, demonstrate the solution by an experiment.) Exercise 52 Show that the circle is the closed plane curve that has minimum length among all closed plane curves (without self-intersections) that enclose a fixed prescribed area. Hint: The problem is an equivalent formulation of the isoperimetric problem. Exercise 53 In a territory a farmer gets an offer of making himself a rectangular field that is bounded by a road on one side. The fertility of the soil increases proportionally to the distance from the road, but is otherwise constant along parallels to the road. The only other restriction is that the total perimeter of the field must have the fixed length of 1200 meters. How shall the

Exercises

105

farmer choose the edge lengths of his rectangular field so that the yield gets as large as possible? Hint: First provide reasons that it all boils down to maximizing the integral /•600-r

f(x) = x I ydy, when 0 < x < 600, Jo and then solve the problem. Do also the problem if the fertility of the soil increases proportionally to the square of the distance from the road. Exercise 54 A ray of light is emitted from a point source at a point P in three dimensional space and hits slantwise a screen in which it is reflected. Consider the plane containing P, the ray of light, and the normal to the screen in the point A in which the ray of light hits the screen. After reflection the reflected ray of light will take a course in this plane. The acute angle i between the ray of light and the normal to the screen is called the angle of incidence, and the acute angle u between the reflected ray of light and the normal to the screen is called the angle of reflection. Make a plane figure that shows the path of rays. According to Fermat's principle for light propagation, the light follows the fastest path. In a homogenous medium as here this is the same as the shortest path. Make use of Fermat's principle to show that the angle of incidence is equal to the angle of reflection.

Chapter 4 Exercise 55 Consider a circle £ in the Euclidean plane with centre in the point O and radius r. Let P and Q be two different points in the interior of £ not on a diagonal in £; in particular, both P and Q are then different from O. Denote by P* and Q* the inversion of respectively P and Q in £. Show that the triangles AOPQ* and AOQP* are similar. Exercise 56 Install a system of rectangular ary-coordinates in the Euclidean plane and consider the circle £ with centre in the point O and radius r. Find an analytical expression for the inversion P* in £ of an arbitrary point P in the interior of £ different from O. Exercise 57 Consider a circle £ in the Euclidean plane with centre in the point O and radius r. Which circles in the plane are mapped into tangent lines to £ by inversion in £? Exercise 58 Find an expression for the hyperbolic distance along a diameter in Poincare's model <J> of the hyperbolic plane expressed by the Euclidean distance from the origo.

106

Exercises

Exercise 59 Given two different points P and Q in Poincare's model $ of the hyperbolic plane. Provide a geometrical construction of the hyperbolic reflection that maps P into Q. Exercise 60 Provide a geometrical construction of the hyperbolic perpendicular bisector to a hyperbolic line segment in Poincare's model <3> of the hyperbolic plane. Exercise 61 Consider a hyperbolic line £ in Poincare's model $ of the hyperbolic plane. The collection of hyperbolic lines perpendicular to £ is called the hyperparallel family with base line £. Consider a point P in $ outside £. By hyperbolic reflection of P in all the hyperbolic lines in the hyperparallel family with base line £ we get a curve in <3>, a so-called equidistant curve with base line £. Why is it called an equidistant curve? Is it a hyperbolic line? [Equidistant curves occur in M.C. Escher: Circle limit III.] Exercise 62 Consider the collection of all hyperbolic lines through a fixed point Po in Poincare's model $ of the hyperbolic plane; the so-called line bundle through Po- Furthermore, let P be an arbitrary point in $ different from Po • Show that by hyperbolic reflection of P in all hyperbolic lines in the line bundle through Po one gets a hyperbolic circle. Exercise 63 Show that every isometry in Poincare's model $ of the hyperbolic plane can be written as a composition of at most three hyperbolic reflections. Exercise 64 In this exercise we use the open unit disc in the Euclidean plane as the basic set in Poincare's model $ of the hyperbolic plane. In the usual way we install a rectangular system of zy-coordinates in the Euclidean plane. In differential geometry one introduces the notion of a Riemannian metric, which measures the length of an infinitesimal curve segment ds. In Poincare's model of the hyperbolic plane the Riemannian metric is given by the expression rfj2=

4(dx2 + dy2) (1 — x2 — y2)2

1) Find a formula for the distance between two points on a diameter in $ computed by the Riemannian metric in <J> and expressed by the Euclidean distance from the origo. Compare the formula with the expression found in Exercise 58. 2) Show, at least in special cases, that the Riemannian metric is invariant (preserved) by a hyperbolic reflection. 3) A geodesic curve in Poincare's model $ of the hyperbolic plane can be defined as a curve, which on sufficiently small segments of the curve defines

Exercises

107

the shortest connection between arbitrary two points of the segment measured by the Riemannian metric. In differential geometry it is proved (and we take it here for a known fact), that geodesic curves do exist and that they are uniquely defined on small segments. Use this information to determine the geodesic curves in $ . Exercise 65 Let ABCD be a hyperbolic quadrilateral in Poincare's model $ of the hyperbolic plane for which the corners lie on a hyperbolic circle. Show that the sum of one pair of opposite angles in the quadrilateral is equal to the sum of the other pair of opposite angles in the quadrilateral. Exercise 66 Given two pairs of mutually different points A and A', respectively B and B', placed on each its diagonal in Poincare's model $ of the hyperbolic plane such that all the points are different from the origo O in $ . Now consider the hyperbolic triangles AAOB and AA'OB'. Assume that the hyperbolic angles in the corners A and A' in the two hyperbolic triangles have equal measures, and that \OA'\ > \OA\. 1) Prove that \OB'\ > \OB\. Hint: Otherwise one can find a hyperbolic triangle with sum of the angles greater than n. 2) Prove next that the hyperbolic angle in the corner B' in AA'OB' is smaller than the hyperbolic angle in the corner B in AAOB. Hint: Look at the sum of angles in a suitable hyperbolic quadrilateral. Exercise 67 Prove that two similar hyperbolic triangles AABC and AA'B'C in Poincare's model $ of the hyperbolic plane are isometric. This result can be used to introduce the hyperbolic area of hyperbolic ngons. If the radius in $ is r and the hyperbolic triangle AABC has the hyperbolic angles a, 0 and 7 in the corners A, B and C, respectively, then the hyperbolic area of the triangle can be defined as the number A = r2(iv - a - 0 - 7 ) . By triangulation we can then define the area of a hyperbolic n-gon. Is this definition of hyperbolic area reasonable? Exercise 68 Provide a geometrical construction of the regular hyperbolic 8gon with the sum of angles 2w in Poincare's model $ of the hyperbolic plane. Hint: First do the construction ignoring the radius in $ . Exercise 69 In the complex plane C we define the closed unit disc E, the unit circle £, and the open unit disc $ , by E={zeC\

\z\
£={z£C\

\z\ = l},

$ = {zGC|

|Z|<1}.

Exercises

108

We use $ as the basic set in Poincare's model of the hyperbolic plane. An isometry in Poincare's model $ of the hyperbolic plane is said to be orientation-preserving, if it preserves the sense of direction of an oriented circle in $ . Otherwise it is said to be orientation-reversing. We intend to establish a description of the orientation-preserving isometries (the motions) in Poincare's model <J> of the hyperbolic plane in terms of the linear fractional transformations in C, the so-called Mobius transformations. We shall need the Mobius transformations of the form az + b w= - — — , bz + a where a and b are complex numbers so that aa — bb > 0. 1) Show that every Mobius transformation as above defines a bijective mapping of E onto itself, which maps £ onto itself and <J> onto itself. 2) Show that every Mobius transformation as above maps hyperbolic lines in <J> into hyperbolic lines. Hint: One can use the rewriting (b ^ 0) az + b a bb — aa r = r + 771 rr> bz + a b b(bz + a) to prove that a Mobius transformation preserves circles and angles. 3) Show that every Mobius transformation as above defines an isometry in Poincare's model <J> of the hyperbolic plane. 4) Show that every Mobius transformation as above has two fixed points (which may coincide) in C Show that either both of these fixed points lie on S, or, that exactly one of them lies in $ . Extract from the above the following result. w=

Theorem. A Mobius transformation as above defines: (i) a parallel translation in $, if the fixed points are different and on £; (ii) a limit rotation in <£, if the fixed points coincide on £; (iii) a rotation in $, if it has a fixed point in $ . 5) Establish a similar description of the orientation-reversing isometries (the reversions) in Poincare's model $ of the hyperbolic plane.

Index Algebraic curve of degree two, Analytic geometry, 2, 72 Apollonius, 1, 72 Archimedes, 46 Bernoulli, Jakob, 67 Bernoulli, Johann, 67 Bolyai, J., 74 Brahe, T., 43 Cantor, G., 51 Centre ellipse, 101 hyperbola, 102 Chord, 4, 98 Compact point set, 53 Cone, 3, 23 nappes, 23 generators, 24 angular aperture, 24 Confocal family of ellipses, 98 Conic section, 25 degenerate, 41 directrix, 26 eccentricity, 26 focal point (focus), 26 Coxeter, H.S.M., 75 Dandelin, G.P., 12 m e t h o d of, 12, 27 Decca Navigator System, 35

Deccometer, 35 Dedekind, R., 51 Descartes, R., 2 Dido, 45 legend of, 45 problem of, 65 Elleipsis, 27 Ellipse, 4 as a locus, 8, 10 conjugate diameters, 99 directrix, 10 eccentricity, 4 equation, 6 focal point (focus), 7 focal radii, 7 major axis, 4 minor axis, 4 parameter, 102 parametrization, 7 tangents, 13 Equidistant curves, 75, 106 Escher, M.C., 75 Euclid, 1, 72 Elements, 1, 72 postulates, 73 Euclidean geometry, 72 Eudoxus, 72 Euler, L., 67 Fermat, P. de, 2 Fermat's principle, 68, 105 109

index

110 Gauss, K.F., 74 Gear wheel movements, 17 Geodesic curve, 106 Geometria, 71 Herodotus, 71 Hilbert, D., 73 Hyperbola, 25 as a locus, 31, 34 asymptotes, 33 branches, 25 conjugate semiaxis, 32 directrix, 31 eccentricity, 26 equation, 33 focal point (focus), 31 focal radii, 28 parameter, 100 parametrization, 101 tangents, 34 transverse semiaxis, 31 vertices, 33 Hyperbole, 27 Hyperbolic plane, 75 isometries of, 89 Hyperbolic area, 107 circle, 91 distance, 86 functions, 100 line, 75 navigation, 37 navigational systems, 35 n-gon, 92 parallel translation, 91 reflection, 89 rotation, 90 tilings, 93 triangle, 91 Instantaneous centre of rotation, 17

Inversion in a circle, 77 as a mapping, 79 Involutes of a circle, 20 Isometry, 89 in hyperbolic plane, 89 Isoperimetric problem, 46, 68 closed curves, 66 closed polygons, 60 isosceles triangles, 47 quadrilaterals, 59 triangles, 57 Jordan, C , 63 Jordan curve, 63 rectifiable, 63 Jordan curve theorem, 63 Kant, I., 75 Kepler, J., 2, 43 Kidney stone crusher, 17 Klein, F., 74 Leibniz, G.W., 67 Lindemann, F., 46 Lobachevsky, N.I., 74 Maxima and minima of functions, 51 Meyer, C M . , 51 Morse, M., 67 Mobius transformations, 108 Newton, I., 43, 67 Non-Euclidean geometry, 74 elliptic, 74, 95 hyperbolic, 74 parabolic, 74 parallel axiom, 74 triangle, sum of angles, 74, 91, 95

index Parabola, 25, 29 as a locus, 29 directrix, 26, 29 eccentricity, 26 equation, 30 focal point (focus), 26, 29 focal radius, 30 parameter, 29 tangents, 30 vertex, 101 Parabole, 27 Parallel axiom, 74 Pasch, M., 73 Peano, G., 73 Perron's paradox, 48 Playfair, 74 Poincare, H., 75 disc, 75 half-plane, 93 Pole curve, 17 fixed, 17 rolling, 17 Principle of nested intervals, 51 Projective plane, 95 Pythagoras, 72 Regular n-gon, 59 area, 62 radius circumscribed circle, 62 Riemannian metric, 106 Saccheri, G., 74 Schwarz, H.A., 68 Spherical triangle, 95 Synthetic geometry, 72 Shockwave lithotripsy, 17 Steiner, J., 63 Thales, 72

Veblen, O., 63 Weierstrass, K.

Zenodor, 46