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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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 + 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

'


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( 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 (.

,_

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;)
„)

An=2n--hn2 "

2ir _

L 2-7T

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

K — = — 2n 47T tan(• 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*\'

80

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

81

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

82

The Poincare disc model of non-Euclidean

geometry

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[

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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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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.

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(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 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

«*