What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

PART II. VARIOUS METHODS FOR PERFORMING CONSTRUCTIONS

§6. MORE ABOUT INVERSION AND ITS APPLICATIONS

1. Invariance of Angles. Families of Circles

Although inversion in a circle greatly changes the appearance of geometrical figures, it is a remarkable fact that the new figures continue to possess many of the properties of the old. These are the properties which are unchanged, “invariant,” under the transformation. As we already know, inversion transforms circles and straight lines into circles and straight lines. We now add another important property: The angle between two lines or curves is invariant under inversion. By this we mean that any two intersecting curves are transformed by an inversion into two other curves which still intersect at the same angle. By the angle between two curves we mean, of course, the angle between their tangents.

The proof may be understood from Figure 60, which illustrates the special case of a curve C intersecting a straight line OL at a point P. The inverse C′ of C meets OL in the inverse point P′, which, since OL

Fig. 60. Invariance of angles under inversion.

is its own inverse, lies on OL. We shall show that the angle x₀ between OL and the tangent to C at P is equal in magnitude to the corresponding angle y₀. To do this we choose a point A on the curve C near P, and draw the secant AP. The inverse of A is a point A′ which, being on both the lineOA and the curve C′, must be at their intersection. We draw the secant A′P′. By the definition of inversion,

r² = OP·OP′ = OA·OA′,

or

i.e. the triangles OAP and OA′P′ are similar. Hence angle x is equal to angle OA′P′, which we call y. Our final step consists in letting the point A move along C and approach the point P. This causes the secant line AP to revolve into the position of the tangent line to C at P, while the angle xtends to x₀. At the same time A′ will approach P′, and A′P′ will revolve into the tangent at P′. The angle y approaches y₀. Since x is equal to y at every position of A, we must have in the limit, x₀ = y₀.

Our proof is only partially completed, however, since we have considered only the case of a curve intersecting a line through O. The general case of two curves C, C* forming an angle z at P is now easily disposed of. For it is evident that the line OPP′ divides z into two angles, each of which we know to be preserved by the inversion.

It should be noted that although inversion preserves the magnitude of angles, it reverses their sense; i.e. if a ray through P sweeps out the angle x₀ in a counterclockwise direction, its image will sweep out angle y₀ in a clockwise direction.

A particular consequence of the invariance of angle under inversion is that two circles or lines that are orthogonal, i.e. that intersect at right angles, remain orthogonal after an inversion, while two circles which are tangent, i.e. intersect at the angle zero, remain tangent.

Let us consider the family of all circles that pass through the center of inversion O and through another fixed point A of the plane. From §4, Article 2, we know that this family of circles is transformed into a family of straight lines that radiate from A′, the image of A. The family of circles orthogonal to the original family goes over into circles orthogonal to the lines through A′, as shown in Figure 61. (The orthogonal circles are shown by broken lines.) The simple picture of the radiating straight lines appears to be quite different from that of the circles, yet we see that they are closely related – indeed from the standpoint of the theory of inversion they are entirely equivalent.

Fig. 61. Two systems of orthogonal circled related by inversion.

Another example of the effect of inversion is given by a family of circles tangent to each other at the center of inversion. After the transformation they become a system of parallel lines. For the images of the circles are straight lines, and no two of these lines intersect, since the original circles meet only at O.

Fig. 62. Tangent circles transformed into parallel lines.

2. Application to the Problem of Apollonius

A good illustration of the usefulness of the theory of inversion is the following simple geometrical solution of the problem of Apollonius. By inversion with respect to any center, the Apollonius problem for three given circles can be transformed into the corresponding problem for three other circles (why is this?). Hence, if we can solve the problem for any one triple of circles, then it is solved for any other triple of circles obtained from the first by inversion. We shall exploit this fact by selecting among all these equivalent triples of circles one for which the problem is almost trivially simple.

We start with three circles having centers A, B, C, and we shall suppose the required circle U with center O and radius ρ to be externally tangent to the three given circles. If we increase the radii of the three given circles by the same quantity d, then the circle with the same center O and the radius ρ − d will obviously solve the new problem. By way of preparation we make use of this fact in order to replace the three given circles by three others such that two of them are tangent to each other at a point K (Fig. 63). Next we invert the whole figure in some circle with center K. The circles around B and C become parallel lines b and c, while the third circle becomes another circle a (Fig. 64). We know that a, b, c can all be constructed by ruler and compass. The unknown circle is transformed into a circle u which touches a, b, c. Its radius r is evidently half the distance between b and c. Its center O′ is one of the two intersections of the line midway between b and c with the circle about A′ (the center of a) having the radius r + s (s being the radius of a). Finally, by constructing the circle inverse to u we find the center of the desired Apollonius circleU. (Its center, O, will be the inverse in the circle of inversion of the point inverse to K in u.)

Fig. 63. Preliminary to Apollonius’ construction.

Fig. 64. Solution of Apollonius’ problem.

*3. Repeated Reflections

Everyone is familiar with the strange reflection phenomena that occur when more than one mirror is used. If the four walls of a rectangular room were covered with ideal non-absorbing mirrors, a lighted point would have infinitely many images, one corresponding to each congruent room obtained by reflection (Fig. 65). A less regular constellation of mirrors, e.g. three mirrors, gives a much more complicated series of images. The resulting configuration can be described easily only when the reflected triangles form a non-overlapping covering of the plane. This occurs only for the case of the rectangular isosceles triangle, the equilateral triangle, and the rectangular half of the latter; see Figure 66.

Fig. 65. Repeated reflection in rectangular walls.

Fig. 66. Regular constellations of triangular mirrors.

The situation becomes much more interesting if we consider repeated inversion in a pair of circles. Standing between two concentric circular mirrors one would see an infinite number of other circles concentric with them. One sequence of these circles tends to infinity, while the other concentrates around the center. The case of two external circles is a little more complicated. Here the circles and their images reflect successively into one another, growing smaller with each reflection, until they narrow down to two points, one in each circle. (These points have the property of being mutually inverse with respect to both circles.) The situation is shown in Figure 67. The use of three circles leads to the beautiful pattern shown in Figure 68.

Fig. 67. Repeated reflection in systems of two circles.

Fig. 68. Reflection in a system of three circles.