This series of articles showcases a variety of applications around the theme of inversion, which constitutes a strategic way of manipulating configurations of circles and lines in the plane. This article includes the Riemann sphere, rings of four tangent circles, and inverting the Sierpinski sieve.

Introduction

The story of inversion, propelled by the study of conic sections and stereographic projection, is quite complex [1]. Although Vietá (1540–1603) already spoke of mutually inverted points, the proper development started with a plethora of notable geometers including L’Huilier (1750–1840), Dandelin (1794–1847), Quetelet (1796–1874), Steiner (1796–1863), Magnus (1790–1861), Plücker (1801–1868), Bellavitis (1803–1880), and Simpson (1849–1924). Steiner, in a manuscript published in 1913, is considered to have been the first to formulate inversion as a method to systematically simplify the study of complex geometric figures where circles play a prominent role. This work culminated in 1855 with the studies of Möbius (1790–1868) (hence the choice of the letter for the inversive circle in this article) [2]. Peaucellier (1832–1913) also applied inversion to his famous linkage [3], and Lord Kelvin (1824–1907) applied inversion to elasticity.

Let denote either the line or the segment from to , depending on the context. The length of the segment is denoted by . The circle with center and radius is denoted by . Denote by the inverse of in the current circle of inversion , where is either a symbolic point or often ; in Manipulate results, is often shown as a red dashed circle.

The following functions encapsulate the basic properties of inversive geometry developed in the first part of this series [4].

The function squareDistance computes the square of the Euclidean distance between two given points. (It is more convenient to use the following definition than the built-in Mathematica function SquaredEuclideanDistance.)

The function collinearQ tests whether three given points are collinear. When exactly two of them are equal, it gives True, and when all three are equal, it gives False, because there is no unique line through them.

The function computes the circle passing through the points , , and . If the points are collinear, it gives the line through them; if all three points are the same, it returns an error message, as there is no meaningful definition of inversion in a circle of zero radius.

The function computes the inverse of in the circle . The object can be a point (including the special point that inverts to the center of ), a circle, or a line (specified by two points).

Here is an example.

Since a circle can be inverted into a line, define a generalized circle to be either an ordinary circle or a line, as in the first article in this series. A consequence of theorem 2 in the next section is that the set of generalized circles is closed under inversion in a circle. The following Manipulate shows that as , the circle tends to the line , and the inverse of the circle tends to its reflection in the line , the circle .

Therefore, it makes sense to define inversion in a line to be reflection.

Here is an example.

The function squareDistanceToLine computes the square of the distance of a given point to a given line.

The function exCircles computes the four circles tangent to the lines forming the sides of a triangle having given vertices.

The function inCircle computes the incircle of a triangle having given vertices as the excircle of smallest radius.

The function redPoint is used to mark a given point in red.

The function draws an arc of radius r and center b from the line ab to the line bc.

The function intersections computes the possible points of intersection of a given line and circle. (There may be zero, one, or two points.)

The following Manipulate shows several of the functions introduced so far.

First Properties

The concept of orthogonality between circles plays an important role in inversive geometry. Two intersecting circles are orthogonal if they have perpendicular tangents at either point of intersection. It is rather amusing to consider orthogonality of circles without mentioning any notion of perpendicularity [3, 4], as is done in the following definition.

Definition

Two intersecting circles and are orthogonal if one of them (say ) belongs to a set of three mutually tangent circles that touch each other at three distinct points of .

The following Manipulate shows the central circle is orthogonal to the black one by showing the other two circles (blue). Some values of the positions of the black circle or the radii of the blue circles make it impossible to close the three circles, so adjustments to close them are done on the fly.

Theorems 1–11 review the basic properties of inversion and introduce some of its remarkable properties. Consider all inversions to be with respect to the circle .

Theorem 1

Two pairs of inverse points are either collinear with or concyclic on a circle orthogonal to .

The following Manipulate illustrates this property and shows that triangles and are similar. The gray circle is orthogonal to .

The following interesting variation of theorem 1 was presented at the Swiss Mathematical Contest
in 1999 [5, 6]:

Two circles intersect in points and . An arbitrary point of the first circle, which is not or , is joined to and , and the straight lines and intersect the second circle again in the points and . Prove that the tangent to the first circle at is parallel to the straight line .

Theorem 2

The inverse of a circle not passing through is also a circle not passing through . (In this case, the images of concyclic points are concyclic.) The inverse of a circle passing through is a straight line not passing through . (In this case, the images of concyclic points are collinear.)

For instance, invert the circle in .

The set of circles and lines, called generalized circles, is therefore closed under inversion.

Theorem 3

A straight line passing through is self-inverse. The inverse of a straight line not through is a circle through .

Theorem 4

Any circle through a pair of inverse points is orthogonal to the circle of inversion. Conversely, any circle cutting the circle of inversion orthogonally and passing through a point also passes through the inverse [7].

Theorem 5

Theorem 6

(The effect of inversion on length.) If inversion sends point to and point to , then

Proof

By the definition of inversion, . Then . Since , the result follows. □

Theorem 7

(The effect of inversion on the size of circles.) If the circle inverts into a circle of radius , then

where , the distance between the centers of and .

To verify this result, notice that in the following expression (the condition implies does not pass through and thus indeed inverts into a circle).

Theorem 8

Two circles of radii and touching externally at the point invert into two parallel lines separated by the distance

.

Subtracting the first points of the following lines obtained by inversion gives the result.

Theorem 9

If the circle of radius inverts into a circle of radius , then

where is the length of the tangent from to [8].

The following Manipulate shows the terms mentioned in theorem 9.

Proof

Let and be the ends of the diameter of shown in the figure above. The point is outside the segment , as otherwise would not be defined. Then

and the result follows. □

For instance, consider inverting in .

Then , which we verify as follows.

In the case of theorem 7, this is the corresponding comparison.

In order to verify this result, assume that and , with and (otherwise would not be a circle).

From this theorem, it follows that the product of the lengths of the tangents from to and
must be .

Theorem 10

(Ptolemy) Let , , , and be arbitrary points. Then , with equality if and only if the points are on a generalized circle.

Proof

Invert , , and in the circle . Then and (by theorem 6). Substituting this and similar results for , , , and in the inequality gives

which reduces to ; this always applies in the triangle . Equality holds if and only if the points , , and are collinear, that is, if and only if points , , , and are concyclic or collinear. □

Ptolemy (circa 127 AD) compiled much of what today is the pseudoscience of astrology. His Earth-centered universe held sway for 1500 years. In the words of Carl Sagan, in the first episode of his TV series Cosmos, “… showing that intellectual brilliance is no guarantee against being dead wrong.” You can find fascinating applications of Ptolemy’s theorem in [9, 10].

Theorem 11

(A fixed-point theorem.) Suppose two nonintersecting circles have centers and . Any circle that cuts those two circles orthogonally passes through the same two points on the line [10].

This result is useful when inverting to concentric circles elsewhere. In order to verify it, assume without loss of generality that the two circles are and , with , and that the orthogonal circle is . Then the following quantity is independent of , and it is equal to the position of one of the two fixed points mentioned.

The Riemann Sphere and Inversion

The following provides a framework with which to interpret inversion. Consider a sphere of unit radius centered at the origin. Draw a line from the north pole to a point on the sphere. The point at which intersects the – plane defines a one-to-one correspondence between points on the sphere and points in the plane; is called stereographic projection. The image of the south pole is the origin. The image of the north pole is not defined, but is introduced as a new point to serve as the image of ; this makes the mapping continuous and one-to-one. In the context of stereographic projection, the sphere is referred to as the Riemann sphere after the prominent mathematician Bernhard Riemann (1826–1866), who studied under Steiner and earned his PhD degree under Gauss [11, 12].

Similar triangles give

By using these formulas, it is possible to prove that the stereographic projection of a circle on is a generalized circle in the plane. (Circles through go to lines and other circles go to circles.) So inversion in the unit circle in the plane induces a map from circles to circles on .

This defines stereographic projection.

An inversive pair of points maps to points reflected in the – plane.

Moreover, as the following Manipulate shows, an inversive pair of generalized circles maps to circles on that are reflections across the – plane. You can vary the radius and center of the control circle in the – plane. The circle inverts to in the unit circle, which is the equator of . You can show the stereographic cone joining to through its stereographic projection and the stereographic cone joining to through its stereographic projection , which is the reflection of in the – plane [13].

Amusing applications related to stereographic projection are found in [14, 15], and interesting Demonstrations in [16–18].

A Ring of Four Tangent Circles

Consider a ring of four cyclically externally tangent circles as in the following Manipulate. The four points of tangency are concyclic (pink circle), even when some of the circles are internally tangent. You can drag the four points around this circle to alter the shape of the arrangement. To see many other similar patterns, see [19]. In contrast to the case of three tangent circles, the circle through the points of tangency is not necessarily orthogonal to the other four circles.

The function perturb slightly varies three points that are coincident or collinear.

Let us apply inversion to deduce the concyclic property. Invert in a circle centered at one of the tangent points, for instance, the red dashed circle in the following Manipulate. That inversion transforms the circles centered at and into two parallel lines and two circles in between these lines. These four inversions are sequentially tangent in three points. The problem then reduces to show that the three points of tangency lie on the same line (shown in blue).

Let us take a closer look at the arrangement formed by the inversion of the chain of circles, starting from a different point of view. The following Manipulate shows an arrangement of two parallel lines, one slanted line, and two circles with the tangencies indicated. The two orange disks indicate adjustable tangency points and of the circles and the parallel lines. It is easy to show that the slanted line always passes through the tangency point of the circles regardless of the positions of and and the value of one of the radii, say, of the lower circle. In fact, the sum of the radii remains constant for fixed positions of and . For some values of the radii there exists a third circle also tangent to the parallel lines and the circles. Selecting the appropriate option in the Manipulate, the radii are adjusted according to the position of the slanted line for this third (blue) circle to exist. When the slanted line is vertical and the circles are congruent, three of those third circles exist.

The next Manipulate inverts the previous arrangement. It shows that the tangent points of the set of (blue) pairwise tangent circles are concyclic. It also shows that there exist up to three circles tangent to the circles of , and sets of up to four circles orthogonal to the circles of . You can vary the center of the inversive circle. Similar examples can be found at [20, 21].

The quadrilateral joining the centers of the circles forming the ring has an inscribed circle. This is easily seen using Pitot’s theorem (1695–1771): a convex quadrilateral with consecutive side lengths , , , is tangential, that is, it has an inscribed circle, if and only if . Oddly enough, in the case of the quadrilateral, its incircle does not necessarily coincide with the circle passing through the four points of tangency, as the following result shows.

Inverting the Sierpinski Sieve

The Sierpinski sieve (or Sierpinski gasket, or Sierpinski triangle) [5, 6], named after the prolific Polish mathematician Waclaw Sierpinski (1882–1969), is a self-similar subdivision of a triangle [22, 23]. The function Sierpinski constructs the iteration corresponding to the recursive definition of this subdivision.

Inverting the vertices of the triangles from the first four iterations in the unit circle gives rise to stunning patterns.

The following Manipulate lets you vary the radius and center of the inverting circle, the number of iterations, and the type of transformation applied to the vertices forming the triangles forming the Sierpinski sieve.

Acknowledgments

I would like to thank the anonymous referee whose thoughtful and detailed comments on this article greatly improved its presentation, and also Queretaro’s Institute of Technology, which provided essential support for the completion of this second part.

References

About the Author

Jaime Rangel-Mondragón received M.Sc. and Ph.D. degrees in Applied Mathematics and Computation from the School of Mathematics and Computer Science at the University College of North Wales in Bangor, UK. He was a visiting scholar at Wolfram Research, Inc. and held positions in the Faculty of Informatics at UCNW, the Center of Literary and Linguistic Studies at the College of Mexico, the Department of Electrical Engineering at the Center of Research and Advanced Studies, the Center of Computational Engineering (of which he was director) at the Monterrey Institute of Technology, the Department of Mechatronics at the Queretaro Institute of Technology and the Autonomous University of Queretaro in Mexico, where he was a member of the Department of Informatics and in charge of the Academic Body of Algorithms, Computation, and Networks. His research included combinatorics, the theory of computing, computational geometry, and recreational mathematics. Jaime Rangel-Mondragón died in 2015.