Let C be a circle, with center O and radius r. If P is a point different from O, then we define the inverse of P in O to be the point P' which is on the ray from O passing through P, such that |OP'| = r2/|OP|. Inversive geometry is just geometry that concentrates on inversions and the properties of geometrical figures that are invariant under inversions.
Some properties of inverse points:
It is convenient to imagine a "point at infinity" which is the inverse of the center of C.
The inverse of any line, curve, or other set of points is the set of all the inverses of those points. For example:
Because straight lines and circles are mixed up with each other by inversion, when we do inversive geometry we may tend to think of lines as just a special kind of circle. They are the circles through that point at infinity. I shall use the term inversive segment to denote either a straight line segment or an arc of a circle.
There is a useful trick by which we can use inversions to prove theorems in ordinary geometry. Given a figure about which we want to prove something, we can sometimes use an inversion (or maybe several inversions) to transform it into a simpler or more special figure. Then the thing we are trying to prove may become easier in this special case. Here is an example, which is useful for studying kaleidoscopes:
Let ABC be a triangle composed of inversive segments, such that its interior angles add up to less than π. Then there is a triangle H such that each of the sides of ABC, if extended, meets H at right angles; ABC is interior to H.
We begin the proof by reducing the general statement to the case in which AB and AC are straight line segments, if they are not so already. This reduction is performed by an inversion. If either AB or AC is a circular arc, then AB and AC can be extended to meet at a point D.
We invert ABC in any circle centered at D. to get a triangle A'B'C' of inversive segments, where A'B' and A'C' are straight lines. Now the inversive segment B'C' must be a circular arc, and in fact an arc of a circle G', to which A' is exterior; this follows from the fact that the sum of the internal angles is less than π.
Let the tangent lines drawn from A' to that circle G' meet it at E' and F'. Let H' be the circle with center at A', passing through E' and F'. Let H be the inverse of H' in the circle centered at D. Then H is orthogonal to the sides of ABC. Why? Because inversion preserves angles, in particular it preserves the relation of orthogonality; and H' is orthogonal to the sides of A'B'C'. Indeed, H' is orthogonal to A'B' and A'C' because these segments are on lines through the center of H'. Also, H'. is orthogonal to A'E' and A'F' at E' and F'; but these lines are tangent to G' at those points, so G' is orthogonal to H'. Finally, it is easy to see that A'B'C' lies inside H', so ABC lies inside H. This completes the proof.
So, let us suppose that we have constructed a triangle out of inversive segments, and we want to make a kaleidoscope from it. The theorem we have just (informally) proved allows us to put a circle H around the triangle, which is orthogonal to the kaleidoscope mirrors. Therefore H is left invariant by those reflections. This implies that our original triangle will be reflected by each mirror into another triangle within H. In fact, every sequence of these reflections will move the triangle into another one inside H. I think of H as the horizon for this kaleidoscope. Every point inside the horizon is covered by at least one such triangle.
Given a fixed triangle H, I will define an orthogonal segment to be an inversive segment that, suitably extended, meets H at right angles. Any combination of reflections in orthogonal segments leaves H invariant, and is a transformation of the interior of H that preserves angles. The reflections in two diameters of H combine to produce a rotation. The product of the reflections in any two intersecting orthogonal segments is an inversive rotation. Other combinations of reflections are analogous to translations of the plane. The applet on this page can be used to see how these reflections and combinations thereof move things around inside H.
The applet can be used in two modes. In the first, you can set up an arbitrary triangle bounded by orthogonal segments and see how it is transformed by series of reflections in its sides. In the second mode, the applet draws a target-like combination of lines and circles, and you can see how this is affected by reflections, inversive rotations, and general translations. This popup menu controls the mode:
There is one book that I know of about inversive geometry. Some other books that touch on the subject include those of Schwerdtfeger and Pedoe.
The disc shown in this applet, and its "reflections," "rotations," and "translations," are part of a model of hyperbolic geometry, which is a kind of non-Euclidean geometry. My favorite book on this subject is that of Coxeter. Bonola is an older but durable reference, containing classic source material by Bolyai and Lobachevskii. Somerville is another classic, recently republished. Some modern surveys are Brannan, Esplen, and Gray; Greenberg; and Ryan. Hyperbolic geometry, in particular, is the subject of Anderson.
Anderson, James W.: Hyperbolic Geometry, New York, Springer (Springer Undergraduate Mathematics Series), 2005.
Bonola, Robert: Non-Euclidean Geometry. Translated by H. S. Carslaw and George Bruce Halsted. New York, Dover Publications, 1955.
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy: Geometry, Cambridge, Cambridge University Press, 1999.
Coxeter, H. S. M.: Non-Euclidean Geometry, sixth edition, Washington, D. C., Mathematical Association of America, 1998.
Greenberg, Marvin Jay: Euclidean and Non-Euclidean Geometries: Development and History, third edition, New York, Freeman, 1993.
Morley, Frank Vigor: Inversive Geometry, Boston, Ginn and Company, 1933; reprint New York, Chelsea, 1954.
Pedoe, Dan: Geometry - a Comprehensive Course, New York, Dover Publications, 1988.
Ryan, Patrick: Euclidean and Non-Euclidean Geometry, an Analytical Approach, Cambridge University Press, 1986.
Somerville, Duncan M'Laren Young: The Elements of Non-Euclidean Geometry, New York, Dover Publications, 2005.
Schwerdtfeger, Hans: Geometry of Complex Numbers, Toronto, University of Toronto Press, 1962; reprint New York, Dover Publications, 1980.