Inversion is the geometrical process by which points P are transformed to their corresponding inverses P'. Inversion is performed around a circle of inversion, in which all points outside the circle go inside, and all inside go outside. Points on the circle stay put.

To invert a point P outside the circle centered at O with radius r, (you might want to get out a piece of paper for this) draw the line segment OP. Now draw a tangent line to the circle that includes P (call the point it touches the circle T). Finally, draw the line segment OT. The foot of the altitude of OTP from T is the inverted point P' (That is, where the perpendicular to TP intersects OP).

Similarly, to invert a point P inside the circle, first draw the line containing OP. Draw a perpendicular line to OP at P. The point at which this intersects the circle is T. Finally, draw a tangent from the circle at T, and the point where T intersects the line containing OP is P', the inverted point.

Practice it a few times, just to get a feel for it. I'd include a diagram if I could.

Inversion does a lot of cool stuff. For instance, inverting an (infinitely long) line outside the circle of inversion turns it into a circle inside the inversion circle. Also, when inverting anything, intersecting points still intersect. For instance, inverting two intersecting lines gives two tangential circles.

Try inverting a square.

Analytically, inversion about the origin with radius r can be described with the vector equation:

x' = r2x / |x|2    (Eric Weisstein's World of Mathematics,

Naturally, this can be extended to any circle of inversion through simple translation.

If lines are treated as circles of infinite radius, all circles invert to circles.

Many geometrical proofs can be done by inverting the figure in question about a suitable circle of inversion. It can turn massively complex concepts into relatively simple ones.

Inverting a parabola about its focus gives a cardioid. Inverting a logarithmic spiral about its center gives yet another logarithmic spiral.

What happens when you invert a hyperbola?