Important Properties

In the diagram we can see other properties of inversion as well. For example:

• Q is a point on the circle such that OQ is perpendicular to PQ

• P' is the foot of the altitude of the triangle OQP where QP' is perpendicular to OP

• We can now see that triangle OP'Q is similar to triangle OQP

From these properties we are able to deduce the equation which is equivalent to our first equation: .

Notice therefore that a point on the inside of the circle is inverted to a point outside the circle and vice versa, a point on the circle is inverted to itself, and the point at the center of the circle is inverted to the point at infinity and vice versa.

This concept of a point at infinity becomes clear when we observe using the point at the center of the circle as P. OP is therefore equal to zero so it follows that as well. Since , we now have . Solving the equation where k is a constant, it follows that OP' must be infinitely large and P' must be the point at infinity.

The point at infinity can be thought of as very close to points extremely far out in all directions. So because lines go out to infinity in both directions you can think of their end points meeting at the point at infinity. If then this line's end points meet then you can think of it as a circle.

Inversion of Curves

This applet can be seen on its own page at: Inversion of A Shape
This inversion transformation can also be used to invert entire curves like the ones in the main image of this page. This is done by inverting all of the points along the original curve to get the inverted curve. The applet to the left shows how the red circle is inverted about the black circle to get the blue inversion curve. Let's call the center of the black circle 'O' for clarity's sake.

By moving around the red curve in the applet on the left, we notice that inversion of a circle that passes through the O will always be a line that does not pass through O. The inversion of a line not passing through O is always a circle that does pass through O.

Based on the equations and what we observe about the point at the center of the circle being inverted to the point at infinity and vice versa, we can think of a line as a curve where the two ends meet at the point at infinity.

Circles that do not pass through O invert to another circle that does not pass through O. Circles and lines invert to themselves if and only if they are orthogonalCircles are orthogonal to each other if the angles where they intersect measure 90 degrees. to the circle of inversion.

3D Inversion

3D Reciprocation