Inverse curve

In inversive geometry, an inverse curve of a given curve $C$ is the result of applying an inverse operation to $C$ . Specifically, with respect to a fixed circle with center $O$ and radius $k$ the inverse of a point $Q$ is the point $P$ for which $P$ lies on the ray $OQ$ and $OP \cdot OQ = k 2$ . The inverse of the curve $C$ is then the locus of $P$ as $Q$ runs over $C$ . The point $O$ in this construction is called the center of inversion, the circle the circle of inversion, and $k$ the radius of inversion.

An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself.

The inverse of the point $(x, y)$ with respect to the unit circle is $(X, Y)$ where

or equivalently

So the inverse of the curve determined by $f (x, y) = 0$ with respect to the unit circle is

It is clear from this that inverting an algebraic curve of degree $n$ with respect to a circle produces an algebraic curve of degree at most $2 n$ .

Similarly, the inverse of the curve defined parametrically by the equations

with respect to the unit circle is given parametrically as

This implies that the circular inverse of a rational curve is also rational.

More generally, the inverse of the curve determined by $f (x, y) = 0$ with respect to the circle with center $(a, b)$ and radius $k$ is

...
Wikipedia