Inverse Curve - Equations

Equations

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 the inverse of an algebraic curve of degree n is also algebraic of degree at most 2n.

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 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

The inverse of the curve defined parametrically by

with respect to the same circle is given parametrically as

In polar coordinates, the equations are simpler when the circle of inversion is the unit circle. The inverse of the point (r, θ) with respect to the unit circle is (R, Θ) where

or equivalently

So the inverse of the curve f(r, θ) = 0 is determined by f(1/R, Θ) = 0 and the inverse of the curve r = g(θ) is r = 1/g(θ).