Definitions

# Inverse curve

In geometry, the inverse curve of a given curve C with respect to a fixed circle with center O and radius a is the locus of points P for which OPQ are collinear and OPPQ=a2 as Q runs over the original curve C.

We may invert a plane algebraic curve given by a single polynomial equation f(xy) = 0 by setting

$u = frac\left\{x\right\}\left\{x^2+y^2\right\}, v=frac\left\{y\right\}\left\{x^2+y^2\right\}.$

Clearing denominators, we have the polynomial equations $ux^2+uy^2-x = 0, vx^2+vy^2-y=0$, and eliminating x and y from the system of three equations in four unknowns consisting of these two equations and f (for instance, by using resultants) we can readily find the equation of the curve inverted in the unit circle. Now $x=u/\left(u^2+v^2\right), y=v/\left(u^2+v^2\right)$ and applying the transformation again leads back to the original curve.

In polar coordinates centered at O, the centre of the circle of inversion, ,the curve C with equation f(r,θ) = 0 has inverse curve with equation f(a2/r,θ)=0.

For a parametrically defined curve, its inverse curve with respect to a circle with center in (0;0) and radius a is defined as

$X\left[x,y\right]=frac\left\{a^2x\right\}\left\{x^2 + y^2\right\}$

$Y\left[x,y\right]=frac\left\{a^2y\right\}\left\{x^2 + y^2\right\}$

## Examples

Applying the above transformation to the lemniscate

$\left(x^2 + y^2\right)^2 = a^2 \left(x^2 - y^2\right),$

gives us

$a^2\left(u^2-v^2\right) = 1,,$

the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply it to the Fermat curve xn + yn = 1, where n is odd, we obtain

$\left(u^2+v^2\right)^n = u^n+v^n.,$

Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.

## Anallagmatic curves

An anallagmatic curve is one which inverts into itself. Examples include the circle, cardioid, oval of Cassini, strophoid, and trisectrix of Maclaurin.