We may invert a plane algebraic curve given by a single polynomial equation f(x, y) = 0 by setting
Clearing denominators, we have the polynomial equations , 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 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
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
Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.
Geometric inversion of two-dimensional stokes flows--application to the flow between parallel planes.(Report)
Oct 01, 2010; 1. Introduction Geometric inversion is a type of transformation of the Euclidean plane. This transformation preserves angles and...