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 OP•PQ=a² as Q runs over the original curve C.

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

u = frac{x}{x^2+y^2}, v=frac{y}{x^2+y^2}.

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 $x=u/(u^2+v^2), y=v/(u^2+v^2)$ 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[x,y]=frac{a^2x}{x^2 + y^2}

Y[x,y]=frac{a^2y}{x^2 + y^2}

Examples

Applying the above transformation to the lemniscate

(x^2 + y^2)^2 = a^2 (x^2 - y^2),

gives us

a^2(u^2-v^2) = 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

(u^2+v^2)^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.

References

J. W. Stubbs (1843). "On the application of a new Method to the Geometry of Curves and Curve Surfaces". Philosophical Magazine Series 3 23 338-347.
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications.