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·OQ = k². 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.

Equations

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

${\displaystyle X={\frac {x}{x^{2}+y^{2}}},\qquad Y={\frac {y}{x^{2}+y^{2}}},}$

or equivalently

${\displaystyle x={\frac {X}{X^{2}+Y^{2}}},\qquad y={\frac {Y}{X^{2}+Y^{2}}}.}$

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

${\displaystyle f\left({\frac {X}{X^{2}+Y^{2}}},{\frac {Y}{X^{2}+Y^{2}}}\right)=0.}$

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 2n.

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

${\displaystyle x=x(t),\qquad y=y(t)}$

with respect to the unit circle is given parametrically as

${\displaystyle {\begin{aligned}X=X(t)&={\frac {x(t)}{x(t)^{2}+y(t)^{2}}},\\Y=Y(t)&={\frac {y(t)}{x(t)^{2}+y(t)^{2}}}.\end{aligned}}}$

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

${\displaystyle f\left(a+{\frac {k^{2}(X-a)}{(X-a)^{2}+(Y-b)^{2}}},b+{\frac {k^{2}(Y-b)}{(X-a)^{2}+(Y-b)^{2}}}\right)=0.}$

The inverse of the curve defined parametrically by

${\displaystyle x=x(t),\qquad y=y(t)}$

with respect to the same circle is given parametrically as

${\displaystyle {\begin{aligned}X=X(t)&=a+{\frac {k^{2}{\bigl (}x(t)-a{\bigr )}}{{\bigl (}x(t)-a{\bigr )}^{2}+{\bigl (}y(t)-b{\bigr )}^{2}}},\\Y=Y(t)&=b+{\frac {k^{2}{\bigl (}y(t)-b{\bigr )}}{{\bigl (}x(t)-a{\bigr )}^{2}+{\bigl (}y(t)-b{\bigr )}^{2}}}.\end{aligned}}}$

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

${\displaystyle R={\frac {1}{r}},\qquad \Theta =\theta .}$

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).

Degrees

As noted above, the inverse with respect to a circle of a curve of degree n has degree at most 2n. The degree is exactly 2n unless the original curve passes through the point of inversion or it is circular, meaning that it contains the circular points, (1, ±i, 0), when considered as a curve in the complex projective plane. In general, inversion with respect to an arbitrary curve may produce an algebraic curve with proportionally larger degree.

Specifically, if C is p-circular of degree n, and if the center of inversion is a singularity of order q on C, then the inverse curve will be an (n − p − q)-circular curve of degree 2n − 2p − q and the center of inversion is a singularity of order n − 2p on the inverse curve. Here q = 0 if the curve does not contain the center of inversion and q = 1 if the center of inversion is a nonsingular point on it; similarly the circular points, (1, ±i, 0), are singularities of order p on C. The value k can be eliminated from these relations to show that the set of p-circular curves of degree p + k, where p may vary but k is a fixed positive integer, is invariant under inversion.

Examples

Applying the above transformation to the lemniscate of Bernoulli

${\displaystyle \left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right)}$

gives us

${\displaystyle {}^{}}$Zdroj:https://en.wikipedia.org?pojem=Inverse_curve
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