## Inversion and n-circular curves

Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles or lines. Inversion is defined with respect to a center O and a radius ρ (in French, the inversion is said to have pôle O and puissance (power) ρ²).

If M’ denotes the image of M, we have

$\overrightarrow{OM'} = \rho^2 \frac{\overrightarrow{OM}}{OM^2}$

which can be expressed in complex coordinates as
$z' = \omega + \frac{\rho^2}{\bar{z} - \bar\omega}$ and in affine coordinates (centered at O) as

$(x,y) \mapsto \big( \frac{\rho^2 x}{x^2+y^2}, \frac{\rho^2 y}{x^2+y^2} \big).$

The fact that inversions map circles to circles is immediate once you notice that if $O$ is a point of power $\rho^2$ with respect to a circle $\mathcal C$, the inversion of center O and radius ρ exchanges the intersection points of lines through O with $\mathcal C$, thus leaves $\mathcal C$ globally invariant.

Since inversion formulae can be written in affine coordinates using rational functions, it can also be considered as a rational transformation of the projective plane ${[x:y:z] \to [xz:yz:x^2+y^2]}$ for an appropriate choice of coordinates. This transformation is birational, since iterating twice gives ${[xz(x^2+y^2) : yz(x^2+y^2) : z^2(x^2+y^2)]}$ which is the formula for the identity map.

However, it is only defined out of exceptional points defined by xz = yz = x²+y² = 0, which are $O$, $I$, $J$ (the center and the cyclic points). It is easier to describe how to resolve indeterminacies by writing inversion as ${[u:v:z] \mapsto [uz:vz:uv]}$ where $u = x+iy$ and $v = x-iy$. Then $I$ and $J$ are defined by $u=z=0$ and $v=z=0$.

One can recognize the standard quadratic transformation of the projective plane except for the exchange of the first two coordinates : thus the total image of $O$ is the line at infinity, the image of $I$ is the line $OI$ and $J$ is mapped to $OJ$. Inversion is best defined on the blowup of the plane at O, I and J.

Inversion is a rational transformation of degree two, thus the proper inverse transform of a line is a conic, and a curve of degree d should map to a curve of degree 2d.
In particular, a circle C goes through $I$ and $J$, thus maps to a degree 4 curve containing the lines $OI$ and $OJ$. Its proper transform $\iota(C)$ is thus a conic, and since C intersects $OI$ (resp. $OJ$) at a second point, $\iota(C)$ should go through $I$ (resp. $J$), thus is a circle. If C contains O, its image has an additional irreducible component, and its proper transform is a line.

A general curve should intersect $OI$ and $OJ$ in d points, thus should have $I$ and $J$ as points of multiplicity d (these curves are called d-circular). A great amount of information on circular cubics and bicircular quartics can be found in Salmon’s Treatise on the higher Plane Curves. Find it on Europeana or Gallica (Bibliothèque Nationale de France).