Generalised circle

A circle ${\displaystyle \Gamma }$ is the set of points ${\displaystyle z}$ in a plane that lie at radius ${\displaystyle r}$ from a center point ${\displaystyle \gamma .}$

${\displaystyle \Gamma (\gamma ,r)=\{z:{\text{the distance between }}z{\text{ and }}\gamma {\text{ is }}r\}}$

In the complex plane, ${\displaystyle \gamma }$ is a complex number and ${\displaystyle \Gamma }$ is a set of complex numbers. Using the property that a complex number multiplied by its conjugate is the square of its modulus (its Euclidean distance from the origin), an implicit equation for ${\displaystyle \Gamma }$ is:

${\displaystyle {\begin{aligned}r^{2}&=\left|z-\gamma \right|^{2}=(z-\gamma ){\overline {(z-\gamma )}}\\[5mu]0&=z{\bar {z}}-{\bar {\gamma }}z-\gamma {\bar {z}}+\left(\gamma {\bar {\gamma }}-r^{2}\right).\end{aligned}}}$

This is a homogeneous bivariate linear polynomial equation in terms of the complex variable ${\displaystyle z}$ and its conjugate ${\displaystyle {\bar {z}},}$ of the form

${\displaystyle Az{\bar {z}}+Bz+C{\bar {z}}+D=0,}$

where coefficients ${\displaystyle A}$ and ${\displaystyle D}$ are real, and ${\displaystyle B}$ and ${\displaystyle C}$ are complex conjugates.

By dividing by ${\displaystyle A}$ and then reversing the steps above, the radius ${\displaystyle r}$ and center ${\displaystyle \gamma }$ can be recovered from any equation of this form. The equation represents a generalized circle in the plane when ${\displaystyle r}$ is real, which occurs when ${\displaystyle AD<BC}$ so that the squared radius ${\displaystyle r^{2}=(BC-AD)/A^{2}}$ is positive. When ${\displaystyle A}$ is zero, the equation defines a straight line.

That the reciprocal transformation ${\displaystyle z\mapsto w=1/z}$ maps generalized circles to generalized circles is straight-forward to verify:

${\displaystyle {\begin{aligned}0&=Az{\bar {z}}+Bz+C{\bar {z}}+D\\[5mu]&={\frac {A}{w{\bar {w}}}}+{\frac {B}{w}}+{\frac {C}{\bar {w}}}+D\\[5mu]&=A+B{\bar {w}}+Cw+Dw{\bar {w}}\\[5mu]&=D{\bar {w}}w+Cw+B{\bar {w}}+A.\end{aligned}}}$

Lines through the origin (${\displaystyle A=D=0}$) map to lines through the origin; lines not through the origin (${\displaystyle A=0,D\neq 0}$) map to circles through the origin; circles through the origin (${\displaystyle A\neq 0,D=0}$) map to lines not through the origin; and circles not through the origin (${\displaystyle A\neq 0,D\neq 0}$) map to circles not through the origin.

The defining equation of a generalized circle

${\displaystyle 0=Az{\bar {z}}+Bz+C{\bar {z}}+D}$

can be written as a matrix equation

${\displaystyle 0={\begin{pmatrix}z&1\end{pmatrix}}{\begin{pmatrix}A&B\\C&D\end{pmatrix}}{\begin{pmatrix}{\bar {z}}\\1\end{pmatrix}}.}$

Symbolically,

${\displaystyle 0=\mathbf {z} ^{\text{T}}{\mathfrak {C}}\,{\bar {\mathbf {z} }},}$

with coefficients placed into an invertible hermitian matrix ${\displaystyle {\mathfrak {C}}={\mathfrak {C}}^{\dagger }}$ representing the circle, and ${\displaystyle \mathbf {z} ={\begin{pmatrix}z&1\end{pmatrix}}^{\text{T}}}$ a vector representing an extended complex number.

Two such matrices specify the same generalized circle if and only if one is a scalar multiple of the other.

To transform the generalized circle represented by ${\displaystyle {\mathfrak {C}}}$ by the Möbius transformation ${\displaystyle {\mathfrak {H}},}$ apply the inverse of the Möbius transformation ${\displaystyle {\mathfrak {G}}={\mathfrak {H}}^{-1}}$ to the vector ${\displaystyle \mathbf {z} }$ in the implicit equation,

${\displaystyle {\begin{aligned}0&=\left({\mathfrak {G}}\mathbf {z} \right)^{\text{T}}{\mathfrak {C}}\,{\overline {({\mathfrak {G}}\mathbf {z} )}}\\[5mu]&=\mathbf {z} ^{\text{T}}\left({\mathfrak {G}}^{\text{T}}{\mathfrak {C}}{\bar {\mathfrak {G}}}\right){\bar {\mathbf {z} }},\end{aligned}}}$

so the new circle can be represented by the matrix ${\displaystyle {\mathfrak {G}}^{\text{T}}{\mathfrak {C}}{\bar {\mathfrak {G}}}.}$