Generalised circle

 

The extended Euclidean plane can be identified with the extended complex plane, so that equations of complex numbers can be used to describe lines, circles and inversions.

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 ${\displaystyle {\mathfrak {G}}^{\text{T}}{\mathfrak {C}}{\bar {\mathfrak {G}}}.}$ 

• Hans Schwerdtfeger, Geometry of Complex Numbers, Courier Dover Publications, 1979
• Michael Henle, "Modern Geometry: Non-Euclidean, Projective, and Discrete", 2nd edition, Prentice Hall, 2001
• David W. Lyons (2021) Möbius Geometry from LibreTexts