Generalised circle

From Wikipedia, the free encyclopedia

A circle Γ is the set of points p that lie at radius r from a center point γ.

$\Gamma = \{ p : \mathrm {\ the\ distance\ between\ } p \mathrm {\ and\ } \gamma \mathrm {\ is\ } r \}$

Using the complex plane, we can treat γ as a complex number and circle Γ as a set of complex numbers.

Using the property that a complex number multiplied by its conjugate gives us the square of the modulus of the number, and that this is the same as its euclidean distance from the origin, we can we can express Γ like so:

$r^2 = {\left | p-\gamma \right |} ^2$

$r^2 = (p-\gamma)\overline{(p-\gamma)}$

$r^2 = p \bar p - p \bar \gamma - \bar p \gamma + \gamma \bar \gamma$

$p \bar p - p \bar \gamma - \bar p \gamma + (\gamma \bar \gamma - r^2) = 0$

We can multiply this by a real constant A to get an equation in the form

$A p \bar p + B p + C \bar p + D = 0$

where A and D are real, and B and C are complex conjugate. Note that when A is zero, this equation defines a straight line - a circle with a radius of infinity.

This can be usefully put into a hermitian matrix

$\mathfrak C = \begin{pmatrix}A & B \\ C & D \end{pmatrix} = \mathfrak C ^{\dagger} .$