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Circle in the Gaussian Plane

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The equation

$\displaystyle \vert z-a\vert = s \vert z-b\vert,\quad s\ne 1\, , $

describes a circle in the Gaussian plane. Its center is

$\displaystyle w=\frac{1}{1-s^2}a-\frac{s^2}{1-s^2}b $

and the radius given by

$\displaystyle r=\frac{s}{\vert 1-s^2\vert}\vert b-a\vert \,. $

If $ s< 1$, $ a$ is located in the interior of the circle, while $ b$ is lies on its outside, vice versa for $ s > 1$.

\includegraphics[width=10cm]{kreis_komplexe_ebene}

The circle's parametric representation is given by

$\displaystyle w + r e^{\mathrm{i}t},\quad t\in[0,2\pi) \,. $

(Authors: Höllig/Kopf/Abele)

Example:

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  automatically generated 10/ 8/2007