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Symmetric Points


Two points z and z^S in C^* are symmetric with respect to a circle or straight line L if all circles and straight lines passing through z and z^S are orthogonal to L. Möbius transformations preserve symmetry. Let a straight line be given by a point z_0 and a unit vector e^(itheta), then

 z^S=e^(2itheta)z-z_0^_+z_0,

where z^_ is the complex conjugate. Let a circle be given by center z_0 and radius r, then

 z^S=z_0+(r^2)/(z-z_0^_).

See also

Möbius Transformation

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Cite this as:

Weisstein, Eric W. "Symmetric Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricPoints.html

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