Is a linear conformal mapping same as a similarity transformation?
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For a mapping between two Euclidean spaces, is it a linear conformal mapping if and only if it is a similarity transformation?
My answer is yes, because the Jacobian matrix of a conformal transformation is everywhere a scalar times a rotation matrix.
Note that both allow reflection, i.e. change of orientation.
- Is it correct that a conformal mapping may not be an affine nor projective transformation, because it may not be linear?
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asked 2012-02-17
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Yes, for elementary reasons. Let $f$ be a linear conformal map and apply this to any triangle $ABC$. Then $f(AB),f(BC),f(CA)$ will be lines by linearity, and by conformality $f(ABC)$ will have the angles of $ABC$ so they will be similar therefore $f$ is a similarity mapping.
Yes, consider inversion with respect to a fixed circle.
( http://en.wikipedia.org/wiki/Inversive_geometry )
asked 2012-02-19
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