Is a linear conformal mapping same as a similarity transformation?

Question

1. 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.

2. Is it correct that a conformal mapping may not be an affine nor projective transformation, because it may not be linear?

Thanks and regards!

Answer 1

1. 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.

2. Yes, consider inversion with respect to a fixed circle.

( http://en.wikipedia.org/wiki/Inversive_geometry )