Quasiconformal Dilatation of Projective Transformations and Discrete Conformal Maps

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circumcircle preserving piecewise projective interpolation between discretely conformally equivalent triangulations. We show that another interpolation scheme, angle bisector preserving piecewise projective interpolation, is in a sense optimal with respect to dilatation. These two interpolation schemes belong to a one-parameter family.