Quasiconformal distortion of projective transformations and discrete conformal maps

TL;DR

This paper studies the quasiconformal distortion of projective transformations in the real projective plane, analyzing dilatation behavior and comparing interpolation schemes for discretely conformal triangulations.

Contribution

It introduces a detailed analysis of dilatation contours for non-affine projective transformations and compares two interpolation schemes, identifying an optimal one in terms of dilatation.

Findings

Contour lines of dilatation form a hyperbolic pencil of circles.

Angle bisector preserving interpolation is optimal regarding dilatation.

The two interpolation schemes form a one-parameter family.

Abstract

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.