  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.
