  We propose a novel Moebius interpolator that takes as an input a discrete map
between the vertices of two planar triangle meshes, and outputs a smooth map on
the input domain. The output map interpolates the discrete map, is continuous
between triangles, and has low quasi-conformal distortion when the input map is
discrete conformal. Our map leads to considerably smoother texture transfer
compared to the alternatives, even on very coarse triangulations. Furthermore,
our approach has a closed-form expression, is local, applicable to any discrete
map, and leads to smooth results even for extreme deformations. Finally, by
working with local intrinsic coordinates, our approach is easily generalizable
to discrete maps between a surface triangle mesh and a planar mesh, i.e., a
planar parameterization. We compare our method with existing approaches, and
demonstrate better texture transfer results, and lower quasi-conformal errors.
