Real‐Time Bijective Planar Maps

We describe a very simple algorithm to bijectively map between two simply‐connected bounded planar domains given a set of n point correspondences constraining the homeomorphic boundary mapping. The bijective map Φ is the composition of two harmonic maps h 1 and h 2 , which we call a double harmonic map. To compute Φ, we first map both domains to the unit disk using two conformal maps f 1 and f 2 and then determine a harmonic map g between the two disks that matches the boundary constraints. Composing f 1 with g gives h 1 , and h 2 is the inverse of f 2 . For the special case of n = 3 , g is a Möbius transformation and Φ reduces to the classical conformal map. Our method allows a graceful tradeoff between conformality and harmonicity, and is efficient enough to be used in an interactive shape deformation system. All the maps involved are defined exclusively through their boundary maps and extended to piecewise linear maps with harmonic coordinates after triangulating the domains.