Smooth bijective maps between arbitrary planar polygons

Mapping from one planar polygonal domain to another is a fundamental problem in computer graphics and geometric modelling. Exploiting the properties of harmonic maps, we define smooth and bijective maps with prescribed behaviour along the domain boundary. These maps can be approximated in different ways, and we discuss the respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples of applications in image warping and surface cross-parameterization. Moreover, we briefly discuss the extension of our construction to 3D and its application to volumetric shape deformation.