r-Adaptive Parameterization of Surfaces

Surface mesh parameterization is a fundamental tool in computer graphics, required for applications like e.g. texture mapping, remeshing and morphing. Linear parameterization methods are of particular interest. Their simplicity, efficiency and robustness, enables the processing of detailed, large meshes. In practice, however, the current linear schemes are limited to producing (quasi-)conformal parameterizations and hence may suffer from considerable distortion e.g. in length and area. We present a novel approach to effectively reduce parametric distortion. Our algorithm is simple, efficient and robust, as it requires only the solution of a sparse linear system. We smoothly adapt an existing quasi-conformal parameterization with respect to different flow quantities such as areas, angles or edge lengths. Our adaptive strategy offers a flexible means for controlling distortion based on error equidistribution. We show how this method can be extended from the planar setting to spherical parameterizations, a problem which recently attracted a lot of interest.