Seamless Parametrization with Cone and Partial Loop Control

We present a method for constructing seamless parametrization for surfaces of any genus, which can handle any feasible cone configuration with any type of cones. The mapping is guaranteed to be locally injective, which is due to careful construction of a simple domain boundary polygon. The polygon’s complexity depends on the cones in the field, and it is independent of mesh geometry. The result is a small polygon that can be optimized prior to the interior mapping, which contributes to the robustness of the pipeline. For a surface of genus > 0, non-contractible loops play an important role, and their holonomies significantly affect mapping quality. We enable holonomy prescription, where local injectivity is guaranteed. Our prescription, however, is limited and cannot handle all feasible holonomies due to monotonicity constraints that keep our polygon simple. Yet, this work is an important step towards fully solving the holonomy prescription problem.