Geometry and Topology of Polyhedral Surfaces

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Polyhedral surfaces are produced at an alarming rate ranging from 3-D scanning to medical imaging in the digital world these days. Almost all surfaces appeared in any computer screen are polyhedral surfaces. An urgent task is to produce a coarse categorization and classification of these surfaces. A mile stone result in mathematics dealing with surfaces is the classical uniformization theorem of Poincare-Koebe. It is an extremely powerful tool to classify smooth surfaces in the 3-space up to angle preserving diffeomorphisms (i.e., conformality). However, to algorithmically implement the classical uniformization theorem is very difficult. The PI and his collaborators introduced a discrete counterpart of conformality and proved a discrete version of uniformization theorem for polyhedral surfaces recently. This proposal investigates theoretically whether the discrete conformal geometry converges to the smooth conformal geometry as triangulation meshes become finer and finer. The numerical evidences for the convergence are very strong. The successful resolution of the convergence issue will have impacts on the applications of the discrete uniformization theorem in many fields.The classical uniformization theorem of Poincare and Koebe is the most powerful tool to classify smooth surfaces with Riemannian metrics according to conformal diffeomorphisms. This theorem is one of the pillars of the 20th century mathematics and has a wide range of applications within and outside mathematics. However, it is very difficult to use it algorithmically for categorizing polyhedral surfaces. Luo and his collaborators introduced a notion of discrete conformality for polyhedral surfaces and proved a discrete uniformization theorem recently. Two main features of the discrete conformality are the following. First, the discrete conformality is algorithmic and second, there exists a finite dimensional (convex) variational principle to find the discrete uniformization metric. The goal of the proposal is to investigate several major remaining issues in the discrete conformal geometry of polyhedral surfaces. For instance, there are strong numerical evidences suggesting that the discrete conformality converges to smooth (classical) conformality when the triangulations are suitably chosen. However, a theoretical proof of it is still lacking. This is the main problem to be resolved in the proposal. Luo plans to use the finite dimensional variational principles that he developed in the past 10 years to approach the convergence problem.