IGS’23

Abstract

Structure-preserving discretization in the field of geometry is the paradigm of discrete differential geometry. Here, of course, deciding which of the structures are to be taken into account is a nontrivial problem. In some aspects, the discrete theory turns out to be even richer than its smooth counterpart. It focuses on developing constructive methods, which allow one to prove natural existence and uniqueness results, as well as the corresponding convergence statements. The well-established theory of discrete conformal maps and circle patterns has already found numerous applications in geometry processing. We present their generalizations beyond the conformal limit: decorated discrete conformal maps and ring patterns, which share the corresponding existence and uniqueness statements. The theory and construction methods are based on variational principles. We also briefly explain how structure preserving discretizations recently helped to answer the long-standing question whether a surface in three-space is uniquely determined by its metric and curvatures (Bonnet problem).