Discrete differential geometry

Summary

Discrete differential geometry builds discrete counterparts of notions of differential geometry, such as curvature, geodesics and conformal maps. It works with polyhedral surfaces, meshes and other finite structures. A good discrete notion keeps the key theorems of the smooth theory exactly, not only in the limit. The angle defect at a vertex of a polyhedron plays the role of Gaussian curvature, and circle packings serve as discrete conformal maps. The field supplies the mathematics behind computer graphics, geometry processing and architectural design.