Differential geometry on discrete surfaces

Point clouds and meshes are ubiquitous in computational geometry and its applications. These subsets of Euclidean space represent in general smooth objects with or without singularities. It is then natural to study their geometry by mimicking the differential geometry techniques adapted for smooth surfaces. The aim of the following pages is to list some geometric quantities (length, area, curvatures) classically defined on smooth curves or surfaces, and to define their analog for discrete objects, justifying our definition by a continuity property: if a sequence of discrete objects tends (in a certain sense) to a smooth object, do the corresponding geometric quantities tend to the ones of the smooth object?