Polygon properties calculated from the vertex neighborhoods

摘要

Calculating properties of polyhedra given only the set of the locations and neighborhoods of the vertices is easy. Possible properties include volume, surface area, and point containment testing. No global topological information at all is explicitly needed (although the complete global topology could be recovered). The neighborhood of the vertex means the directions of the edges and faces on it but not their extents. These vertex-based formulae are dual to the usual formulae that use the faces. They have been implemented and the stability against inconsistent data tested. Alternative data structures and formulae for polyhedron calculation are important since special cases are a function partly of the data structure, and because different methods have different numerical accuracy and error detection properties.