CSG-BRep duality and compression

Abstract

Solid Modeling technology has been traditionally divided into two camps: CSG and BRep. Constructive Solid Geometry (CSG) represents a shape as a Boolean combination of half-spaces. A Boundary Representations (BRep) specifies the location of the vertices their connectivity, and a description of how they should be interpolated or approximated by a piecewise simple surface (such as a polyhedon, a subdivision surface, a Bspline, or a trimmed implicit or parametric patch). We will investigate the equivalence between CSG and BRep (using a simple duality) and will show that for a large class of polyhedral models, both can be encoded using (3k+4)N bits, where N represents the number of primitives in a CSG model or equivalently the number of vertices in the dual BRep,nd where k represents the number of bits used to represent a quantization of each coordinate of vectors that define each either a vertex of the BRep or a plane of the CSG primitive. We will review recent advances in lossless and lossy compression and in selective and progressive transmission over error-prone connections. In particular, we will describe in detail the Corner Table, a simple and compact data structure for processing triangle meshes, and the Edgebreaker 3D connectivity compression algorithm, whose simplicity (2 pages of code) and effectiveness (between 1 and 1.8 bits per triangle) surpasses other compression techniques and standards. Details and source code may be found at http://www.gvu.gatech.edu/~jarek/edgebreaker/eb/.