Constructive non-regularized geometry

Abstract

Solid modelling is concerned with the construction and manipulation of unambiguous computer representations of solid objects. These representations permit us to distinguish the interior, the boundary and the complement of a solid. They are conveniently specified in Constructive Solid Geometry (CSG) by a construction tree that has solid primitives as leaves and rigid body motions or regularized Boolean operations as internal nodes. Algoriths for classifying sets with respect to CSG trees and for evaluating the boundaries of the corresponding solids are known, at least for simple geometric domains. Emerging CAD applications require that we extend the domain of solid modellers to support more general and more structured geometric objects. The focus is on dimensionally non-homogeneous, non-closed pointsets with internal structures. These entities are well suited for dealing with mixed-dimensional (‘non-manifold’) objects in Rn that have dangling or missing boundary elements, and that may be composed of several regions. A boundary representation for such objects has been described elsewhere. We propose to specify and represent inhomogeneous objects in terms of Constructive Non-Regularized Geometry (CNRG) trees that extend the domain of CSG by supporting non-regularized primitive shapes as leaves, and by providing more general set-theoretic and topological operators at interior nodes. Filtering operations are also provided that construct CNRG objects from aggregates of selected regions of other CNRG objects. A syntax and semantics of the operators in CNRG are presented, and some basic algorithms for classifying pointsets with respect to the regions of objects represented by CNRG trees are outlined.