Extended Octtrees, between CSG Trees and Boundary Representations

Abstract

Besides the most widely used models in the Geometric Modeling Systems, Constructive Solid Geometry (CSG) and Boundary Representations (BR), Octtrees have appeared as an alternative representation scheme which is particularly well suited for the solid boolean operation algorithms. Extended Octtrees, which incorporates three additional node types containing part of the surface of the object, are much more compact and allow the exact representation of plane faced objects, while supporting also low complexity algorithms for boolean operations. Due to the limitations of the two main models, CSG and Boundary Representation , a number of Hybrid Systems have appeared, which support both schemes and perform every operation in the most suitable model. However, algorithms for the boundary evaluation of CSG trees are complex, and at the moment little is known on algorithms for the inverse conversion, from BR to CSG. In the present paper, the use of the Extended Octtree model as an intermediate tool in the conversions between. CSG trees and BR is studied. In the case of CSG trees build from primitives with plane faces, an algorithm for the conversion from the CSG model to the Extended Octtree representation is presented. Its complexity is linear with respect to the numbe of nodes in the Octtree. The use of this algorithm in model-to-model conversions is discussed, together with the BR to Extended Octtree and Extended Octtree to BR conversion algorithms, that present also linear complexity with respect to the total number of nodes.