Relationship Between S-bounds and Active Zones in Constructive Solid Geometry

Abstract

Active zones were introduced by Rossignac and Voelcker to speed certain geometric computations over Constructive Solid Geometry (CSG) representations. The active zone associated with a CSG node is, essentially, the region in which the shape of the set associated with the node is important. An active zone is defined algebraically as the intersection of certain nodes of the whole CSG tree and thus its CSG expression is always readily available for each node. Alternate representations of active zone (such as boundaries) need never be precomputed, and each calculation on active zones is carried out using the exact sets. On the other hand, the performance of most application algorithms (whether they are based on active zones or not) can be considerably improved by precomputing enclosing boxes around the contribution of each primitive. Cameron studied the generation of such optimal boxes, which he calls S-bounds. His approach assigns regions of space to nodes, but is couched in terms of an inductive refinement process rather than algebra. Active zones and S-bounds have been used to improve the efficiency of various geometric algorithms, such as boundary evaluation, shading, and null object detection. By establishing the relation between these two schemes, we justify the use of S-bounds as approximations for various computations on active zones, and provide a simpler characterisation for S-bounds.