Ray-reps for Free-form Modeling: Line Membership Classi cation on CSRs

Low degree (typically 2, 3) tetrahedral implicit algebraic patches are gaining in popularity for modeling boundaries of free-form (sculptured) solids. A Constructive Shell Representation (CSR) is a union of truncated tetrahedra, called \trunctets", that form a thick shell containing the boundary of such a free-form solid. One bounding face of each trunctet is an algebraic patch, and the other faces are planes belonging to the associated tetrahedron. Recently, ray representations (ray-reps) have been shown to be extremely versatile for solid modeling, particularly when computed on the RayCasting Engine (RCE) { a highly parallel custom-VLSI computer for CSG processing. A ray-rep is the set of \in" segments resulting from the classiication of a grid of parallel rays against a solid. This paper merges the CSR and ray-rep streams of research and shows how CSRs can be processed directly to produce ray-reps. The algorithm developed here exploits the hybrid Brep/CSG character of CSRs, and uses a CSG-like divide and conquer approach to infer patch intercepts, which are then interpreted in a Brep fashion. Subtleties related to processing special points (singular, interpolating, planar, and interior) are handled in a neighborhood-free fashion. Consequently, this algorithm can be mapped onto the RCE with minor modiications. This provides an RCE-based access to all ray-rep applications for free-form solids by direct CSR processing, thus obviating expensive CSG conversions. Applications include rendering, particle transport, sweeping, Minkowski operations and NC machining.