Computational geometry: an introduction
Computational geometry: an introduction
ACM Transactions on Graphics (TOG)
Generalizing active zones for set-theoretic solid models
The Computer Journal
Efficient intersection tests for objects defined constructively
International Journal of Robotics Research
Relationship between s-bounds and active zones in constructive solid geometry
Theory and practice of geometric modeling
A null-object detection algorithm for constructive solid geometry
Communications of the ACM
Boundary Evaluation Using Inner and Outer Sets: the ISOS Method
IEEE Computer Graphics and Applications
Approximation hierarchies and S-bounds
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
Computing CSG tree boundaries as algebraic expressions
SMA '93 Proceedings on the second ACM symposium on Solid modeling and applications
Orthogonal polyhedra as geometric bounds in constructive solid geometry
SMA '97 Proceedings of the fourth ACM symposium on Solid modeling and applications
Efficient Bounds in Constructive Solid Geometry
IEEE Computer Graphics and Applications
Blister: GPU-based rendering of Boolean combinations of free-form triangulated shapes
ACM SIGGRAPH 2005 Papers
Solid modelling based on sixth order partial differential equations
Computer-Aided Design
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In constructive solid geometry, geometric solids are represented as trees whose leaves are labeled by primitive solids and whose internal nodes are labeled by set-theoretic operations. A bounding function in this context is an upper or lower estimate on the extent of the constituent sets; such bounds are commonly used to speed up algorithms based on such trees. We introduce the class of totally consistent bounding functions, which have the desirable properties of allowing surprisingly good bounds to be built quickly. Both outer and inner bounds can be refined using a set of rewrite rules, for which we give some complexity and convergence results. We have implemented the refinement rules for outer bounds within a solid modeling system, where they have proved especially useful for intersection testing in three and four dimensions. Our implementations have used boxes as bounds, but different classes (shapes) of bounds are also explored. The rewrite rules are also applicable to relatively slow, exact operations, which we explore for their theoretical insight, and to general Boolean algebras. Results concerning the relationship between these bounds and active zones are also noted.