Triangulated Polyhedral Surface Boolean Operations
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Solution Overview
Problem
Existing computer-aided geometric design (CAGD) systems face challenges in robustly computing Boolean operations on polyhedral solids due to sensitivity to numerical rounding errors, leading to self-intersections and inconsistencies in boundary representations.
Innovation Solution
A computer-implemented process for computing a resulting closed triangulated polyhedral surface by intersecting triangles, splitting them into polygonal facets, calculating winding numbers, and selecting facets based on computed coefficients to ensure robustness and accuracy, using exact arithmetic and simplicial 2-cycle representations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If floating point representations and epsilon-programming heuristics are used for Boolean operations on polyhedral solids, then computational speed is improved, but numerical rounding errors cause self-intersections and inconsistencies in boundary representations
Solution Approach 1:
The patent replaces the traditional floating-point arithmetic system with a symbolic computation system based on exact predicates. Instead of using numerical approximations (floating point) that are fast but inaccurate, the invention uses symbolic representation of geometric computations that are slower but exact, thereby eliminating rounding errors while maintaining computational feasibility through optimized predicate evaluation
Solution Approach 2:
The patent changes the fundamental parameter representation from floating-point numbers to exact arithmetic representations. By transforming the computational parameters from approximate numerical values to exact symbolic forms, the system achieves robust Boolean operations that are immune to numerical rounding errors, directly resolving the reliability issue while managing computational complexity through efficient predicate design
2Measurement precision
If exact arithmetic is used for computing Boolean operations, then numerical accuracy is improved, but computational complexity and processing time increase
Solution Approach 1:
The patent segments the complex Boolean operation computation into distinct phases: construction phase using exact arithmetic to build a precise boundary representation, and simplification phase to reduce topological complexity. This segmentation allows exact computation to be applied only where necessary for accuracy, while avoiding unnecessary computational overhead in phases where simpler operations suffice
Solution Approach 2:
The patent performs preliminary construction of the boundary representation using exact arithmetic predicates to ensure numerical accuracy from the start. By establishing a topologically correct boundary structure upfront through exact computation, the system avoids the need for repeated corrections and validations that would otherwise be required, thereby reducing overall processing time despite the initial computational cost
Data Source
AI summary
The invention is directed to a computer-implemented process, in a computer-aided geometric design system, for computing a resulting closed triangulated polyhedral surface from a first and a second modeled objects. The first modeled object is modeled by a first closed triangulated polyhedral surface and the second modeled object is modeled by a second closed triangulated polyhedral surface. The process according to the invention comprises:computing intersections between triangles of the first modeled object and triangles of the second modeled object;splitting triangles of the first and second modeled objects into polygonal facets adjacent to said intersections;computing for each triangle and each polygonal facet two couples of winding numbers p and q, wherein the first winding number of each couple of winding numbers is computed from the first closed triangulated polyhedral surface and the second winding number of each couple of winding numbers is computed from the second closed triangulated polyhedral surface;computing a coefficient k for each triangle and each polygonal facet, the coefficient being computed with a function φ having as inputs the two couples of winding numbers p and q of each triangle and each polygonal facet;selecting triangles and polygonal facets according to the computed coefficients;triangulating said selected polygonal facets; andobtaining the resulting closed triangulated polyhedral surface with the selected triangles and the triangulated selected polygonal facets.