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G.: Subdivision termination criteria in subdivision multivariate solvers using dual hyperplanes representations (2007)
| Content Provider | CiteSeerX |
|---|---|
| Author | Hanniel, Iddo Elber, Gershon |
| Abstract | Abstract. The need for robust solutions for sets of non-linear multivariate constraints or equations needs no motivation. Subdivision-based multivariate constraint solvers [1–3] typically employ the convex hull and subdivision/domain clipping properties of the Bézier/B-spline representation to detect all regions that may contain a feasible solution. Once such a region has been identified, a numerical improvement method is usually applied, which quickly converges to the root. Termination criteria for this subdivision/domain clipping approach are necessary so that, for example, no two roots reside in the same sub-domain (root isolation). This work presents two such termination criteria. The first theoretical criterion identifies sub-domains with at most a single solution. This criterion is based on the analysis of the normal cones of the multiviarates and has been known for some time [1]. Yet, a computationally tractable algorithm to examine this criterion has never been proposed. In this paper, we present such an algorithm for identifying sub-domains with at most a single solution that is based on a dual representation of the normal cones as parallel hyper-planes over the unit hyper-sphere. Further, we also offer a second termination criterion, based on the representation of bounding parallel hyper-plane pairs, to identify and reject sub-domains that contain no solution. We implemented both algorithms in the multivariate solver of the IRIT [4] solid modeling system and present examples using our implementation. 1 |
| File Format | |
| Volume Number | 39 |
| Journal | Comput. Aided Des |
| Language | English |
| Publisher Date | 2007-01-01 |
| Access Restriction | Open |
| Subject Keyword | Subdivision Termination Criterion Subdivision Multivariate Solver Dual Hyperplanes Representation Termination Criterion Subdivision Domain Normal Cone Single Solution Feasible Solution Parallel Hyper-plane Pair Multivariate Solver Parallel Hyper-planes Present Example Solid Modeling System Dual Representation Tractable Algorithm Convex Hull Non-linear Multivariate Constraint First Theoretical Criterion Identifies Sub-domains Subdivision-based Multivariate Constraint Solver Numerical Improvement Method Robust Solution Zier B-spline Representation Second Termination Criterion Root Isolation |
| Content Type | Text |
| Resource Type | Article |
