A bisection method for systems of nonlinear equations
ACM Transactions on Mathematical Software (TOMS)
Loop detection in surface patch intersections
Computer Aided Geometric Design
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Matrix computations (3rd ed.)
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code
Real-Time Rendering
Introduction to Algorithms
Optimal bounding cones of vectors in three dimensions
Information Processing Letters
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Surface self-intersection computation via algebraic decomposition
Computer-Aided Design
Hausdorff and minimal distances between parametric freeforms in R2and R3
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
GPU-based parallel solver via the Kantorovich theorem for the nonlinear Bernstein polynomial systems
Computers & Mathematics with Applications
A hybrid parallel solver for systems of multivariate polynomials using CPUs and GPUs
Computer-Aided Design
Solving polynomial systems using no-root elimination blending schemes
Computer-Aided Design
Solving polynomial systems using no-root elimination blending schemes
Computer-Aided Design
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
Computer Aided Geometric Design
Computer Aided Geometric Design
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The need for robust solutions for sets of nonlinear multivariate constraints or equations needs no motivation. Subdivision-based multivariate constraint solvers typically employ the convex hull and subdivision/domain clipping properties of the Bezier/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 subdomains 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. Yet, a computationally tractable algorithm to examine this criterion has never been proposed. In this paper, we present a dual representation of the normal cones as parallel hyperplanes over the unit hypersphere, which enables us to construct an algorithm for identifying subdomains with at most a single solution. Further, we also offer a second termination criterion, based on the representation of bounding parallel hyperplane pairs, to identify and reject subdomains that contain no solution. We implemented both algorithms in the multivariate solver of the IRIT solid modelling system and present examples using our implementation.