Direct methods for sparse matrices27100
Direct methods for sparse matrices27100
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Solving geometric constraints by homotopy
SMA '95 Proceedings of the third ACM symposium on Solid modeling 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 CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
Geometric Constraint Solving and Applications
Geometric Constraint Solving and Applications
Symbolic and numerical techniques for constraint solving
Symbolic and numerical techniques for constraint solving
Topology preserving surface extraction using adaptive subdivision
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Using Cayley-Menger determinants for geometric constraint solving
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Solving nonlinear polynomial systems in the barycentric Bernstein basis
The Visual Computer: International Journal of Computer Graphics
Interrogating witnesses for geometric constraint solving
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Polytope-based computation of polynomial ranges
Proceedings of the 2010 ACM Symposium on Applied Computing
Using the witness method to detect rigid subsystems of geometric constraints in CAD
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Linear programming for Bernstein based solvers
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Polytope-based computation of polynomial ranges
Computer Aided Geometric Design
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This paper presents a new solver for systems of nonlinear equations. Such systems occur in Geometric Constraint Solving, e.g., when dimensioning parts in CAD-CAM, or when computing the topology of sets defined by nonlinear inequalities. The paper does not consider the problem of decomposing the system and assembling solutions of subsystems. It focuses on the numerical resolution of well-constrained systems. Instead of computing an exponential number of coefficients in the tensorial Bernstein basis, we resort to linear programming for computing range bounds of system equations or domain reductions of system variables. Linear programming is performed on a so called Bernstein polytope: though, it has an exponential number of vertices (each vertex corresponds to a Bernstein polynomial in the tensorial Bernstein basis), its number of hyperplanes is polynomial: O(n2) for a system in n unknowns and equations, and total degree at most two. An advantage of our solver is that it can be extended to non-algebraic equations. In this paper, we present the Bernstein and LP polytope construction, and how to cope with floating point inaccuracy so that a standard LP code can be used. The solver has been implemented with a primal-dual simplex LP code, and some implementation variants have been analyzed. Furthermore, we show geometric-constraint-solving applications, as well as numerical intersection and distance computation examples.