Efficiently approximating the minimum-volume bounding box of a point set in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Parallel Robots (Solid Mechanics and Its Applications)
Parallel Robots (Solid Mechanics and Its Applications)
Introduction to Interval Analysis
Introduction to Interval Analysis
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees
IEEE Transactions on Visualization and Computer Graphics
Nonlinear systems solver in floating-point arithmetic using LP reduction
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Computation of singularities and intersections of offsets of planar curves
Computer Aided Geometric Design
Interval Analysis for Certified Numerical Solution of Problems in Robotics
International Journal of Applied Mathematics and Computer Science - Verified Methods: Applications in Medicine and Engineering
Subdivision termination criteria in subdivision multivariate solvers
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
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Solving polynomial systems of equations is an important problem in many fields such as computer-aided design, manufacturing and robotics. In recent years, subdivision-based solvers, which typically make use of the properties of the Bezier/B-spline representation, have proven successful in solving such systems of polynomial constraints. A major drawback in using subdivision solvers is their lack of scalability. When the given constraint is represented as a tensor product of its variables, it grows exponentially in size as a function of the number of variables. In this paper, we present a new method for solving systems of polynomial constraints, which scales nicely for systems with a large number of variables and relatively low degree. Such systems appear in many application domains. The method is based on the concept of bounding hyperplane arithmetic, which can be viewed as a generalization of interval arithmetic. We construct bounding hyperplanes, which are then passed to a linear programming solver in order to reduce the root domain. We have implemented our method and present experimental results. The method is compared to previous methods and its advantages are discussed.