Algorithm for algebraic curve intersection
Computer-Aided Design
Recursive subdivision and iteration in intersections and related problems
Mathematical methods in computer aided geometric design
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
Exploiting topological and geometric properties for selective subdivision
SCG '85 Proceedings of the first annual symposium on Computational geometry
Solving nonlinear polynomial systems in the barycentric Bernstein basis
The Visual Computer: International Journal of Computer Graphics
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
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
GPU-based parallel solver via the Kantorovich theorem for the nonlinear Bernstein polynomial systems
Computers & Mathematics with Applications
Computer Aided Geometric Design
A New Approach for Solving Nonlinear Equations Systems
IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans
Finding intersections of B-spline represented geometries using recursive subdivision techniques
Computer Aided Geometric Design
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
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Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its Bernstein-Bezier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surface-surface-surface intersection (SSSI) and surface-curve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks.