Improperly parametrized rational curves
Computer Aided Geometric Design
Improved test for closed loops in surface intersections
Computer-Aided Design
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
Implicit Curves and Surfaces in CAGD
IEEE Computer Graphics and Applications - Special issue on computer-aided geometric design
Solving polynomial systems for curve, surface and solid modeling
SMA '93 Proceedings on the second ACM symposium on Solid modeling and applications
Algorithms for intersecting parametric and algebraic curves I: simple intersections
ACM Transactions on Graphics (TOG)
Algorithms for intersecting parametric and algebraic curves II: multiple intersections
Graphical Models and Image Processing
A new approach to the surface intersection problem
Computer Aided Geometric Design
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Solving Geometric Constraints By Homotopy
IEEE Transactions on Visualization and Computer Graphics
Implicit and parametric curves and surfaces for computer aided geometric design
Implicit and parametric curves and surfaces for computer aided geometric design
Computer Aided Geometric Design
Computer Aided Geometric Design
HOM4PS-2.0para: Parallelization of HOM4PS-2.0 for solving polynomial systems
Parallel Computing
Comparison of three curve intersection algorithms
Computer-Aided Design
A new class of algorithms for the processing of parametric curves
Computer-Aided 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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Intersection problems are fundamental in computational geometry, geometric modeling and design and manufacturing applications, and can be reduced to solving polynomial systems. This paper introduces two homotopy methods, i.e. polyhedral homotopy method and linear homotopy method, to compute the intersections of two plane rational parametric curves. Extensive numerical examples show that computing curve intersection by homotopy methods has better accuracy, efficiency and robustness than by the Ehrlich-Aberth iteration method. Finally, some other applications of homotopy methods are also presented.