Computing curve intersection by homotopy methods

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.