Isotopic ε-Approximation of Algebraic Curves

Abstract

AbstractIn this paper, we will introduce our implementation for isotopic approximation of plane and space algebraic curves. The important basic algorithm used in our implementation is real solving of zero-dimensional polynomial systems, especially for bivariate polynomial systems. For the topology computation of plane curves, compared to other symbolic methods, the novelty of our method is that we can get many simple roots on the fibers when computing the critical points of the plane curve, which greatly improves the lifting step. After the topology is computed, we also give a certified approximation for the plane curve, which is a basic operation for approximating a space curve and further for an algebraic surface. As to space curve case, the topology and approximation are recovered from that of their projection plane curves. We implemented our algorithms in Maple 15. The benchmarks show the high efficiency of the implementation.KeywordsBivariate polynomial systemreal roots isolationplane (space) algebraic curvestopologyisotopic approximation