On the Topology and Visualization of Plane Algebraic Curves

In this paper, we present a symbolic algorithm to compute the topology of a plane curve. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. The novelty of this paper is that we use a technique of interval polynomials to solve the system $\big\{f\alpha,\,y=\frac{\partial f}{\partial y}\alpha,\,y=0\big\}$ and at the same time, get the simple roots of fα, y=0 on the α fiber. It greatly improves the efficiency of the lifting step since we need not compute the simple roots of fα, y=0 any more. After the topology is computed, we use a revised Newton's method to compute the visualization of the plane algebraic curve. We ensure that the meshing is topologically correct. Many nontrivial examples show our implementation works well.