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 {f(alpha, y) = partial derivative f/partial derivative y (alpha, y) = 0} and at the same time, get the simple roots of f(alpha, y) = 0 on the a fiber. It greatly improves the efficiency of the lifting step since we need not compute the simple roots of f(alpha, 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.