An algebraic framework for computing the topology of offsets to rational curves

摘要

A new algebraic framework is introduced for computing the topology of the offset C_ δ at distance δ to a rational plane curve C defined by a parameterization (x(t), y(t)). The focus is on computing the topology of C_δ by analyzing the image of the parameterization of C_δ which involves square roots. This framework is mainly intended to deal with curves that bring initially complicated singularities or with curves such that the offset to compute introduces such singularities making approximation techniques difficult to apply in these cases. In this framework the topology of C_δ is determined by computing, among other notable points, its singular, discontinuity and self-intersection points together with analyzing the ordering of these points, according to the values of the parameter t, obtaining in this way the final branching producing the searched topology for C_δ. The computation of the singular and discontinuity points requires determining the real roots of two univariate polynomials. Self-intersection points are characterized as the intersection of two auxiliary algebraic curves and require to compute only one sequence of subresultants. This approach requires only the manipulation of x(t) and y(t) without computing and dealing with the implicit equation of C_δ (known to be typically a huge polynomial difficult to deal with).