A new method to compute the singularities of offsets to rational plane curves

Abstract

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

The square-freeness of the offset equation to a rational planar curve, computed via resultants

“…Finally, we recall the next lemma, which is proven in Appendix I of [4]; although in [4] one works with properly parametrized curves, one can check that the proof of the lemma does not depend on the properness of the curve, and therefore it is also valid in the case of non-proper curves.…”

“…Roughly speaking, the implicit equation of O d (C) is found by eliminating the variable t in the system formed by (6) and (7). However, in order to avoid as many extraneous components as possible, we first divideP (x, y, t), Q(x, y, t) by their contents with respect to t. 4 The polynomials obtained 4 Let f (x1, . .…”

Reduced difference polynomials and self-intersection computations

“…The following example shows that the algorithm can also accommodate self-intersections of higher order. It is well known that an "ordinary" m-fold point (i.e., a point that the curve traverses m times, with distinct tangents) is equivalent to 1 2 m(m−1) double points. In the presence of an m-fold point, the algorithm will identify This curve has a triple point at the origin, corresponding to the three distinct parameter values t = 1 4 , 1 2 , 3 4 ( Figure 8).…”

“…When p(t) is specified in Bernstein form on t ∈ [ 0, 1 ], the Bernstein coefficients of the tensor-product form of f (u, v) on (u, v) ∈ [ 0, 1 ] × [ 0, 1 ] can be easily obtained from those of p(t), and can be specialized to any rectangular subdomain [ u 1 , u 2 ] × [ v 1 , v 2 ] by standard subdivision algorithms. These ideas also generalize to a product p 1 (t) p 2 (t) of polynomials, in which case the reduced difference polynomial is defined as f (u, v) = (p 1…”

“…Section 2 describes the construction of the bivariate difference polynomial associated with a given univariate polynomial. Section 3 then shows how its Bernstein coefficients on any given subdomain [ 1 ] can be obtained by matrix multiplications. These results are used in Section 4 to generate a quadtree decomposition of (u, v) ∈ [ 0, 1 ] × [ 0, 1 ] to a prescribed resolution, governed by reduced difference polynomials f (u, v) = 0 and g(u, v) = 0 that characterize the self-intersections of planar polynomial curves.…”