Computing the topology of a plane or space hyperelliptic curve

Abstract

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a Maple implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.

On the geometry and the topology of parametric curves

“…The larger the number of sampled parameters is, the more likely it is for the embedding to be isotopic to C. However, we might need a prohibitive large number of points to sample; their number is related to the distance between two branches of the curve. We show that by introducing a few additional points, we can replace the parametric arcs of the embedded graph with straight line segments and count on it being isotopic to C. Following closely Alcázar et al (2020), if X, Y ⊂ R n are one dimensional, then being isotopic implies that one of them can be deformed into the other without removing or introducing self-intersections.…”

“…To overcome this issue we need to segment some edges of G to two or more edges. To find the extra vertices that we need to add to the graph we follow a common approach (Alcázar and Díaz-Toca, 2010;Alcázar et al, 2020;Kahoui, 2008;Daouda et al, 2008 (green) and its orthogonal projection on the xy-plane (blue).…”

“…If the (proper) parametrization of the curve consists of polynomials of degree d and bitsize τ, then PTOPO outputs a graph isotopic (Boissonnat and Teillaud, 2006, p.184), Alcázar et al (2020) to the curve in the embedding space, by performing…”