Isotopic approximation of implicit curves and surfaces

Abstract

Abstract

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…There are also approaches [24] that achieve locally quadratic convergence towards the simple roots of polynomial systems and there very efficient in practice. Another interesting direction of subdivision algorithms, of more geometric nature, concerns the approximation of algebraic varieties [30,25,5,29,37,21], and the computation of the approximate Voronoi diagrams [39]. There are also quite important applications of these algorithms to the problem of robot motion planning [33].…”

“…The PV Algorithm [25] Let f ∈ R[x, y] or R[x, y, z] be a square-free polynomial such that its real zero set V R (f ) is smooth and bounded. The PV algorithm recursively subdivides an initial bounding square or cube I for the variety with a quad-tree or oct-tree data structure until at least one of the following two tests holds on each subregion J.…”

“…We consider the following problem: Plantinga and Vegter [25] presented a subdivisionbased algorithm for correctly approximating curves and surfaces, see Figure 1. We call this the PV algorithm.…”

“…The approximation has the correct topology in the sense that there is an ambient isotopy between the approximation and the zero set; additionally, by further subdivisions, the Hausdorff distance between the approximation and the zero set can be made as small as desired. The authors of [25] claim that the PV algorithm is efficient in practice, but, to our knowledge, there is no prior complexity analysis of the PV algorithm.…”

“…(b) The approximation produced by the PV algorithm [25] as well as the boxes constructed by the algorithm.…”