Implicitization and Distance Bounds

Abstract

We address the following problem: given a curve in parametric form, compute the implicit representation of another one that approximates the parametric curve on a certain domain of interest. We study this problem from the numerical point of view: what happens with the output curve if the input curve is slightly changed? It is shown that for any approximate parameterization of the given curve, the curve obtained by an approximate implicitization with a given precision is contained within a certain perturbation region.

Fat Arcs for Implicitly Defined Curves

“…Fig. 5 visualizes the relation between the width of the fat arc and the size of the box for the three polynomials in (14). For sufficiently large values of k, the slopes of the three curves in the doubly-logarithmic plot are all three, thus confirming the expected approximation order.…”

“…In order to construct a fat arc, we need to bound the distance between the median arc and the curve using a result from [14], see also [15]. On the one hand, we consider the medial arc as a parametric curve g : t → g(t) with parameter domain t ∈ [a, b], which traces the point set…”

An Evolution-Based Approach for Approximate Parameterization of Implicitly Defined Curves by Polynomial Parametric Spline Curves

“…Finally, one may use the techniques described in [2] in order to obtain a bound on the (one-sided) Hausdorff distance between a curve and its approximation. Essentially, the maximum distance between the points of the spline curve and the corresponding closest points of the algebraic curve Z(f ) can be bounded by V /G, where V is the maximum value of f on the spline curve and G is the minimum length of the gradient in the (suitably chosen) subregion of the region of interest.…”