Matrix representations by means of interpolation

TL;DR

This paper introduces a novel interpolation matrix-based method for implicit surface and curve representation, enabling efficient ray shooting and intersection computation, with practical algorithms demonstrated in Maple.

Contribution

It extends interpolation matrix techniques to parametric space curves and higher codimension objects, proposing a randomized algorithm using Chow forms for accurate intersection computation.

Findings

Efficient ray shooting for parametric surfaces using interpolation matrices.

Extension of the method to space curves and higher codimension objects.

Maple implementation shows robustness and fewer equations compared to existing methods.

Abstract

We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a surface patch, including the case of high-multiplicity intersections. Most matrix operations are executed during pre-processing since they solely depend on the surface. For a given ray, the bottleneck is equation solving. Our Maple code handles bicubic patches in $\le 1$~sec, though numerical issues might arise. Our second contribution is to extend the method to parametric space curves and, generally, to codimension $> 1$, by computing the equations of (hyper)surfaces intersecting precisely at the given object. By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, we obtain 3 surfaces whose polynomials are of near-optimal degree; in this case, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple prototype with other methods implemented in Maple i.e., Groebner basis and implicit matrix representations. Our Maple prototype is not faster but yields fewer equations and seems more robust than Maple's {\tt implicitize}; it is also comparable with the other methods for degrees up to 6.