Matrix Representations by Means of Interpolation

We examine two different matrix representations of curves and surfaces based on or constructed by interpolation through points. Both are essentially implicit representations of objects given as parametric models or given as a point cloud, and both are quite powerful since they reduce geometric operations to linear algebra. First, we examine a representation by interpolation matrices, developed for plane curves, surfaces, and hypersurfaces, which is not sensitive to base points. We extend the method to parametric space curves and, generally, objects of codimension higher than 1, by computing the equations of (hyper)surfaces intersecting precisely at the given object. We propose a practical algorithm that uses interpolation and always produces correct output but possibly with a number of surfaces higher than minimal. Our experiments indicate that, typically, a small number of computed surfaces defines exactly any given space curve. Our second contribution concerns the same representation and completes its properties by showing how it can be used for ray shooting of a parametric ray with a surface patch, given by a parametric or point cloud model. Most matrix operations are executed in pre-processing since they solely depend on the surface but not on the ray. For a given ray, the bottleneck is equation solving. We present experiments in Maple: ray shooting for a bicubic patch takes ≤ 1 sec, though numerical issues might arise. Our approach covers the case of intersection points of high multiplicity. It is extendable to surface-surface intersections yielding an implicit representation of the intersection curve in the plane of one surface’s parameters. The second representation examined is based on syzygies. The corresponding theory, including μ-bases, has been developed for parametric models. Our third contribution is to show how to compute the required syzygies by interpolation, when the input curve or surface is given by a point cloud whose sampling satisfies mild assumptions; this significantly generalizes the method. Even when noise corrupts the sampling, it is possible to estimate the necessary degree of the syzygies and interpolate them numerically, thus constructing the matrix representing the implicit object.