Sparse implicitization by interpolation: Geometric computations using matrix representations

TL;DR

This paper introduces a sparse interpolation method for implicitizing hyper-surfaces, enabling efficient geometric computations like membership and sidedness tests through matrix nullspace analysis.

Contribution

It presents a novel approach leveraging sparsity to reduce implicitization and geometric predicate computations to simple matrix operations.

Findings

Efficient implicitization via sparse matrix nullspace computation.

Simplified geometric predicates using matrix-based numerical tests.

Validated with examples implemented in Maple.

Abstract

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation.