Abstract
Based on the computation of a superset of the implicit support, implicitization of a parametrically given hypersurface is reduced to computing the nullspace of a numeric matrix. Our approach predicts the Newton polytope of the implicit equation by exploiting the sparseness of the given parametric equations and of the implicit polynomial, without being affected by the presence of any base points. In this work, we study how this interpolation matrix expresses the implicit equation as a matrix determinant, which is useful for certain operations such as ray shooting, and how it can be used to reduce some key geometric predicates on the hypersurface, namely membership and sidedness for given query points, to simple numerical operations on the matrix, without need to develop the implicit equation. We illustrate our results with examples based on our Maple implementation.
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