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Journal Articles Computer-Aided Design Year : 2019

Implicitizing rational curves by the method of moving quadrics

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Abstract

A new technique for finding implicit matrix-based representations of rational curves in arbitrary dimension is introduced. It relies on the use of moving quadrics following curve parameterizations, providing a high-order extension of the implicit matrix representations built from their linear counterparts, the moving planes. The matrices we obtain offer new, more compact, implicit representations of rational curves. Their entries are filled by linear and quadratic forms in the space variables and their ranks drop exactly on the curve. Typically, for a general rational curve of degree d we obtain a matrix whose size is half of the size of the corresponding matrix obtained with the moving planes method. We illustrate the advantages of these new matrices with some examples, including the computation of the singularities of a rational curve.
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Dates and versions

hal-02112538 , version 1 (26-04-2019)
hal-02112538 , version 2 (29-04-2019)

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Laurent Busé, Clément Laroche, Fatmanur Yildirim. Implicitizing rational curves by the method of moving quadrics. Computer-Aided Design, 2019, Proceedings of the Symposium on Physical and Solid Modeling (SPM) 2019, 114, pp.101-111. ⟨10.1016/j.cad.2019.05.019⟩. ⟨hal-02112538v2⟩
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