Implicitizing rational curves by the method of moving quadrics. (September 2019) - Busé, Laurent; Laroche, Clément; Yıldırım, Fatmanur
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. Highlights: Moving quadrics are used for implicitizing a parameterized space curve. Matrices filled by linear and quadratic forms. Their ranks drop exactly on the curve. Matrices about half size of implicitization matrices built with moving hyper-planes. Advantages are illustrated with examples and computation of singularities is given.