Implicitization of curves and (hyper)surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive

Using projection operators in computer aided geometric design

We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention different formulations of resultants, including the projective resultant, the toric resultant, and the residual resultant. In the second part we illustrate these tools, and other projection operators, on typical problems of CAGD such as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface

A partial solution to the problem of proper reparametrization for rational surfaces

Given an algebraically closed field K, and a rational parametrization P of an algebraic surface V ⊂ K3 , we consider the problem of computing a proper rational parametrization Q from P (reparametrization problem). More precisely, we present an algorithm that computes a rational parametrization Q of V such that the degree of the rational map induced by Q is less than the degree induced by P. The properness of the output parametrization Q is analyzed. In particular, if the degree of the map induced by Q is one, then Q is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from P is not proper

On the problem of proper reparametrization for rational curves and surfaces

A rational parametrization of an algebraic curve (resp. surface) establishes a rational correspondence of this curve (resp. surface) with the affine or projective line (resp. affine or projective plane). This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational proper parametrization trace the curve or surface once. We consider the problem of computing a proper rational parametrization from a given improper one. For the case of curves we generalize, improve and reinterpret some previous results. For surfaces, we solve the problem for some special surface's parametrizations

The Bernstein basis in set-theoretic geometric modelling

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