Representing rational curve segments and surface patches using semi-algebraic sets

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.National Natural Science Foundation of ChinaMinisterio de Ciencia, Innovación y Universidade

Implicitization of rational maps

Motivated by the interest in computing explicit formulas for resultants and discriminants initiated by B\'ezout, Cayley and Sylvester in the eighteenth and nineteenth centuries, and emphasized in the latest years due to the increase of computing power, we focus on the implicitization of hypersurfaces in several contexts. Our approach is based on the use of linear syzygies by means of approximation complexes, following [Bus\'e Jouanolou 03], where they develop the theory for a rational map $f:P^{n-1}\dashrightarrow P^n$. Approximation complexes were first introduced by Herzog, Simis and Vasconcelos in [Herzog Simis Vasconcelos 82] almost 30 years ago. The main obstruction for this approximation complex-based method comes from the bad behavior of the base locus of $f$. Thus, it is natural to try different compatifications of $A^{n-1}$, that are better suited to the map $f$, in order to avoid unwanted base points. With this purpose, in this thesis we study toric compactifications $T$ for $A^{n-1}$. We provide resolutions $Z.$ for $Sym_I(A)$, such that $\det((Z.)_\nu)$ gives a multiple of the implicit equation, for a graded strand $\nu\gg 0$. Precisely, we give specific bounds $\nu$ on all these settings which depend on the regularity of \SIA. Starting from the homogeneous structure of the Cox ring of a toric variety, graded by the divisor class group of $T$, we give a general definition of Castelnuovo-Mumford regularity for a polynomial ring $R$ over a commutative ring $k$, graded by a finitely generated abelian group $G$, in terms of the support of some local cohomology modules. As in the standard case, for a $G$-graded $R$-module $M$ and an homogeneous ideal $B$ of $R$, we relate the support of $H_B^i(M)$ with the support of $Tor_j^R(M,k)$.Comment: PhD. Thesis of the author, from Universit\'e de Paris VI and Univesidad de Buenos Aires. Advisors: Marc Chardin and Alicia Dickenstein. Defended the 29th september 2010. 163 pages 15 figure

Intersecting biquadratic Bézier surface patches

International audienceWe present three symbolic–numeric techniques for computing the in- tersection and self–intersection curve(s) of two Bézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods

Intersecting biquadratic Bézier surface patches

International audienceWe present three symbolic–numeric techniques for computing the in- tersection and self–intersection curve(s) of two Bézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods

The implicit equation of a canal surface

A canal surface is an envelope of a one parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the \mu-basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the the dual variety, the canal surface and any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio