相关论文: Implicitization of rational surfaces using toric v…

We provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant. As a consequence, the symmetry axes, symmetry planes and symmetry center of the surface, if any, can be determined…

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…

It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve.…

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit…

We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general rational surfaces.

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are…

Neural implicit functions have emerged as a powerful representation for surfaces in 3D. Such a function can encode a high quality surface with intricate details into the parameters of a deep neural network. However, optimizing for the…

This paper shows that the multiplicity of the base points locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree…

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as…

We present an approach of computing the intersection curve $\mathcal{C}$ of two rational parametric surface $\S_1(u,s)$ and $\S_2(v,t)$, one being projectable and hence can easily be implicitized. Plugging the parametric surface to the…

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…

We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…

The revised version has two additional references and a shorter proof of Proposition 5.7. This version also makes numerous small changes and has an appendix containing a proof of the degree formula for a parametrized surface.

The advancements in neural rendering have increased the need for techniques that enable intuitive editing of 3D objects represented as neural implicit surfaces. This paper introduces a novel neural algorithm for parameterizing neural…

A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered,…

Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we…

We discuss how Dokken's methods of approximate implicitization can be applied to triangular B\'ezier surfaces in both the original and weak forms. The matrices $\mathbf{D}$ and $\mathbf{M}$ that are fundamental to the respective forms of…

We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.

Given a $4$-dimensional vector subspace $U=\{ f_{0},\ldots,f_{3}\}$ of $H^{0}(\mathbb{P}^1 \times \mathbb{P}^1,\mathcal{O}(a,b))$, a tensor product surface, denoted by $X_{U}$, is the closure of the image of the rational map…

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the…