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

The μ-basis of improper rational parametric surface and its application

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.Ministerio de Ciencia, Innovación y Universidade

Computing the form of highest degree of the implicit equation of a rational surface

A method is presented for computing the form of highest degree of the implicit equation of a rational surface, defined by means of a rational parametrization. Determining the form of highest degree is useful to study the asymptotic behavior of the surface, to perform surface recognition, or to study symmetries of surfaces, among other applications. The method is efficient, and works generally better than known algorithms for implicitizing the whole surface, in the absence of base points blowing up to a curve at infinity. Possibilities to compute the form of highest degree of the implicit equation under the presence of such base points are also discussed. We provide timings to compare our method with known methods for computing the whole implicit equation of the surface, both in absence and in presence of base points blowing up to a curve at infinity.Agencia Estatal de Investigació

Improving Filtering for Computer Graphics

When drawing images onto a computer screen, the information in the scene is typically more detailed than can be displayed. Most objects, however, will not be close to the camera, so details have to be filtered out, or anti-aliased, when the objects are drawn on the screen. I describe new methods for filtering images and shapes with high fidelity while using computational resources as efficiently as possible. Vector graphics are everywhere, from drawing 3D polygons to 2D text and maps for navigation software. Because of its numerous applications, having a fast, high-quality rasterizer is important. I developed a method for analytically rasterizing shapes using wavelets. This approach allows me to produce accurate 2D rasterizations of images and 3D voxelizations of objects, which is the first step in 3D printing. I later improved my method to handle more filters. The resulting algorithm creates higher-quality images than commercial software such as Adobe Acrobat and is several times faster than the most highly optimized commercial products. The quality of texture filtering also has a dramatic impact on the quality of a rendered image. Textures are images that are applied to 3D surfaces, which typically cannot be mapped to the 2D space of an image without introducing distortions. For situations in which it is impossible to change the rendering pipeline, I developed a method for precomputing image filters over 3D surfaces. If I can also change the pipeline, I show that it is possible to improve the quality of texture sampling significantly in real-time rendering while using the same memory bandwidth as used in traditional methods

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

Core foundations, algorithms, and language design for symbolic computation in physics

This thesis presents three contributions to the field of symbolic computation, followed by their application to symbolic physics computations. The first contribution is to interfacing systems. The Notation package, which is developed in this thesis, allows the entry and the creation of advanced notations in the Mathematica symbolic computation system. In particular, a complete and functioning notation for both Dirac's BraKet notation as well as a full tensorial notation, are given herein. The second part of the thesis introduces a prototype based rule inheritance language paradigm that is applicable to certain advanced pattern matching rewrite rule language models. In particular, an implementation is presented for Mathematica. After detailing this language extension, it is adopted throughout the rest of the thesis. Finally, the third major contribution is a highly efficient algorithm to canonicalize tensorial expressions. By an innovative technique this algorithm avoids the dummy index relabeling problem. Further algorithmic optimizations are then presented. The complete algorithm handles linear symmetries such as the Bianchi identities. It also fully accommodates partial derivatives as well as mixed index classes. These advances in language and notations are extensively demonstrated on problems in quantum mechanics, angular momentum, general relativity, and quasi-spin. It is shown that the developments in this thesis lead to an extremely flexible, extensible, and powerful working environment for the expression and ensuing calculation of symbolic physics computations