Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

Parametric Interpolation To Scattered Data [QA281. A995 2008 f rb].

Dua skema interpolasi berparameter yang mengandungi interpolasi global untuk data tersebar am dan interpolasi pengekalan-kepositifan setempat data tersebar positif dibincangkan. Two schemes of parametric interpolation consisting of a global scheme to interpolate general scattered data and a local positivity-preserving scheme to interpolate positive scattered data are described

Representation and application of spline-based finite elements

Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements. This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems. The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation

Automatic G1 Parametric Fitting Of Curves And Surfaces To Outlines Of Images

Rapid advancement in imaging technologies produces massive amount of data which can be harnessed for information and knowledge gathering. Mathematical representations of objects of interest from these images are amenable to manipulation of shapes and sizes, thus aiding analysis and design. As a process in reverse engineering, we aim to automatically reproduce a mathematical outline of a 2D contour based image of an object. Next we will reconstruct a 3D object (surface) from its cross-sectional images. It is our objective to have a representation which is reliable, reasonably fast and with flexible accuracy

Flexible G1 Interpolation of Quad Meshes

International audienceTransforming an arbitrary mesh into a smooth G1 surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G1-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

A code for surface modeling and grid generation coupled to a panel method for aerodynamic configuration design

An integrated platform has been developed which features a geometric, a grid generation and an aerodynamic analysis module. The main intent is to execute a quick though reliable preliminary aerodynamic analysis on a generic complex aerodynamic configuration and, at the same time, provide a mean of exporting the defined geometry or grid to leading CAE/CAD, meshing and analysis softwares, for deep detail modifications or more accurate, although time consuming, analysis. In the geometric module, the process of shape definition is easily and intuitively achieved with the aid of specific features and tools. The geometric description relies on NURBS, a flexible, accurate and efficient parametric form. Once the configuration has been defined, the user is ready to move on the grid generation module, or to export it to IGES standard format in order to use CAE/CAD, meshing or aerodynamic analysis programs. The grid generation module is capable to build structured or unstructured meshes. Both of the processes are automatized, even if the user can easily set and control grid parameters. The structured grid generator is oriented to LaWGS description standard, while the unstructured grid can be exported to different formats. The user is now ready to launch Pan Air, a panel method, as the aerodynamic solver. The preprocessor and postprocessor aid to the definition of the flow parameters and to the graphical visualization of the results. One of the strength of this code is the user friendly GUI organization of each module: the user is aided throughout all the steps. Besides this, every module relies on fast computational algorithms to speed up the overall process. For all these reasons, this code has a natural lean to be used in pair with an optimization tool

ShapeWright--finite element based free-form shape design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1990.Includes bibliographical references (p. 179-192).by George Celniker.Ph.D