Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 The Authors.EPSRC, Chinese Government (PhD studentship) and Cambridge Trust

Algorithms for curve design and accurate computations with totally positive matrices

Esta tesis doctoral se enmarca dentro de la teoría de la Positividad Total. Las matrices totalmente positivas han aparecido en aplicaciones de campos tan diversos como la Teoría de la Aproximación, la Biología, la Economía, la Combinatoria, la Estadística, las Ecuaciones Diferenciales, la Mecánica, el Diseño Geométrico Asistido por Ordenador o el Álgebra Numérica Lineal. En esta tesis nos centraremos en dos de los campos que están relacionados con matrices totalmente positivas.This doctoral thesis is framed within the theory of Total Positivity. Totally positive matrices have appeared in applications from fields as diverse as Approximation Theory, Biology, Economics, Combinatorics, Statistics, Differential Equations, Mechanics, Computer Aided Geometric Design or Linear Numerical Algebra. In this thesis, we will focus on two of the fields that are related to totally positive matrices.<br /

Neural-network-based curve fitting using totally positive rational bases

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational Bézier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation

Curve Reconstruction By Metaheuristics Algorithms On Cubic Rational Bézier Function

Curve reconstruction regularly used in reverse engineering. Meanwhile, curve fitting is one of the main compositions of curve reconstruction that is usually represented by mathematical functions, most suitable for representing a set of data points, and may need to meet some constraints. Various of curve fitting studies had been done by many researchers specifically using optimisation technique. The optimisation technique consists of exact algorithm, and approximate algorithm. The approximate algorithm is a good technique to be highlighted since it is a feasible way to develop an easier, more convenient curve fitting method, that will save great computation, solve a large scale problem and produce a better quality end result. Metaheuristics has strong and intelligent mechanisms to avoid being trapped in the local minimum