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Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

JOURNAL OF SYMBOLIC COMPUTATION 2012年第2期47卷 131-152页

作者： Burr, Michael Choi, Sung Woo Galehouse, Ben Yap, Chee K. NYU Courant Inst New York NY 10012 USA Duksung Womens Univ Dept Math Seoul 132714 South Korea Korea Inst Adv Study Seoul 130722 South Korea

Given a real valued function f (X, Y), a box region B-0 subset of R-2 and epsilon > 0, we want to compute an E-isotopic polygonal approximation to the restriction of the curve S = f(-1) (0) = {p is an element of R-2 : f (p) = 0} to B-0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Meshing Singularity Root bound Evaluation bound Implicit algebraic curve Complete numerical algorithm subdivision algorithm

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Study on algorithm for Grid subdivision and Encryption Based on Technology of Three Dimensional Geological Modeling

Study on Algorithm for Grid Subdivision and Encryption Based...

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2nd International Conference on Measurement, Instrumentation and Automation (ICMIA 2013)

作者： He, Wei Wu, Wenli Chinese Acad Geol Sci Inst Geophys & Geochem Explorat Langfang 065000 Hebei Peoples R China

ISBN: (纸本)9783037857519

To achieve 3D grid models which have a non-uniform size and varying properties, we proposed the algorithm of grid subdivision and encryption by human-computer interaction. This algorithm was the technology based on 3D geological modeling, and achieving process has following three steps. Firstly, we converted many 2D cross sections to 3D space, and reconstructed 3D vector models using the algorithm of optimal path suture, and set the property of abnormal body and surrounding rock. Then, achieving 3D grids subdivided according to the relationship between the center of 3D grid and 3D vector models, the properties of 3D grids were determined. Finally, we encrypted grids in the survey area and expansion area, and modified the properties. The results show that the algorithm can realize the conversion from 3D vector models to 3D grid models, and this process is reliable and efficient.

关键词： computer application three dimensional modeling subdivision algorithm encryption algorithm table structure

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CURVE AND SURFACE INTERPOLATIONBY subdivision algorithmS

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Computer Aided Drafting,Design and Manufacturing 1994年第2期4卷 28-39页

作者： Ruibin Qu

Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control *** parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also ***(MOS) : 65D05 , 65D15 , 65D17.

关键词： curve and surface interpolation subdivision algorithm divided difference generationpolynomial uniform triangulation wavelet

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A subdivision approach to the construction of approximate solutions of boundary-value problems with deviating arguments

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1998年第11期35卷 121-135页

作者： Qu, R Agarwal, RP Natl Univ Singapore Dept Math Singapore 119260 Singapore

Using the ideas employed in the construction of subdivision algorithms, we offer here a high-accuracy algorithm to compute numerical solutions for two point boundary-value problems of differential equations with deviating arguments. Numerical examples are included to demonstrate the fast convergence and high accuracy of the algorithm. This paper is a further development to our previous works for solving various types of boundary-value problems. (C) 1998 Elsevier Science Ltd. All rights reserved.

关键词： boundary-value problem delay differential equation approximation method of collocation interpolation iterative method subdivision algorithm

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A subdivision Approach to Planar Semi-algebraic Sets

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6th International Conference on Geometric Modeling and Processing (GMP 2010)

作者： Mantzaflaris, Angelos Mourrain, Bernard INRIA Mediterranee GALAAD F-06902 Sophia Antipolis France

ISBN: (纸本)9783642134104

Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a. certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given precision, tins algorithm can also provide a polygonal and isotopic approximation of the exact set. The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information. Then a systematic traversal of the boundary curve graph yields polygonal regions isotopic to the connected components of the semi-algebraic set. Space subdivision is supported by a kd-tree structure and localization is done using Bernstein representation. We conclude by demonstrating our C++ implementation in the CAS MATHEMAGIX.

关键词： subdivision algorithm semi-algebraic set connected component algebraic curve topology computation

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A Single-Ring Absolute Encoder subdivision Method 7

A Single-Ring Absolute Encoder Subdivision Method

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Seventh International Conference on Measuring Technology and Mechatronics Automation (ICMTMA 2015)

作者： Su Xiaogang Hu Xiaodong Zhang Xiaodong Chinese Acad Sci Xian Inst Opt & Precis Mech Xian 710119 Shaanxi Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China

ISBN: (纸本)9781467371438

The basic working principle of single-ring absolute photoelectric shaft encoder is to use light-emitting diode as light source to illuminate the encoder, at the same time using linear CCD as photoelectric signal receiving device and subdivision device. So the essence of single ring absolute encoder subdivision technology is based on line array CCD subdivision technology. This paper proposes a linear CCD subdivision algorithm Based on polynomial interpolation method. By computing the center of the bar code to get value of accurate positioning, this calculates the center of the coded disc line. As a result of the existence of encoder's characterization error, this article adopts the average operation of multiple line center position to get the precise localization of encoder center line. So as to achieve the aim of improve the barcode location accuracy, and can realize in the application of engineering to achieve the precision of 1 ".

关键词： Single ring encoder subdivision algorithm polynomial interpolation

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A collocation method for solving a class of singular nonlinear two-point boundary value problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1997年第2期83卷 147-163页

作者： Qu, RB Agarwal, RP NATL UNIV SINGAPORE DEPT MATHSINGAPORE 119260SINGAPORE

In this paper, an iterative algorithm for serving singular nonlinear two-point boundary value problems is formulated. This method is basically a collocation method for nonlinear second-order two-point boundary value problems with singularities at either one or both of the boundary points. It is proved that the iterative algorithm converges to a smooth approximate solution of the BVP provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates for uniform partitions are also investigated. It has been shown that, for sufficiently smooth solutions, the method produces order h(4) approximations. Numerical examples are provided to show the effectiveness of the algorithm.

关键词： singular boundary value problems approximation method of collocation interpolation iterative method subdivision algorithm

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THE GEOMETRY OF TCHEBYCHEFFIAN SPLINES

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COMPUTER AIDED GEOMETRIC DESIGN 1993年第3-4期10卷 181-210页

作者： POTTMANN, H Institut f&uuml r Geometrie Technische Universit&auml t Wien A-1040 Wien Austria

This paper shows that many properties of Bezier and B-spline curves hold for a much wider class of curves. Using a ''normal curve'' associated with an extended Tchebycheff space, we derive a Bezier representation of Tchebycheffian spline curve segments. These are affine or projective images of segments of the normal curve. A generalization of the blossoming method allows us to study Tchebycheffian B-spline curves and their segments in a simple geometric way. The basic algorithms such as knot insertion and construction of the Bezier points are described. Whereas the generation of tensor product surfaces is straightforward, some preliminary studies indicate that a similarly natural generalization of Bezier triangles does not exist.

关键词： FREE-FORM CURVE B-SPLINE TCHEBYCHEFFIAN SPLINE PROJECTIVE INVARIANCE RATIONAL CURVE GEOMETRIC ORDER BLOSSOMING VARIATION DIMINISHING PROPERTY subdivision algorithm TENSOR PRODUCT SURFACE

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An iterative scheme for solving nonlinear two point boundary value problems

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 1997年第3-4期64卷 285-302页

作者： Qu, RB Agarwal, RP Department of Mathematics National University of Singapore Singapore 119260 Singapore

In this paper, by using the ideas employed in the analysis of interpolatory subdivision algorithms for the generation of smooth curves, an iterative scheme for solving nonlinear two point boundary value problems is formulated. This method is basically a collocation method for nonlinear second order two point boundary value problems. It is proved that the iterative algorithm converges to a smooth approximate solution provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates in the case of uniform partitions are also investigated. Some numerical examples are included to show the convergence of the proposed algorithm.

关键词： boundary value problems approximation method of collocation interpolation iterative method subdivision algorithm

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A new approach to numerical differentiation and integration

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MATHEMATICAL AND COMPUTER MODELLING 1996年第10期24卷 55-68页

作者： Qu, R Department of Mathematics National University of Singapore Singapore 119260

Some new formulae for numerical differentiation and integration are derived by using interpolatory subdivision algorithms. These interpolatory subdivision algorithms are originally designed for the generation of smooth curves. The main advantage of these numerical formulae is that they produce better numerical results if the data comes from functions with fractal-like derivatives. The main disadvantage of these formulae is that they normally do not have the best approximation orders. By using different interpolatory subdivision algorithms, higher order approximation formulae can be obtained. Some numerical examples are given to compare these formulae with the traditional high accuracy formulae.

关键词： numerical differentiation numerical integration interpolation subdivision algorithm

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