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Three Kinds of Geometric Approximation in Curves and Surfaces Modeling

Author: ZhouLian
Tutor: WangGuoZuo
School: Zhejiang University
Course: Applied Mathematics
Keywords: Computer Aided Geometric Design Bézier curves Tensor product Bézier surfaces Triangular Bézier Rational parametric curves and surfaces Reduced-order Jacobi polynomials Least Squares Derivative vector sector Reparameterization Piecewise linear approximation
CLC: TP391.72
Type: PhD thesis
Year: 2010
Downloads: 156
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Abstract


Compressed information or calculated convenient and relatively simple, commonly used form curves and surfaces in computer-aided geometric design, in order to approximately instead of the known curves and surfaces, and such that the geometric error between the two is as little as possible. This approximation with the traditional function approximation different geometric images approximation object geometric position error of approximation error, it is referred to as the geometry of curves and surfaces approximation, it is an important research topic in geometric design paper do three types of geometric approximation problem the theoretical study of the system, Reduced-order approximation of parametric curves and surfaces and derivative vector sector approach, as well as the rational triangular surfaces piecewise linear approximation. mainly made innovative theoretical results: 1. Reduction Approximation of Bezier curve constraints: to Bezier curve in Paul endpoint parameters consecutive drop and Paul geometric continuous two constraints explicit multi-step algorithm. continuous constraints of Paul endpoint parameters, application sub-rule ideology, degree-reduced surface to be seeking control the vertex divided into constrained control vertex control points and is not constrained. constraints first obtained constraints control points. then the use of a single variable Jacobi polynomials with the conversion relationship between Bernstein polynomials and Jacobi polynomials orthogonal , the reduction problem is converted to a range accuracy of the least squares problem, then find the unconstrained control points the algorithm has approximation error minimum degree-reduced surface control vertices explicitly said Paul endpoint higher order interpolation, a drop multi-stage error prediction, less computing time and six other advantages. particular approximation error as the objective function, the algorithm cleverly got Bernstein polynomial continuous constraint best under the conditions of high-end endpoint geometric Paul explicit reduced-order approximation, and further given the best continuous Bezier curves Paul endpoint G1 explicit multi-degree reduction algorithms, completely solve the existing literature in Bulgaria geometric continuous constraints can only be given reduced curve numerical solution the problem of the proposed algorithm is simple and intuitive, CAD / CAM systems, data communications, data compression curve intersection quadrature aspect has important applications .2. tensor product Bezier surface constraints reduced-order approximation: respectively given in the absence of higher order interpolation constraints, conformal point security boundary order continuum three constraints under conditions explicit multi-degree reduction algorithm. unconstrained case, the transition between the Jacobi polynomial with Bernstein polynomials orthogonality relations and Jacobi polynomials, given reduced-order matrix representation of the surface as well as a priori error in the the conformal point higher order interpolation constraints, the use of the idea of ??dimensionality reduction, will control vertex reordering into a one-dimensional order reduction algorithm combined curve, gives the best explicit reduced-order approximation in the security boundary constraints order continuum, the first degree-reduced surface constraint control points determined according to the boundary conditions, and then by the least squares method, seeking A reduced-order surfaces matrix representation ensure continuous blending surface film remains in the reduced-order the original continuous order to adapt to the shape of the CAD / CAM system requirements .3 triangular Bezier surface constraint Reduction Approximation: continuous stitching triangular Bezier surface as well as the discrete surface in the sub-surface patches simultaneously reduced order to achieve the overall C1 continuous best explicit multi-degree reduction algorithm. determined first according to the constraints constraints control points, and then use the drop Invensys want to convert it to curve down order problem, the final conversion relationship between the triangular Jacobi polynomials and triangular Bernstein polynomials and triangular Jacobi polynomials orthogonal, respectively, to obtain the best reduced-order approximation of the sub-surface patches, and gives the first posteriori error of the method is a simple operation, high accuracy, and speed .4. approximation of parametric curves and surfaces derivative vector sector: the use of a particular type of fractional linear parameter transformation, to reparameterize rational parametric curves and surfaces. reparameterization curves and surfaces to maintain control after the vertex and the same domain, but simply changing the distribution of weight factors and parameters. use of technology reparameterization two optimized power factor, the maximum weight factor and minimum weight factor the effect of minimizing the ratio between the two will minimize the variance of the number after the right factors. export the a derivative vector profession rational curves and surfaces tighter on the basis of literature results, which can be further optimized geometric design system efficiency .5 rational triangular surfaces piecewise linear approximation: given the domain of any triangle C2 continuous rational triangular surfaces piecewise linear approximation and the use of technology reparameterization, on the basis of existing achievements further improve the rational triangular Bezier surface piecewise linear approximation effect this has a very high value at the intersection of parametric surfaces, rendering.

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CLC: > Industrial Technology > Automation technology,computer technology > Computing technology,computer technology > Computer applications > Information processing (information processing) > Machine-assisted technology > Machine -aided design (CAD), aided drawing
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