本文選題：toric曲面　切入點：toric退化　出處：《大連理工大學(xué)》2016年博士論文

【摘要】：參數(shù)曲線曲面是計算機(jī)輔助幾何設(shè)計(Computer Aided Geometric Design,簡稱為CAGD)的重要研究內(nèi)容.目前,對參數(shù)曲線曲面的研究主要集中在對Bezier, B樣條與NURBS (Non-Uniform Rational B-Splines)曲線曲面的研究.Toric曲面是有理Bezier曲面的一種多邊形推廣形式,繼承了很多Bezier曲面的造型優(yōu)點.本文研究toric曲面的幾何連續(xù)條件和近似極小toric曲面的構(gòu)造,并將toric曲面應(yīng)用到數(shù)據(jù)擬合和管道拼接中.多元樣條函數(shù)也是幾何造型的一個重要工具,常用的研究方法有光滑余因子方法,B網(wǎng)方法,B樣條方法,同調(diào)方法等.在本文中,我們利用同調(diào)代數(shù)的方法研究代數(shù)曲線剖分下的多元樣條函數(shù)空間.本文主要工作包括：1.在CAGD中,參數(shù)曲面的幾何連續(xù)是一個非常重要的研究內(nèi)容.針對toric曲面的幾何連續(xù)問題,我們推導(dǎo)toric Bernstein基函數(shù)的一階和二階偏導(dǎo)性質(zhì),證明當(dāng)提升函數(shù)滿足提升準(zhǔn)則時,toric曲面沿邊界處的一階和二階偏導(dǎo)在toric退化的過程中保持不變,并由此給出toric曲面的一階幾何連續(xù)和曲率連續(xù)的充要條件.通過指定特殊的提升函數(shù),使得toric曲面沿邊界區(qū)域退化為張量積型或三角型的有理Bezier曲面.由有理Bezier曲面的一階幾何連續(xù)和曲率連續(xù)的已知結(jié)果,給出toric曲面基于控制結(jié)構(gòu)幾何關(guān)系的一階幾何連續(xù)和曲率連續(xù)的充分條件,并給出了一些具體構(gòu)造的實例.2.極小曲面研究中的一個著名問題是求解Plateau問題,即以給定的邊界閉曲線為條件,求解極小曲面.在實際應(yīng)用中,已知的邊界往往是多邊的,我們結(jié)合toric曲面的參數(shù)域是任意凸多邊形,由此考慮Plateau-toric問題,使用Dirichlet泛函代替能量泛函求解,得到近似極小toric曲面的一個構(gòu)造方法,并通過實例驗證了方法的可行性.3.擬合數(shù)據(jù)點集并重構(gòu)曲面是幾何造型中研究的一個重要問題.由于toric曲面是一種多邊參數(shù)曲面,我們使用toric曲面擬合數(shù)據(jù)點集并重構(gòu)曲面,當(dāng)點集的參數(shù)域為凸多邊域時,無需對點集的參數(shù)域進(jìn)行剖分,即可得到一個整體擬合的多邊參數(shù)曲面.其次,構(gòu)造多管道的過渡曲面在模具設(shè)計,工業(yè)零部件制造等領(lǐng)域有著廣泛應(yīng)用.借助幾何連續(xù)條件,我們使用兩片toric曲面來構(gòu)造多管道的過渡曲面.通過這兩個應(yīng)用,可以看出toric曲面不僅保持了有理Bezier曲面構(gòu)造簡單,形狀可調(diào)等優(yōu)點,而且參數(shù)域為任意的凸多邊形,可減少造型中曲面片的個數(shù),避免了多片曲面間的拼接問題.4.幾何造型中的一個重要研究對象是多元樣條函數(shù),而同調(diào)代數(shù)是研究多元樣條函數(shù)的一種有效工具.我們推廣線性剖分下的多元樣條函數(shù)的同調(diào)方法到任意代數(shù)曲線剖分下的多元樣條函數(shù).由Bezout定理可知,2條n次代數(shù)曲線最多可相交于n2個點,本文使用同調(diào)代數(shù)的方法討論了構(gòu)成剖分的N條n次代數(shù)曲線相交于1個點和n2個點的情況.利用交換代數(shù)和toric退化的相關(guān)理論分別證明了這兩種剖分下的多元樣條�？臻g與線性剖分下的多元樣條�？臻g的關(guān)系,分析了這兩種剖分下的樣條�？臻gCT(△)的結(jié)構(gòu),并給出了樣條�？臻gCT(△)的Hilbert多項式與多元樣條函數(shù)空間CTd(△)的維數(shù)公式.
[Abstract]:Parametric curve and surface in computer aided geometric design (Computer Aided Geometric Design, referred to as CAGD) is the important research content. At present, the research of parametric curves and surfaces are mainly focused on Bezier, B and NURBS spline (Non-Uniform Rational B-Splines) on the.Toric surface curve is a polygon generalization of rational Bezier surfaces. Bezier inherits many advantages. The geometric surface modeling of toric surface and the continuity conditions of toric approximate minimal surfaces, and toric surface will be applied to the data fitting and splicing pipeline. A multivariate spline function is an important tool for geometric modeling, the research methods are commonly used smoothing cofactor method, B network method B, spline method, homology method. In this paper, we use multivariate spline space method of algebraic curve section under the homological algebra. In this paper, the main To work includes: 1. in CAGD, the geometric parameters of continuous surface is a very important research content. According to the geometric continuity problem of toric surfaces, we derive the toric Bernstein function one order and two order derivative properties, proved when lifting functions meet the criteria, toric surfaces along the boundary of the first order and two order partial derivatives remain unchanged in the process of toric degradation, and thus gives a second-order geometric surface toric continuous and continuous curvature of sufficient and necessary conditions. By specifying the special lifting function, the toric surface along the boundary area of degradation of rational Bezier surfaces for the tensor product type or triangle. The known results by first order geometric rational Bezier curved continuous and continuous curvature, given toric surface geometric control structure based on the geometric relationship of continuous curvature and sufficient conditions for the continuity, and gives some specific examples of very small structures.2. A famous problem of surface research is to solve the Plateau problem, with the given boundary conditions for closed curve, solving the minimal surface. In practical application, the known boundary is often multilateral, we combine the parameter domain of toric surface is a convex polygon, which account for Plateau-toric, the use of Dirichlet function instead of solving the energy functional and get a method to construct approximate minimal toric surfaces, and proves the feasibility of the method of.3. fitting data points and surface reconstruction is an important research problem in geometric modeling. The toric surface is a multilateral parametric surface, we use toric surface fitting data points and surface reconstruction, when the parameter domain the point set is convex multi domain, the parameter domain without the need for the point set is divided, can get a whole multilateral parametric surface fitting. Secondly, construct the transitional curved pipe In the mold design, parts manufacturing industry has been widely used. By means of geometric continuous conditions, the transition surface we use two pieces of toric surface to construct multiple pipelines. Through the two application, we can see that the toric surface not only keeps the rational Bezier surface has the advantages of simple structure, adjustable shape etc., and the parameter domain is convex a polygon, which can reduce the number of patches in the other, to avoid an important research object in the tiling problem.4. geometric modeling several surfaces between the multivariate spline, and homological algebra is an effective tool to study the multivariate spline function. We generalize homological methods of multivariate spline function for linear cutting under the multiple samples of arbitrary algebraic curves based on the partition function. By Bezout theorem, 2 n algebraic curves can intersect at N2 points, this paper uses the structure of homological algebra is discussed Split N n algebraic curves intersect at 1 points and N2 points. Using the exchange theory of algebra and toric degradation respectively prove the relationship of multivariate spline space model of the two division of the multivariate spline space and linear model based on the partition, and analyzes the two kinds of based on the partition of the spline space model CT (delta) structure, and gives the spline space model CT (delta) Hilbert polynomial and multivariate spline space CTd (delta) dimension formula.

【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：TP391.7

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