本文選題：細分曲面 + 幾何連續(xù)��； 參考：《南京航空航天大學》2015年碩士論文

【摘要】：本文基于幾何造型和逆向工程的相關理論,研究了細分曲面的1G連續(xù)性和2G連續(xù)性,尤其是對于曲面中含有奇異點的情形,文中給出了相應的1G和2G算法,解決了工業(yè)設計中出現(xiàn)的奇異點處無法連續(xù)問題.首先,討論了三種經(jīng)典的細分方法,提出了一種形狀可調(diào)的細分算法,并且給出了新算法的幾何規(guī)則和拓撲規(guī)則.通過引入形狀調(diào)節(jié)參數(shù)t(0?t?1)對Catmull-Clark細分進行改進,達到形狀可調(diào)的目的.其次,對于曲面重建中奇異點處的一階幾何連續(xù),本文給出了一種樣條曲面重建算法.先采用Hoppe的三角網(wǎng)格重建算法,由散亂點集生成初始網(wǎng)格;再運用改進的Harmonic參數(shù)化方法對初始網(wǎng)格參數(shù)化生成新的三角網(wǎng)格;然后利用四邊界區(qū)域劃分法得到四邊形網(wǎng)格;最后,采用B樣條進行擬合,計算出了曲面片的所有控制頂點,使各曲面片之間滿足1G連續(xù).運用該方法,本文推導出了B樣條曲面片的控制頂點,與以往的方法相比,該方法可以在保證1G的情況下,采用低階樣條進行擬合,降低了算法復雜度,并且重建后的樣條曲面自然滿足切平面連續(xù).再次,對于細分曲面中奇異點處的二階幾何連續(xù),本文將第二章給出的新算法作為C-C細分的前置方法,進行“混合細分”,構造出奇異點的2-環(huán);再以奇異點處的2-環(huán)作為控制網(wǎng)格,采用循環(huán)映射的方法得到二階幾何連續(xù)的約束方程組;然后引入快速傅里葉變換(FFT),利用循環(huán)矩陣和能量函數(shù)最優(yōu)化方法推導出了Bezier控制點的顯式解,使奇異點處各曲面片之間滿足2G連續(xù).與以往的方法相比,本文不僅給出了曲面中奇異點處的2G處理方法,而且生成的曲面還具有一定的可調(diào)性.最后,本文給出了部分算法流程和相關數(shù)據(jù)結構,針對文中提出的算法也給出了相應的實例進行驗證.除此之外,在總結全文研究成果的基礎上,對未來研究工作進行了展望.
[Abstract]:In this paper, based on the theory of geometric modeling and reverse engineering, the 1G continuity and 2G continuity of subdivision surfaces are studied, especially in the case of singular points in the surfaces, and the corresponding 1G and 2G algorithms are given.The problem of discontinuity of singularity in industrial design is solved.Firstly, three classical subdivision methods are discussed, and a new subdivision algorithm with adjustable shape is proposed, and the geometric and topological rules of the new algorithm are given.The Catmull-Clark subdivision is improved by introducing the shape adjustment parameter t0 / t ~ (1) to achieve the purpose of adjustable shape.Secondly, for the first order geometric continuity of singular points in surface reconstruction, a spline surface reconstruction algorithm is presented in this paper.First, the triangular mesh reconstruction algorithm of Hoppe is used to generate the initial mesh from the scattered point set, then the improved Harmonic parameterization method is used to generate the new triangular mesh, and then the quadrilateral mesh is obtained by using the quadrilateral region partition method.Finally, all the control vertices of the surface slice are calculated by using B-spline fitting, and the continuity between the surfaces is 1G.By using this method, the control vertices of B-spline patches are derived. Compared with the previous methods, this method can be fitted with low-order splines to ensure 1G, thus reducing the complexity of the algorithm.And the reconstructed spline surface naturally satisfies the tangent plane continuity.Thirdly, for the second order geometric continuity of singular points in subdivision surfaces, this paper uses the new algorithm in Chapter 2 as the prepositioning method of C-C subdivision, carries out "mixed subdivision", and constructs the 2-ring of singular points.Then the 2-ring at the singular point is taken as the control grid and the second-order geometric continuous constraint equations are obtained by the method of cyclic mapping.Then, the explicit solution of Bezier control points is derived by using the cyclic matrix and the energy function optimization method by introducing the fast Fourier transform (FFT), which satisfies the 2G continuity between the surfaces at the singular points.Compared with the previous methods, this paper not only gives the 2G processing method of the singular point in the surface, but also the generated surface has some tunability.Finally, part of the algorithm flow and related data structure are given, and the corresponding examples are given to verify the proposed algorithm.In addition, on the basis of summarizing the full-text research results, the future research work is prospected.
【學位授予單位】：南京航空航天大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：TP391.7

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