细分曲面中奇异点处的G~2连续性研究

发布时间：2018-04-17 23:15

本文选题：细分曲面 + 几何连续　；参考：《南京航空航天大学》2015年硕士论文

【摘要】：本文基于几何造型和逆向工程的相关理论,研究了细分曲面的1G连续性和2G连续性,尤其是对于曲面中含有奇异点的情形,文中给出了相应的1G和2G算法,解决了工业设计中出现的奇异点处无法连续问题.首先,讨论了三种经典的细分方法,提出了一种形状可调的细分算法,并且给出了新算法的几何规则和拓扑规则.通过引入形状调节参数t(0?t?1)对Catmull-Clark细分进行改进,达到形状可调的目的.其次,对于曲面重建中奇异点处的一阶几何连续,本文给出了一种样条曲面重建算法.先采用Hoppe的三角网格重建算法,由散乱点集生成初始网格;再运用改进的Harmonic参数化方法对初始网格参数化生成新的三角网格;然后利用四边界区域划分法得到四边形网格;最后,采用B样条进行拟合,计算出了曲面片的所有控制顶点,使各曲面片之间满足1G连续.运用该方法,本文推导出了B样条曲面片的控制顶点,与以往的方法相比,该方法可以在保证1G的情况下,采用低阶样条进行拟合,降低了算法复杂度,并且重建后的样条曲面自然满足切平面连续.再次,对于细分曲面中奇异点处的二阶几何连续,本文将第二章给出的新算法作为C-C细分的前置方法,进行“混合细分”,构造出奇异点的2-环;再以奇异点处的2-环作为控制网格,采用循环映射的方法得到二阶几何连续的约束方程组;然后引入快速傅里叶变换(FFT),利用循环矩阵和能量函数最优化方法推导出了Bezier控制点的显式解,使奇异点处各曲面片之间满足2G连续.与以往的方法相比,本文不仅给出了曲面中奇异点处的2G处理方法,而且生成的曲面还具有一定的可调性.最后,本文给出了部分算法流程和相关数据结构,针对文中提出的算法也给出了相应的实例进行验证.除此之外,在总结全文研究成果的基础上,对未来研究工作进行了展望.
[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.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TP391.7

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