﻿ 细分曲面奇异点处的光滑过渡 Smooth Connection near Singular Points on Subdivision Surfaces

Vol.06 No.09(2017), Article ID:23137,11 pages
10.12677/AAM.2017.69141

Smooth Connection near Singular Points on Subdivision Surfaces

Yuehong Tang1, Sen Li1, Hao Liu2, Yuping Gu1

1Department of Mathematics, College of Science, Nanjing University Aeronautics and Astronautics, Nanjing Jiangsu

2Department of Mechanical and Engineering, College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing Jiangsu

Received: Nov. 30th, 2017; accepted: Dec. 15th, 2017; published: Dec. 22nd, 2017

ABSTRACT

This paper discusses G2 smooth connection near singular points on subdivision surfaces. A shape adjustable Catmull-Clark subdivision algorithm is present by introducing subdivision shape adjustment parameter $c\left(0\le c\le 1\right)$ . Based on this, aimed at the singular points in subdivision surface, based on the 2-ring of the singular points as control mesh, adopting the method of cyclic mapping, this paper proposes a G2 surface modeling method which is shape-adjustable. Then we can get the explicit solution of the Bezier control points. Compared with existing methods, the generated surface not only achieves the G2 continuous at singular points, but solves the problem of the curved surface design to be adjustable. The algorithm process and data structure are present. Corresponding examples are also given in this paper.

Keywords:Subdivision Surfaces, Geometric Continuity, Singular Points

1南京航空航天大学理学院数学系，江苏 南京

2南京航空航天大学机电学院机械工程系，江苏 南京

1. 引言

2. 一种形状可调的C-C细分曲面

(1) 几何点的产生

${f}_{i}^{\left(k+1\right)}=\frac{1}{4}\left({v}^{\left(k\right)}+{f}_{i}^{\left(k\right)}+{e}_{i}^{\left(k\right)}+{e}_{i+1}^{\left(k\right)}\right)$

${e}_{i}^{\left(k+1\right)}=\frac{{v}^{\left(k\right)}+{e}_{i}^{\left(k\right)}}{2}c+\frac{{f}_{i-1}^{\left(k+1\right)}+{f}_{i+1}^{\left(k+1\right)}}{2}\left(1-c\right)$

${v}^{\left(k+1\right)}=\frac{4cQ}{n}+\frac{8c\left(1-c\right)R}{n}+\frac{n-4c\left(2-c\right){v}^{\left(k\right)}}{n}$

(2) 拓扑结构的建立

$C=\left[\begin{array}{cccc}\frac{c}{2\left(1+c\right)}& \frac{1}{1+c}& \frac{c}{2\left(1+c\right)}& 0\\ -\frac{c\left(2-c\right)}{1+c}& 0& \frac{c\left(2-c\right)}{1+c}& 0\\ \frac{c\left(5-4c\right)}{2\left(1+c\right)}& \frac{\left(-2c+3\right)\left(c-2\right)}{2\left(1+c\right)}& \frac{4{c}^{2}-11c+6}{2\left(1+c\right)}& \frac{c\left(2c-1\right)}{2\left(1+c\right)}\\ \frac{c\left(c-1\right)}{1+c}& \frac{\left(c-1\right)\left(c-2\right)}{1+c}& -\frac{\left(c-1\right)\left(c-2\right)}{1+c}& \frac{c\left(1-c\right)}{1+c}\end{array}\right]$

3. 细分曲面奇异点处的G2连续性

3.1. 二阶几何连续条件

$\frac{{\partial }^{i+j}}{\partial {u}^{i}\partial {v}^{j}}P\left(t\right)=\frac{{\partial }^{i+j}}{\partial {s}^{i}\partial {r}^{j}}\overline{Q}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i+j=0,1,\cdots ,k.$

$\frac{{\partial }^{2}}{\partial {u}_{}^{2}}{P}_{i}^{}={p}_{2}\left(v\right)\frac{\partial }{\partial u}{P}_{i+1}+{q}_{2}\left(v\right)\frac{\partial }{\partial v}{P}_{i+1}+{p}_{1}^{2}\left(v\right)\frac{{\partial }^{2}}{\partial {u}^{2}}{P}_{i+1}+2{p}_{1}\left(v\right){q}_{1}\left(v\right)\frac{{\partial }^{2}}{\partial u\partial v}{P}_{i+1}+{q}_{1}^{2}\left(v\right)\frac{{\partial }^{2}}{\partial {v}^{2}}{P}_{i+1}.$

3.2. 初始控制网格的“混合细分”

$P\left(u,v\right)=UCH{C}^{\text{T}}{V}^{\text{T}}$

$A=CH{C}^{\text{T}}$$H$ 为双三次系数矩阵，则第 $k$ 个曲面片对应的系数矩阵为 ${A}_{k}={C}_{k}{H}_{k}{C}_{k}^{\text{T}}$ ，相应的参数调节矩阵为

${C}_{k}=\left[\begin{array}{cccc}\frac{{c}_{k}}{2\left(1+{c}_{k}\right)}& \frac{1}{1+{c}_{k}}& \frac{{c}_{k}}{2\left(1+{c}_{k}\right)}& 0\\ -\frac{{c}_{k}\left(2-{c}_{k}\right)}{1+{c}_{k}}& 0& \frac{{c}_{k}\left(2-{c}_{k}\right)}{1+{c}_{k}}& 0\\ \frac{{c}_{k}\left(5-4{c}_{k}\right)}{2\left(1+{c}_{k}\right)}& \frac{\left(-2{c}_{k}+3\right)\left({c}_{k}-2\right)}{2\left(1+{c}_{k}\right)}& \frac{4{c}_{k}^{2}-11{c}_{k}+6}{2\left(1+{c}_{k}\right)}& \frac{{c}_{k}\left(2{c}_{k}-1\right)}{2\left(1+{c}_{k}\right)}\\ \frac{{c}_{k}\left({c}_{k}-1\right)}{1+{c}_{k}}& \frac{\left({c}_{k}-1\right)\left({c}_{k}-2\right)}{1+{c}_{k}}& -\frac{\left({c}_{k}-1\right)\left({c}_{k}-2\right)}{1+{c}_{k}}& \frac{{c}_{k}\left(1-{c}_{k}\right)}{1+{c}_{k}}\end{array}\right]$

${T}_{i}=\left[\begin{array}{cccc}\frac{\partial {P}_{i}}{\partial u}& \frac{\partial {P}_{i}}{\partial v}& \frac{{\partial }^{2}{P}_{i}}{\partial {u}^{2}}& \begin{array}{cccc}\frac{{\partial }^{2}{P}_{i}}{\partial u\partial v}& \frac{{\partial }^{2}{P}_{i}}{\partial {v}^{2}}& \frac{{\partial }^{3}{P}_{i}}{\partial {u}^{2}\partial v}& \begin{array}{cc}\frac{{\partial }^{3}{P}_{i}}{\partial u\partial {v}^{2}}& \frac{{\partial }^{4}{P}_{i}}{\partial {u}^{2}\partial {v}^{2}}\end{array}\end{array}\end{array}\right]$

${T}_{i+1}=\left[\begin{array}{cccc}\frac{\partial {P}_{i+1}}{\partial u}& \frac{\partial {P}_{i+1}}{\partial v}& \frac{{\partial }^{2}{P}_{i+1}}{\partial {u}^{2}}& \begin{array}{cccc}\frac{{\partial }^{2}{P}_{i+1}}{\partial u\partial v}& \frac{{\partial }^{2}{P}_{i+1}}{\partial {v}^{2}}& \frac{{\partial }^{3}{P}_{i+1}}{\partial {u}^{2}\partial v}& \begin{array}{cc}\frac{{\partial }^{3}{P}_{i+1}}{\partial u\partial {v}^{2}}& \frac{{\partial }^{4}{P}_{i+1}}{\partial {u}^{2}\partial {v}^{2}}\end{array}\end{array}\end{array}\right]$

${\Theta }_{i}$ 为曲面片 ${P}_{i}$${P}_{i+1}$ 之间的参数化矩阵，则由二阶几何连续性定义可以得到

Figure 1. N-sided hole formed by mingle subdivision

Figure 2. 2-ring around singular vertex on mesh

${T}_{i}={T}_{i+1}\cdot {\Theta }_{i}$

${T}_{0}={T}_{0}\cdot {\Theta }_{n-1}\cdot {\Theta }_{n-2}\cdots {\Theta }_{1}\cdot {\Theta }_{0}$

$I={\Theta }_{n-1}\cdot {\Theta }_{n-2}\cdots {\Theta }_{1}\cdot {\Theta }_{0}$

3.2.1.内部边界的连续性

$\left\{\begin{array}{l}x=\left(\mathrm{cos}\frac{\text{2π}}{n}\right)\left[\left(1-u\right)v+\mathrm{sec}\frac{\text{2π}}{n}\right]\\ y=\left(\mathrm{sin}\frac{\text{2π}}{n}\right)\left[\left(1-u\right)v+\left(\mathrm{sec}\frac{\text{2π}}{n}\right)uv\right]\end{array}$

${r}_{n}$ 是以原点为中心 $\frac{\text{2π}}{n}$ 为单位的一个逆时针旋转变换

${r}_{n}\left(u,v\right)=\left[\begin{array}{cc}\mathrm{cos}\frac{\text{2π}}{n}& -\mathrm{sin}\frac{\text{2π}}{n}\\ \mathrm{sin}\frac{\text{2π}}{n}& \mathrm{cos}\frac{\text{2π}}{n}\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$

$I={\left({\Phi }_{n}^{-1}\cdot {R}_{n}^{-1}\cdot {\Phi }_{n}\right)}^{n}={\Phi }_{n}^{-1}\cdot {\left({R}_{n}^{-1}\right)}^{n}\cdot {\Phi }_{n}$ (1)

${P}_{i}\left(0,t\right)-{P}_{i+1}\left(t,0\right)=0$ (2)

${c}_{n}\left(1-t\right){P}_{i+1}^{u}\left(t,0\right)-{P}_{i+1}^{v}\left(t,0\right)-{c}_{n}\left(t-1\right){P}_{i}{}^{v}\left(0,t\right)-{P}_{i}^{u}\left(0,t\right)=0$ (3) $\begin{array}{l}2{c}_{n}^{2}\left(1-t\right)\left({P}_{i+1}^{v}\left(t,0\right)-{P}_{i}^{u}\left(0,t\right)\right)+2{c}_{n}\left(1-t\right)\left(1+{c}_{n}\left(1-t\right)\right)\left({P}_{i+1}^{uv}\left(t,0\right)-{P}_{i}^{uv}\left(0,t\right)\right)\\ -\left(1+{c}_{n}\left(1-t\right)\right)\left({P}_{i+1}^{vv}\left(t,0\right)-{P}_{i}^{uu}\left(0,t\right)\right)=0\end{array}$ (4)

${p}_{00}^{i}={p}_{07}^{i+1},{p}_{01}^{i}={p}_{17}^{i+1},{p}_{02}^{i}={p}_{27}^{i+1},{p}_{03}^{i}={p}_{37}^{i+1},$

${p}_{04}^{i}={p}_{47}^{i+1},{p}_{05}^{i}={p}_{57}^{i+1},{p}_{06}^{i}={p}_{77}^{i+1},{p}_{07}^{i}={p}_{77}^{i+1}.$

${c}_{n}\left(-{\alpha }_{1}-2{\alpha }_{2}+3{\alpha }_{3}-7{\alpha }_{4}+5{\alpha }_{5}-21{\alpha }_{6}+7{\alpha }_{7}-{\alpha }_{8}\right)$ $+\left({\beta }_{1}-{\beta }_{2}+{\beta }_{3}-{\beta }_{4}+{\beta }_{5}-{\beta }_{6}+{\beta }_{7}-{\beta }_{8}\right)=0$ .

${c}_{n}\left(7{\alpha }_{1}\text{+}13{\alpha }_{2}-17{\alpha }_{3}+31{\alpha }_{4}-19{\alpha }_{5}+57{\alpha }_{6}-13{\alpha }_{7}+{\alpha }_{8}\right)$ $+\left(-7{\beta }_{1}+6{\beta }_{2}-5{\beta }_{3}+4{\beta }_{4}-3{\beta }_{5}+2{\beta }_{6}-{\beta }_{7}\right)=0.$ ${c}_{n}\left(-21{\alpha }_{1}-36{\alpha }_{2}+40{\alpha }_{3}-54{\alpha }_{4}+27{\alpha }_{5}-51{\alpha }_{6}+6{\alpha }_{7}\right)$ $+\left(21{\beta }_{1}-15{\beta }_{2}+10{\beta }_{3}-6{\beta }_{4}+3{\beta }_{5}-{\beta }_{6}\right)=0.$ ${c}_{n}\left(35{\alpha }_{1}+55{\alpha }_{2}-50{\alpha }_{3}+46{\alpha }_{4}-17{\alpha }_{5}+15{\alpha }_{6}+6{\alpha }_{7}\right)+\left(-35{\beta }_{1}+20{\beta }_{2}-10{\beta }_{3}+4{\beta }_{4}-{\beta }_{5}\right)=0.$ ${c}_{n}\left(-35{\alpha }_{1}-50{\alpha }_{2}+35{\alpha }_{3}-19{\alpha }_{4}+4{\alpha }_{5}\right)+\left(35{\beta }_{1}-15{\beta }_{2}+5{\beta }_{3}-{\beta }_{4}\right)=0.$ ${c}_{n}\left(21{\alpha }_{1}+27{\alpha }_{2}-13{\alpha }_{3}+3{\alpha }_{4}\right)+\left(-21{\beta }_{1}+6{\beta }_{2}-{\beta }_{3}\right)=0.$

${c}_{n}\left(-7{\alpha }_{1}-8{\alpha }_{2}+2{\alpha }_{3}\right)+\left(7{\beta }_{1}-{\beta }_{2}\right)=0.$

${c}_{n}\left({\alpha }_{1}+{\alpha }_{2}\right)-{\beta }_{1}=0.$

${\alpha }_{1}=-7{p}_{00}^{i+1}-7{p}_{00}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{2}=7{p}_{10}^{i+1}+7{p}_{01}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{3}=21{p}_{20}^{i+1}+21{p}_{02}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{4}=35{p}_{30}^{i+1}+35{p}_{03}^{i},$

${\alpha }_{5}=35{p}_{40}^{i+1}+35{p}_{04}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{6}=7{p}_{50}^{i+1}+7{p}_{05}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{7}=7{p}_{60}^{i+1}+7{p}_{06}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{8}=7{p}_{70}^{i+1}+7{p}_{07}^{i}.$

${\beta }_{1}=7\left({p}_{00}^{i+1}-{p}_{01}^{i+1}\right)+7\left({p}_{10}^{i}-{p}_{00}^{i}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{2}=49\left({p}_{10}^{i+1}-{p}_{11}^{i+1}\right)+49\left({p}_{11}^{i}-{p}_{01}^{i}\right),$

${\beta }_{3}=147\left({p}_{20}^{i+1}-{p}_{21}^{i+1}\right)+147\left({p}_{12}^{i}-{p}_{02}^{i}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{4}=245\left({p}_{30}^{i+1}-{p}_{31}^{i+1}\right)+245\left({p}_{13}^{i}-{p}_{03}^{i}\right),$

${\beta }_{5}=245\left({p}_{40}^{i+1}-{p}_{41}^{i+1}\right)+245\left({p}_{14}^{i}-{p}_{04}^{i}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{6}=147\left({p}_{50}^{i+1}-{p}_{51}^{i+1}\right)+147\left({p}_{15}^{i}-{p}_{05}^{i}\right),$

${\beta }_{7}=7\left({p}_{60}^{i+1}-{p}_{61}^{i+1}\right)+7\left({p}_{16}^{i}-{p}_{06}^{i}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{8}=7\left({p}_{70}^{i+1}-{p}_{71}^{i+1}\right)+7\left({p}_{17}^{i}-{p}_{07}^{i}\right).$

3.2.2. 外部边界的连续性

$q\left(u,v\right)=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$$s\left(u,v\right)=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$

$Q={\left[\begin{array}{ccccc}1& 0& & & \\ 0& -1& & & \\ & & \ddots & & \\ & & & 1& 0\\ & & & 0& -1\end{array}\right]}_{n×n}$$S={\left[\begin{array}{ccccc}0& 1& & & \\ 1& 0& & & \\ & & \ddots & & \\ & & & 0& 1\\ & & & 1& 0\end{array}\right]}_{n×n}$

$I=\left(Q\cdot {\psi }_{n}\cdot S\cdot {\left({R}_{n}^{-1}\cdot {\phi }_{n}\right)}^{-1}\cdot {\phi }_{n}\cdot {\psi }_{n}^{-1}\right)\cdot \left(Q\cdot {\psi }_{m}\cdot S\cdot {\left({R}_{m}^{-1}\cdot {\phi }_{m}\right)}^{-1}\cdot {\phi }_{m}\cdot {\psi }_{m}^{-1}\right)$

${\psi }_{n}\left(u,v\right)={\psi }_{n}\left(v,u\right)$ ，则 $S\cdot {\psi }_{n}\cdot S={\psi }_{n}$ ，对于第二类点有

$I={\psi }_{k}\cdot {\left(S\cdot {\psi }_{k}\cdot S\right)}^{-1}\cdot {\psi }_{l}\cdot {\left(S\cdot {\psi }_{l}\cdot S\right)}^{-1}\cdot {\psi }_{m}\cdot {\left(S\cdot {\psi }_{k}\cdot S\right)}^{-1}\cdot {\psi }_{n}\cdot {\left(S\cdot {\psi }_{n}\cdot S\right)}^{-1}$

$y\left(u,v\right)={b}^{4}{\left(v\right)}^{\text{T}}\left[\begin{array}{cccc}0& 0& 0& 0\\ \frac{1}{4}& \frac{5{c}_{n}^{3}-3{c}_{n}^{2}-15{c}_{n}+18}{12\left({c}_{n}-2\right)\left(2{c}_{n}-3\right)}& \frac{{c}_{n}^{2}+2{c}_{n}-6}{12\left({c}_{n}-2\right)}& \frac{1}{4}\\ \frac{1}{2}& \frac{{c}_{n}+3}{6}& \frac{{c}_{n}^{2}+3{c}_{n}-9}{9\left({c}_{n}-2\right)}& \frac{1}{2}\\ \frac{3}{4}& \frac{{c}_{n}+9}{12}& \frac{{c}_{n}+9}{12}& \frac{3}{4}\\ 1& 1& 1& 1\end{array}\right]{b}^{3}\left(u\right)$

${b}^{k}=\frac{1}{36}{Q}^{\text{T}}AQ$

$A$ 为“混合细分”矩阵。

${A}_{k}=\left[\begin{array}{cccc}1& {a}_{10}^{k+2}& {a}_{11}^{k+1}& {a}_{12}^{k+1}\\ {a}_{10}^{k-1}& {a}_{00}^{k}& {a}_{10}^{k+1}& {a}_{20}^{k+1}\\ {a}_{11}^{k-1}& {a}_{10}^{k}& {a}_{11}^{k}& {a}_{12}^{k}\\ {a}_{12}^{k-1}& {a}_{20}^{k}& {a}_{20}^{k}& {a}_{22}^{k}\end{array}\right]$

${B}_{i}\left(u,v\right)=\frac{1}{36}{b}^{3}\left(u\right){Q}^{\text{T}}\left[\begin{array}{cccc}1& {a}_{10}^{k+2}& {a}_{11}^{k+1}& {a}_{12}^{k+1}\\ {a}_{10}^{k-1}& {a}_{00}^{k}& {a}_{10}^{k+1}& {a}_{20}^{k+1}\\ {a}_{11}^{k-1}& {a}_{10}^{k}& {a}_{11}^{k}& {a}_{12}^{k}\\ {a}_{12}^{k-1}& {a}_{20}^{k}& {a}_{21}^{k}& {a}_{22}^{k}\end{array}\right]Q{\left({b}^{3}\left(v\right)\right)}^{\text{T}}$

$\frac{{\partial }^{j}}{\partial {u}^{j}}{P}_{i}\left(1,t\right)=\frac{{\partial }^{j}}{\partial {u}^{j}}\left({B}_{i}\circ {\psi }_{n}\right)\left(1,t\right),\text{\hspace{0.17em}}j=0,1,2.$

${P}_{i}\left(1,t\right)$${B}_{i}\left(1,t\right)$$\frac{\partial }{\partial u}{P}_{i}\left(1,t\right)$$\frac{\partial }{\partial x}{B}_{i}\left(1,t\right)$$\frac{\partial }{\partial y}{B}_{i}\left(1,t\right)$$\frac{{\partial }^{2}}{\partial {x}^{2}}{B}_{i}\left(1,t\right)$$\frac{{\partial }^{2}}{\partial xy}{B}_{i}\left(1,t\right)$$\frac{{\partial }^{2}}{\partial {y}^{2}}{B}_{i}\left(1,t\right)$ 。同时，由已知条件可以得到

$\frac{\partial }{\partial u}{\psi }_{n,x}\left(1,t\right)=-\frac{1}{2}{\left(1-t\right)}^{3}-\frac{3\left({c}_{n}+1\right)}{2}t{\left(1-t\right)}^{2}-\frac{3\left({c}_{n}+1\right)}{2}{t}^{2}\left(1-t\right)-\frac{1}{2}{t}^{3}$

$\frac{\partial }{\partial u}{\psi }_{n,y}=\frac{-{c}_{n}^{2}+{c}_{n}}{{c}_{n}-2}t{\left(1-t\right)}^{3}+\frac{-2{c}_{n}^{2}+\frac{33}{2}{c}_{n}-27}{{c}_{n}-2}{t}^{2}{\left(1-t\right)}^{2}-\frac{3\left({c}_{n}+9\right)}{2}{t}^{3}\left(1-t\right)$

$\frac{{\partial }^{2}}{\partial {u}^{2}}{\psi }_{n,x}\left(1,t\right)=-9{\left(1-t\right)}^{3}-3\left({c}_{n}+9\right)t{\left(1-t\right)}^{2}+\frac{3\left({c}_{n}^{2}+3{c}_{n}-6\right)}{{c}_{n}-2}{t}^{2}\left(1-t\right)-9{t}^{3}$

$\frac{{\partial }^{2}}{\partial {u}^{2}}{\psi }_{n,y}\left(1,t\right)=\frac{2{c}_{n}^{3}+2{c}_{n}^{2}}{\left({c}_{n}-2\right)\left(2{c}_{n}-3\right)}t{\left(1-t\right)}^{3}-\frac{2{c}_{n}^{2}}{{c}_{n}-2}{t}^{2}{\left(1-t\right)}^{2}-3{c}_{n}{t}^{3}\left(1-t\right)$

3.3. Bézier控制点的求解

$CP=WA$

$\left\{\begin{array}{l}\left[\begin{array}{cc}E& {c}_{0}^{\text{T}}\\ {c}_{0}& 0\end{array}\right]\left[\begin{array}{c}{p}_{0}\\ {\lambda }_{0}\end{array}\right]+\left[\begin{array}{cc}0& 0\\ {c}_{1}& 0\end{array}\right]\left[\begin{array}{c}{p}_{1}\\ {\lambda }_{1}\end{array}\right]+\cdots +\left[\begin{array}{cc}0& {c}_{1}^{\text{T}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_{n-1}\\ {\lambda }_{n-1}\end{array}\right]=\left[\begin{array}{c}0\\ {\omega }_{0}\end{array}\right]{a}_{0}+\left[\begin{array}{c}0\\ {\omega }_{1}\end{array}\right]{a}_{1}+\cdots +\left[\begin{array}{c}0\\ {\omega }_{n-1}\end{array}\right]{a}_{n-1}\hfill \\ \left[\begin{array}{cc}0& {c}_{1}^{\text{T}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_{0}\\ {\lambda }_{0}\end{array}\right]+\left[\begin{array}{cc}E& {c}_{0}^{\text{T}}\\ {c}_{0}& 0\end{array}\right]\left[\begin{array}{c}{p}_{1}\\ {\lambda }_{1}\end{array}\right]+\cdots +\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_{n-1}\\ {\lambda }_{n-1}\end{array}\right]=\left[\begin{array}{c}0\\ {\omega }_{n-1}\end{array}\right]{a}_{0}+\left[\begin{array}{c}0\\ {\omega }_{0}\end{array}\right]{a}_{1}+\cdots +\left[\begin{array}{c}0\\ {\omega }_{n-2}\end{array}\right]{a}_{n-1}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\hfill \\ \left[\begin{array}{cc}0& 0\\ {c}_{1}& 0\end{array}\right]\left[\begin{array}{c}{p}_{0}\\ {\lambda }_{0}\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_{1}\\ {\lambda }_{1}\end{array}\right]+\cdots +\left[\begin{array}{cc}0& {c}_{1}^{\text{T}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{p}_{n-2}\\ {\lambda }_{n-2}\end{array}\right]+\left[\begin{array}{cc}E& {c}_{0}^{\text{T}}\\ {c}_{0}& 0\end{array}\right]\left[\begin{array}{c}{p}_{n-1}\\ {\lambda }_{n-1}\end{array}\right]=\left[\begin{array}{c}0\\ {\omega }_{1}\end{array}\right]{a}_{0}+\left[\begin{array}{c}0\\ {\omega }_{2}\end{array}\right]{a}_{1}+\cdots +\left[\begin{array}{c}0\\ {\omega }_{0}\end{array}\right]{a}_{n-1}\hfill \end{array}$

$\left\{\begin{array}{l}\sum _{i=0}^{n-1}{c}_{i}^{\text{T}}{\lambda }_{i}+\sum _{i=0}^{n-1}E{p}_{i}=0\hfill \\ \sum _{i=0}^{n-1}{c}_{i}{p}_{i}=\sum _{i=0}^{n-1}{\omega }_{i}{a}_{i}\hfill \end{array}$

Euler系数 ${C}_{n}^{{k}_{l}}=\mathrm{cos}\frac{2\text{π}{k}_{l}}{n}$${S}_{n}^{{k}_{l}}=\mathrm{sin}\frac{2\text{π}{k}_{l}}{n}$ 则有

${\stackrel{^}{\rho }}_{j}=\frac{1}{n}\sum _{k=0}^{n-1}{\tau }_{k}{\rho }_{k}$ (5)

${p}_{i},{c}_{i},{\lambda }_{i},{\omega }_{i},{a}_{i}$ 代入(5)式即可得到 ${\stackrel{^}{p}}_{j},{\stackrel{^}{c}}_{j},{\stackrel{^}{\lambda }}_{j},{\stackrel{^}{\omega }}_{j},{\stackrel{^}{a}}_{j}$

4. 曲面造型结果

(a)3价奇异点局部网格 (b)3奇异点的控制顶点 (c)3价奇异点的拟合

Figure 3. 3-sided hole filling

(a)5价奇异点局部网格 (b)5奇异点的控制顶点 (c)5价奇异点的拟合

Figure 4. 5-sided hole filling

(a) 进行“混合细分” (b) 对n边洞进行填充 (c) G2连续曲面

Figure 5. Surface modeling with G2 continuity

5. 小结

Smooth Connection near Singular Points on Subdivision Surfaces[J]. 应用数学进展, 2017, 06(09): 1163-1173. http://dx.doi.org/10.12677/AAM.2017.69141

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