曲面加工几何学

1、大连理工大学微分几何作业学院:姓名:学号:授课教师:电子邮箱:完成时间:机械工程学院王晓明2012年6月25日1/18非均匀有理B样条（NURBS曲线和曲面1,非均匀有理B样条曲线（3阶）NURB珈线提供了对标准解析几何和自由曲线、曲面的统一数学描述方法,它可通过调整控制顶点和权因子,方便地改变曲面形状，同时也可方便地转换成对应Bezier曲面。给定n+1个控制点及权因子,则k阶（k-1）次NURB釉线的表达式为:r(x)nNi,k(x)WRi0nNi,k(x)Wii0其中：Ni,k（u）为非均匀有理B样条基函数，按照deBoor-Cox递推公式，基函数满足一下关系:M,k(x)01,当Uiu

2、Ui1其它M,k(x)(xUi)Ni,k1(x)(Uikx)Ni01(x)1.1.曲线的建立Uik1UiUikUi1本文选取四个控制顶点P1,P2,P3,P4,已经四个权因子W,W2,%,WA,通过Matlab计算其基函数，并作出曲线。P控制顶点：0.51；权因子：w20.513；1.52/18非均匀节点：u12456.57.589.3由Matlab绘制曲线如下:-控制多边形A-B-样条曲线在每个点上标注其Frenet标架如图:控制多边形*B-样条曲线alphabetagama3/18在每个点上标注其曲率、挠率如图:控制多边形控制多边形B-样条曲线-挠率2.非均匀有理B样条曲面(3阶)假设名&

3、amp;定R3空间中，齐次坐标下的m1n1个点R(X。,Zj,Wj)(WjXij,WijYj，WjZij,Wij),i0,1,m;j=0,1,n,w。0,则参数曲面：mnR(u,v)(X(u,v),Y(u,v),Z(u,v),w(u,v)2Bm(u)B：(v),u0,v01 0j0称为矩形域0,1x0,1上的mxn次有理B样条曲面。其中巳称为控制顶点，w。称为权因子。NURBS®面的表达式为:nR(u,v)Wi,jR,jNi,k(u)Nj(v)j0nWi,jNi,k(u)N"(v)i01.1. 曲面计算的基本公式曲面第一基本形式:Frurv,Grvi;4/18曲面第二基本形

4、式:Lnruu,Mnruv,Nnrwo高斯曲率计算公式:2LNMEGF2平均曲率计算公式:NE2MFLG2EGF2曲面主曲率计算公式：kiH4HK,k2H4HK1.2. 曲面的建立选才?16个顶点构造B样条曲面，如表1所示：表116个顶点坐标数据厅Pxyz1-24.8849.76-4.352-23.660.560.153-19.4771.915.174-11.9981.099.785-22.8950.75-3.016-21.0361.451.447-16.6572.446.348-9.4781.3510.729-20.2551.91-1.2910-17.6662.433.1811-12.967

5、2.967.9112-6.1581.5711.9513-16.3453.371.0414-12.8163.545.5715-7.6173.4110.0416-1.4581.7213.675/1820.513w权因子：0.310.8211.510.80.5010.51计算结果如图:图2B样条曲面平均曲率H图3E值图4F值图5G值图6L值6/18图9k1值图10k2值3.参考文献1王省富.样条函数及其应用M.西北工业大学出版社，1989,9,西安.2张化光，刘鑫蕊，孙秋野.MATLAB/SIMULINK实用教程M.人民邮电出版社，2009,3,北京.3JohnH.Mathews,KurtisD.F

6、ink.数值方法（MATLAB版）.电子工业出版社，2002,6,北京.7/18附录(Matlab源程序)：非均匀有理B样条曲线(3阶)functionNURBS3clear;clc;symsrP=010;000.5;101;111.5;w=2,0.5,1,3;%插入点权重%w=2,2,2,2;%权重相等时u=12456.57.589.3;%非均匀节点(构造样条基函数)k=3;%三阶非均匀有理B样条曲线n=size(P,1);%控制点数目nu=size(u,2);%非均匀节点数ifnu=n+k+1M=100;r1=zeros(M,3);r2=zeros(M,3);r=zeros(M,3);X=

7、linspace(u(k+1),u(n+1),M);forl=1:M%构造非均匀有理B样条基函数x=X(l);N=zeros(n+k,k+1);fori=1:n+kifx>=u(i)&&x<u(i+1)N(i,1)=1;endendforj=2:k+1fori=1:n+k-j+18/18N(i,j)=(x-u(i)/(u(i+卜1)-u(i)*N(i,j-1)+(u(i+j)-x)/(u(i+j)-u(i+1)*N(i+1,j-1)；endendform1=1:nform2=1:3r1(l,m2)=r1(l,m2)+w(m1)*N(m1,k+1)*P(m1,m2);

8、%分子分母endr2(l,:)=r2(l,:)+w(m1)*N(m1,k+1);%endendr=r1./r2;plot3(P(:,1),P(:,2),P(:,3),'-ko',r(:,1),r(:,2),r(:,3),'-r.');holdon;gridon;axisequallegend('控制多边形','B-样条曲线');elsedisp('输入错误，请核查u的维数是否满足n+k+1');endaxisequal犷阶导xt1=diff(r(:,1)',1);yt1=diff(r(:,2)',1

9、);zt1=diff(r(:,3)',1);ft1=xt1;yt1;zt1'%2阶导xt2=diff(r(:,1)',2);yt2=diff(r(:,2)',2);zt2=diff(r(:,3)',2);ft2=xt2;yt2;zt2'%3阶导xt3=diff(r(:,1)',3);yt3=diff(r(:,2)',3);zt3=diff(r(:,3)',3);ft3=xt3;yt3;zt3'%率计算k=zeros(size(ft2,1),1);9/18fori=1:size(ft2,1)n1=ft1(i,:);n

10、2=ft2(i,:);ka=sqrt(sum(cross(n1,n2).*cross(n1,n2);kb=sqrt(sum(n1.*n1);k(i)=ka/kbA3;end%fc点上表示曲率大小figure(2)plot3(P(:,1),P(:,2),P(:,3),'-ko',r(:,1),r(:,2),r(:,3),'-r.');holdon;gridon;axisequalforj=1:2:M-3quiver3(r(j,1),r(j,2),r(j,3),0,0,k(j),b');endlegend('控制多边形','B-样条曲

11、线','曲率)；%率计算t=zeros(size(ft3,1),1);fori=1:size(ft3,1)nn1=ft1(i,:);nn2=ft2(i,:);nn3=ft3(i,:);ta=det(nn1;nn2;nn3);t(i)=ta/kaA2;end%在点上表示挠率大小figure(3)plot3(P(:,1),P(:,2),P(:,3),'-ko',r(:,1),r(:,2),r(:,3),'-r.');holdon;gridon;axisequalforj=2:2:M-310/18quiver3(r(j,1),r(j,2),r(j,3)

12、,0,0,t(j),b');endlegend('控制多边形','B-样条曲线',挠率)；%fc点上表示其Frenetic标架figure(4)plot3(P(:,1),P(:,2),P(:,3),'-ko',r(:,1),r(:,2),r(:,3),'-r.');holdon;gridon;axisequalforj=1:M-2alpha=ft1(j,:)/norm(ft1(j,:);quiver3(r(j,1),r(j,2),r(j,3),alpha(1),alpha(2),alpha(3),'b');

13、beta=ft2(j,:)/(k(j)*norm(ft2(j,:)A2);beta=beta/norm(beta);quiver3(r(j,1),r(j,2),r(j,3),beta(1),beta(2),beta(3),'g');gama=cross(alpha,beta)/norm(cross(alpha,beta);quiver3(r(j,1),r(j,2),r(j,3),gama(1),gama(2),gama(3),'m');endlegend('控制多边形','B-样条曲线','alpha','

14、beta','gama');非均匀有理B样条曲面(3阶)P1,P2,P3,F4,已经四个权因子w,w，wi,W4,通过Matlab计算其基函数，并作出曲线。Clear；clcfid=fopen('Data.dat','r');k=1;s=fscanf(fid,'%s',1);%读字符串1维num=0;%NUMB行数Dotnum=zeros(1);%保存每个NUM勺点的个数Dot=cell(1);%保存每个NUM勺点的坐标whilek=111/18ifstrcmp(s,'NUM')a=fscanf(fid,&

15、#39;%d',2);%读整型数据2维num=num+1;k=1;elsek=0;break;endb=fscanf(fid,'%f,3*a(2);%读浮点型数据3*a(2)维Dotnum(num)=a(2);b=reshape(b,3,a(2);%将b转化为3行19列Dotnum=b'%clearaclearbend%hold;view(3);s=fscanf(fid,'%s',1);%读字符串1维u=0:0,08:1;v=0:0,04:1;a1=(1-3*u+3*u.*u-u.*u.*u);a2=(3*u-6*u.*u+3*u.*u.*u);a3=(

16、3*u.*u-3*u.*u.*u);a4=u.*u.*u;b1=(1-3*v+3*v.*v-v.*v.*v);b2=(3*v-6*v.*v+3*v.*v.*v);b3=(3*v.*v-3*v.*v.*v);b4=v.*v.*v;aa1=(-3+6*u-3*u.*u);aa2=(3-12*u+9*u.*u);aa3=(6*u-9*u.*u);aa4=3*u.*u;bb1=(-3+6*v-3*v.*v);bb2=(3-12*v+9*v.*v);bb3=(6*v-9*v.*v);bb4=3*v.*v;aaa1=(6-6*u);aaa2=(-12+18*u);aaa3=(6-18*u);aaa4=6*

17、u;bbb1=(6-6*v);bbb2=(-12+18*v);bbb3=(6-18*v);bbb4=6*v;12/18forh=1:length(u)a(h,:)=a1(h)a2(h)a3(h)a4(h);aa(h,:)=aa1(h)aa2(h)aa3(h)aa4(h);aaa(h,:)=aaa1(h)aaa2(h)aaa3(h)aaa4(h);endfork=1:length(v)b(k,:)=b1(k)b2(k)b3(k)b4(k);bb(k,:)=bb1(k)bb2(k)bb3(k)bb4(k);bbb(k,:)=bbb1(k)bbb2(k)bbb3(k)bbb4(k);end%fori

18、=1:num-3forj=1:Dotnum(num)-3xc=Doti(j,1),Doti(j+1,1),Doti(j+2,1),Doti(j+3,1);Doti+1(j,1),Doti+1(j+1,1),Doti+1(j+2,1),Doti+1(j+3,1);Doti+2(j,1),Doti+2(j+1,1),Doti+2(j+2,1),Doti+2(j+3,1);Doti+3(j,1),Doti+3(j+1,1),Doti+3(j+2,1),Doti+3(j+3,1);yc=Doti(j,2),Doti(j+1,2),Doti(j+2,2),Doti(j+3,2);Doti+1(j.2),

19、Doti+1(j+1,2),Doti+1(j+2,2),Doti+1(j+3,2);Doti+2(j,2),Doti+2(j+1,2),Doti+2(j+2,2),Doti+2(j+3,2);Doti+3(j,2),Doti+3(j+1,2),Doti+3(j+2,2),Doti+3(j+3,2);zc=Doti(j,3),Doti(j+1,3),Doti(j+2,3),Doti(j+3,3);Doti+1(j.3),Doti+1(j+1,3),Doti+1(j+2,3),Doti+1(j+3,3);13/18Doti+2(j,3),Doti+2(j+1,3),Doti+2(j+2,3),Do

20、ti+2(j+3,3);Doti+3(j,3),Doti+3(j+1,3),Doti+3(j+2,3),Doti+3(j+3,3);A=a*xc*b'%曲面上点的坐标赋值B=a*yc*b'C=a*zc*b'rA=A;rB=B;rC=C;surf(A,B,C),shadinginterp;colormap(jet)%彩色显示endendfori=1:num-3forj=1:Dotnum(num)-3xc=Doti(j,1),Doti(j+1,1),Doti(j+2,1),Doti(j+3,1);Doti+1(j,1),Doti+1(j+1,1),Doti+1(j+2,1)

21、,Doti+1(j+3,1);Doti+2(j,1),Doti+2(j+1,1),Doti+2(j+2,1),Doti+2(j+3,1);Doti+3(j,1),Doti+3(j+1,1),Doti+3(j+2,1),Doti+3(j+3,1);yc=Doti(j,2),Doti(j+1,2),Doti(j+2,2),Doti(j+3,2);Doti+1(j,2),Doti+1(j+1,2),Doti+1(j+2,2),Doti+1(j+3,2);Doti+2(j,2),Doti+2(j+1,2),Doti+2(j+2,2),Doti+2(j+3,2);Doti+3(j,2),Doti+3(j

22、+1,2),Doti+3(j+2,2),Doti+3(j+3,2);zc=Doti(j,3),Doti(j+1,3),Doti(j+2,3),Doti(j+3,3);Doti+1(j,3),Doti+1(j+1,3),Doti+1(j+2,3),Doti+1(j+3,3);Doti+2(j,3),Doti+2(j+1,3),Doti+2(j+2,3),Doti+2(j+3,3);Doti+3(j,3),Doti+3(j+1,3),Doti+3(j+2,3),Doti+3(j+3,3);forh=1:length(u)fork=1:length(v)14/18A=a(h,:)*xc*b(k,:)

23、'%曲面上点的坐标赋值B=a(h,:)*yc*b(k,:)'C=a(h,:)*zc*b(k,:)'Au=aa(h,:)*xc*b(k,:)'Av=a(h,:)*xc*bb(k,:)'Bu=aa(h,:)*yc*b(k,:)'Bv=a(h,:)*yc*bb(k,:)'Cu=aa(h,:)*zc*b(k,:)'Cv=a(h,:)*zc*bb(k,:)'Auu=aaa(h,:)*xc*b(k,:)'Avv=a(h,:)*xc*bbb(k,:)'Buu=aaa(h,:)*yc*b(k,:)'Bvv=a(h,

24、:)*yc*bbb(k,:)'Cuu=aaa(h,:)*zc*b(k,:)'Cvv=a(h,:)*zc*bbb(k,:)'Auv=aa(h,:)*xc*bb(k,:)'Buv=aa(h,:)*yc*bb(k,:)'Cuv=aa(h,:)*zc*bb(k,:)'Sz=Au,Bu,Cu;Sf=Av,Bv,Cv;Sff=Avv,Bvv,Cvv;Sfz=Auv,Buv,Cuv;Szz=Auu,Buu,Cuu;N=cross(Sz,Sf);n=N/norm(N,2);E(h,k)=dot(Sz,Sz);F(h,k)=dot(Sz,Sf);G(h,k)=do

25、t(Sf,Sf);L(h,k)=dot(Szz,n);M(h,k)=dot(Sfz,n);N(h,k)=dot(Sff,n);f2(h,k)=L(h,k)*N(h,k)-M(h,k)*M(h,k);f1(h,k)=E(h,k)*G(h,k)-F(h,k)*F(h,k);K(h,k)=f2(h,k)/f1(h,k);%高斯曲率H(h,k)=L(h,k)*G(h,k)-2*M(h,k)*F(h,k)+N(h,k)*E(h,k)/(f1(h,k);%平均曲率k1(h,k)=H(h,k)+sqrt(H(h,k)A2-K(h,k);k2(h,k)=H(h,k)-sqrt(H(h,k)A2-K(h,k);

26、%»%bl=1000;if(H(h,k)=NaN)break;15/18elseAD(h,k)=A+bl*1.0/H(h,k)*n(:,1);BD(h,k)=B+bl*1.0/H(h,k)*n(:,2);CD(h,k)=C+bl*1.0/H(h,k)*n(:,3);line(AAD(h,k),BBD(h,k),CCD(h,k),'Color','b');endendendendendfigure(2);surf(rA,rB,rC),shadinginterp;colormap(jet)%彩色显示holdon;gridon;fori=1:size(rA,

27、1)forj=1:size(rA,2)quiver3(rA(i,j),rB(i,j),rC(i,j),0,0,E(i,j),'b');endendfigure(3);surf(rA,rB,rC),shadinginterp;colormap(jet)%彩色显示holdon;gridon;fori=1:size(rA,1)forj=1:size(rA,2)quiver3(rA(i,j),rB(i,j),rC(i,j),0.0,F(i,j)/20,'b');endendfigure(4);surf(rA,rB,rC),shadinginterp;colormap(jet)%彩色显示holdon;gridon;16/18fori=1:size(rA,1)forj=1:size(rA,2)quiver3(rA(i,j),rB(i,j),rC(i,j),0.0,G(i,j)/20,b');endendfigure(5);surf(rA,rB,rC),shadinginterp;colormap(jet)%彩色显示holdon;gridon;fori=1:size(rA,1)forj