样条曲面拟合及其Matlab实现_胡玉兰

Matlab胡玉兰,李沫沫(,650031)　　z=f(x,y)f(xi,yi),Matlab,,,.　;;;Matlab【】O174.4【】A【】1672—8513(2005)02-0168-04SurfaceFittingbySplinesusingMatlabHUYu-lan,LIMo-mo(SchoolofMathematicsandComputerScience,YunnanNationalitiesUniversity,Kunming650031,China)Abstract:Inthispaper,wediscusssurfacefittingbycubicsplinesinterpolationusingMatlabandwediscussthemothedsofcardinalsplinesinterpolationandmore-knotscardinalsplinesinterpolation.FurtherweoffertheMatlab.procedureofsmoothsurfacefittingbycubicsplineinterpolation,whichisnew.Keywords:surface;spline;fitting;Matlab.1　,[a,b]Δ,a=x0x1x2…xn-1xn=b.s(x):(1)[xi-1,xi]k;(2)[a,b]k-1.s(x)x0,x1,…,xnk.x0,x1,xnkSk(Δ),:dimSk(Δ)=n+ks(x):(3)s(xi)=yi(i=0,1,2,…,n),k.k=333.S3(Δ)3,4n,(1)(2)(3)3(n-1)+(n+1)=4n-2,2,,:Ⅰ:s′(x0)=y′0,s′(xn)=y′n;Ⅱ:s″(x0)=y″0,s″(xn)=y′n″;Ⅲ:s′(x0)=s′0,s″(xn)=s″(x0).ΔΔ:x-kx-k+1…x-1x0x1…xnxn+1…xn+kA:{1,x,x2,…,xk,(x-xi)k+,i=1,2,…,n-1}B:{(x-xi)k+,i=-k,-k+1,…,n-1}C:{Ni,k(x),i=-k,-k+1,…,n-1}:xk+=xk,x≥00,x0Nik,(x)=(xi+k-xi)∑i+kj=iwij(xj+x)k-1+wij=∏i+kl=i,l≠j1xj-xlNi,k(x):(i)Ni,k(x)=x-xixi+k+1-xiNi,k-1(x)+168:2004-06-11:(1963～),(),,,.14220054　　()JournalofYunnanNationalitiesUniversity(NaturalSciencesEdition)Vol.14,No.2Apr.2005xi+k-xxi+k-xi+1Ni+1,k-1(x)Ni,1=1,x∈(xi,xi+1)0,others(ii)Ni,k(x)=0,x∈(xi,xi+k)=0,x[xi,xi+k],i=-k+1,…,n-1(iii)xj+i=xi+jh,Ni,k(x)=Ψk(x-xih-k+12)Ψk(x)=∑k+1j=0(-1)jCjk+1(x+k+12-j)k+/k!xj=j-k+12(j=0,1,…,k+1)kB.Ni,k(x)[1,2,3,4](3)ks(x)s(x)=∑n-1i=-kciNi,k(x),(),,,3.2　　　,{Li(x)},Li(j)=1,i=j0,i≠j,Li(x),∑yiLi(x),,.L0(x)=∑+∞-∞cjφj(x),φj(x)=Ψ3(x-j)cjL0(i)=1,i=00,i≠0,,i.,cj:16ci+1+23ci+16ci-1=δ0i=1,i=00,i≠0λ2+4λ+1=0,λ1=1λ2=-2+3,cj=αλj1+βλj2,cj※0(|j|※+∞),cj,cj=αλj1(j≥0),L0(x),c-j=cj.j=0α=3,cj=c-j=3(-2+3)j,(j≥0),Li(x)=∑+∞j=-∞cjφj(x-i)=∑+∞j=-∞cjΨ3(x-i-j){Li(x)}.h,φj(x)=Ψ3(x-xjh)s3(x)=∑n+1i=-1yiLi(x),(x0=a≤x≤b=xn)(1)s3(xi)=yi(i=1,2,…,n)(I):s′3(x0)=y′0,s′3(xn)=y′n,(1)y-1,yn+1..1:{Li(x)}[5,6].,,{xi}ni=0.Ψj/k+12k+1(x)=[Ψ2k+1(x+jk+1)+Ψ2k+1(x-jk+1)]/2,j=0,1,…,k+1.q2k+1(x)=∑k+1j=0βjΨj/k+12k+1(x),βj(j=0,1,…,k+1)q2k+1(x),q2k+1(0)=1,q2k+1(l)=0,l=1,2,…,k,k+1,q2k+1(x)..suppΨwk+1[-k-1,k+1]Ψ2k+1((k+1))=0suppq2k+1[-k-2,k+2]q2k+1((k+2))=0.m=0,β0=1,β1=1;m=1,β0=10/3,β1=-8/3,β2=1/3.,(xi,yi)i=0,1,…,n(xi=x0+ih)()(xi,yi),i=-1,0,1,…,n,n+1,3:s3(x)=∑n+1j=-1yjq3(x-xjh)1692　　　　　　　　　　　　　,:Matlaby-1,yn+1.,::f∈[a,b],|f(x)-s(x)|=0(h4),.3　{(xi,yj)},(i=0,1,2,…,n;j=1,2,…,m){rij},xi=x0+ih,yj=y0+jτ,:s(x,y)=∑n-1i=-k∑m-1j=-lcijNi,k(x)Nj,l(y).,k,l=3.s(x,y)=∑ni=0∑mj=0rijΨk(x-x0h-i)Ψly-y0τ-j)[7]s(x,y)=∑ni=0∑mj=0rijLi(x)Lj(y)s(x,y)=∑ni=0∑mj=0rijq3(x-xih-i)q3y-yiτ-j).2:.Be′zierCoons.3:C2(Ψ)().fs3C2(Ψ),:　　s3(xi,yj)=f(xi,yj),0≤i,j≤n,s3x(xi,yj)=fx(xi,yj),0≤j≤n,i=0,n,s3y(xi,yj)=fy(xi,yj),0≤j≤n,i=0,n,2s3xy(xi,yj)=2fxy(xi,yj),i,j=0,ns3(x,y)=∑n-1i=-3∑n-1j=-3cijNi,3(x)Nj,3(y),‖f-s3‖≤β0h4,f∈C2(Ψ),β0f.[8,9].4　Matlab6.1、,[10];Matlab..Matlab():csapicsapecsapscscvngetcurve　　csapi::values=csapi(x,y,xx)pp=csapi(x,y):x,s(x(j))=y(j)s,csapeconds='not-a-knot'.,csapi(x,y,xx)s(x(j))=y(j)s,xx,values.Caspi(x,y)s(x(j))=y(j)s,pp.ppfnval,fnder.xmx1,…,xm,,{{a1,a2,a3},{a4,a5},{a6,a7,a8,a9},},xn1,…nm,,y,[n1,…,nm],-m,,pp,,xxs(xx),xxm,m.::x=0.0001+[-4:0.2:4];y=-3:0.2:3;[yy,xx]=meshgrid(y,x);r=pi＊sqrt(xx.2+yy.2);z=sin(r+3)./4+3;bcs=csapi({x,y},z);fnplt(bcs),axis([-5,5,-5,5,-0.5,1])1.170()　　　　　　　　　　　　　　14,:　forn=1:1:50　forj=1:1:40　x(n)=(n-25)/10　ifabs((x(n)+2.5)＊10-j)=2　　y(n,j)=0;　elseifabs((x(n)+2.5)＊10-j)=1　　y(n,j)=(1/2)＊(abs((x(n)+2.5)＊10-j))2-((x(n)+2.5)＊10-j)2+(2/3);　elsey(n,j)=(-1/6)＊(abs((x(n)+2.5)＊10-j))3+((x(n)+2.5)＊10-j)2-2＊abs((x(n)+2.5)＊10-j)+(4/3);endendendA=[y(1,:);y(2,:);y(3,:);y(4,:);y(5,:);y(6,:);y(7,:);y(8,:);y(9,:);y(10,:);…　　y(11,:);y(12,:);y(13,:);y(14,:);y(15,:);y(16,:);y(17,:);y(18,:);y(19,:);y(20,:);…y(21,:);y(22,:);y(23,:);y(24,:);y(25,:);y(26,:);y(27,:);y(28,:);y(29,:);y(30,:);…y(31,:);y(32,:);y(33,:);y(34,:);y(35,:);y(36,:);y(37,:);y(38,:);y(39,:);y(40,:);…y(41,:);y(42,:);y(43,:);y(44,:);y(45,:);y(46,:);y(47,:);y(48,:);y(49,:);y(50,:)];u=-2:0.1:1.9;v=-2:0.1:1.9;[uu,vv]=meshgrid(u,v);r=pi＊sqrt(uu.2+vv.2);F=sin(r+3)./(r+3);S=A＊F＊A';mesh(S)2.:[1]　,.[M].:,1979.25-45.[2]　.[M].:,1983.82-105.[3]　.[M].:,1982.149-190.[4]　LarrySchumaker.SplineFunctions-basictheory[M].JohnWiley&SonsInc.Canada.1981.108-188.[5]　SchoenbergIJ.CardinalInterpolationandSplineFunctions:II[J].J.ofApprox.Theory,1972(6):404-420.[6]　.Cardinal[J].(),2001,9(1):7-8.[7]　.-[M].:,1993.1-26.[8]　,,.[M].:,1985.223-273.[9]　P.M..[M].,,.:,1980.110-160.[10]　,.Matlab6.1()[M].:,2002.128-132.(　杨多立)1712　　　　　　　　　　　　　,:Matlab