过控制顶点的B样条曲线

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.()JournalofSouthernYangtzeUniversity(NaturalScienceEdition)Vol.2No.6Dec.2003:1671-7147(2003)06-0553-04:2003-10-09;:2003-11-201:(50105013)1:(1963-),,,,1B1,2,3,3(11,214036;21,515063;31,510090):B,B,,,,1:B;;:TP391.41:ATheB2SplineCurvePassingControlPointsLINYi1,XIONGHan2wei2,LUOShao2ming3,ZHANGXiang2wei3(11SchoolofInformationTechnology,SouthernYangtgeUniversity,Wuxi214036,China;2.SchoolofMechano2elec2tronicEngineering,ShantouUniviersity,Shantou515063,China;3.SchoolofMechano2electronicEngineering,Guang2dongpolytechnicUniversity,Guangzhou510090,China)Abstract:ThegeneralB2splinecurvedoesnotpassthecontrolpoints.ButtheB2splinecurvecanpassthecontrolpointswhencorrectingshapeofcurve.ThisisAnti2calculateandAnequationcanbesolved.However,itisdifficultytocorrectshapeofcurvereal2time.Inthispaper,anewarithmeticabouthowtogetB2splinecurvewhichpassthecontrolpointsandneednottosolveequation,hasbeenproposed.Itiseasywiththearithmetictocorrecttheshapeofcurvereal2time.Keywords:B2spline;controlpoints;anti-calculateBezierBNURBS,[1,2],1,,,[3]1,1,B,1,,1B,,,B;,,,11BBB,B1©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(t-x)k+=(t-x)ktx0tx2Mi,k(x)=[xi,xi+1,,xi+k+1](t-x)k+k(k+1)B,Ni,k(x)=(xi+k+1-xi)Mi,k(x)kB,,B1,3(4)BNi,3(u)=0x(xi,xi+4)16u3x[xi,xi+1]16(1+3u+3u2-3u3)x[xi+1,xi+2]16(4-6u2+3u3)x[xi+2,xi+3]13(1-u)3x[xi+3,xi+4](1)u=x-xj,u[0,1](j=i,i+1,i+2,i+3)B,Bp(u)=ni=0diNi,k(u)di(i=0,1,,n)1B,:,11,BSk(u)=12(1,u,u2)110-2201-21dkdk+1dk+2(2):dk,0u1,k=0,1,,n-21Sk(u)dkdk+1bk,bkdkdk+112Bb0,b1,,bn+1,di(i=1,2,,n),bkdkdk+11d0=(x0,y0,z0),d1=2b0-d0,b0d0d11,dk=2bk-1-dk-1(k=1,,n)bk-1dk-1dk1,dk(k=0,1,,n),dk,ÛS(0)=a0(2)S0(u)=12[(1-2u+u2)d0+(1+2u-2u2)d1+u2d2]=(-2u+2u2)d0+(1+2u-4u2)b0+u2b1,ÛS(u)=(-2+4u)d0+(2-8u)b0+2ub1,ÛS(0)=-2d0+2b0,d0=12(2b0-a0)1,b0,b1,,bn-1,:12(2b0-a0),(2b0-a0),,dk=(2bk-1-dk-1)1,B,S0(u)=12(1,u,u2)110-2201-2112(2b0-a0)b0+12a02b1-b0-12a0Sk(u)=12(1,u,u2)110-2201-21dkdk+1dk+21:dk=2bk-1-dk-1k=1,,n-2;u[0,1];a0;bk(k=0,1,,n)1,13B(1),Sl-1(1)=Sl(0),ÛSl-1(1)=ÛSl(0),Sl-1(1)=Sl(0),C21dlbl,B1,n+1pi(i=0,,n+3)BSi(t)=16(1-3t+3t2-t3)di+16(4-6t2+3t3)di+1+16(1+3t+3t2-3t3)di+2+16t3di+3,Si(0)=pi(i=0,,n),16d0+46d1+16d2=p016d1+46d2+16d3=p116dn-3+46dn-2+16dn-1=pn-3:di,pi1n-2,n,,455()2©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(0)=q0,ÛSn-3(0)=qn-3,12(-d0+3d2)=q012(-dn-3+3dn-1)=qn-312(-d0+3d2)=q016d0+46d1+16d2=p016d1+46d2+16d3=p116dn-3+46dn-2+16dn-1=pn-312(-dn-3+3dn-1)=qn-3-120320464616461629460-12032d0d1d2dn-2dn-1=q0p0+13q0p1pn-3-19qn-3qn-3,31141141141434d2d3dn-3dn-2=6p16p26pn-3-23qn-3:4d1+4d2=6p0+2q0,12(-d0+3d2)=q0,12(-dn-3+3dn-1)=qn-3,LU:L=122n-2n-2,U=11121n-311:i=1(i=2,3,,n-3);n-2=43;1=3;k=4-1k-1(k=2,3,,n-3);n-2=4-43n-3;k=1k(k=1,2,,n-3);n-2=43n-211111111-1=112131n-31232n-31ij=(-1)j-iii+1j-1112131n-443n-3-1=11(-1)1212(-1)2123(-1)2313(-1)n-51n-4(-1)n-62n-4(-1)n-73n-41n-44(-1)n-431n-34(-1)n-532n-34(-1)n-633n-34(-1)3n-4n-31n-35556:B©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=(-1)j-1ij112131n-443n-3-1=112122313233n-41n-42n-43n-4n-443n-3143n-3243n-3343n-3n-4n-3n-3d2d3dn-2=112131n-31232n-31112122313233n-41n-42n-43n-4n-443n-3143n-3243n-3343n-3n-4n-3n-36p16p26pn-3-23qn-1,di(i=1,,n-2),,14,1113Fig.1Thethreetypecurves115GHz,[4],1B,,,;,,,1111Tab.1ThecomparisonofsomealgorithmsBBBB,B,,,1:[1]1B(CAGD&NURBS)[M]1:,19941[2]1[M]1:,20001[3]LAXMIPARIDA.AComputationaltechniqueforgeneralshapedeformationforuseinfontdesign[J].Computer&Graphics,1993,17(4):349-3561[4]1[M]1:,20011(:)655()2