基于周期性延伸的三次B样条闭曲线插值

2009124012B*李学艺1,2王钊2连小珉2曾庆良1(1,266510;2,100084),B,,,,:B:TP39172:AInterpolationofCubicB-splineClosedCurveBasedonPeriodicExtensionLiXueyi1,2WangZhao2LianXiaomin2ZengQingliang1(1SchoolofMechanicalandElectronicEngineering,ShandongUniversityofScience&Technology,Qingdao266510,China2DepartmentofAutomobileEngineering,TsinghuaUniversity,Beijing100084,China)AbstractAccordingtotheperiodicextensionfactforclosedcurve,amethodofcubicB-splineinterpolationforclosedcurvewasproposed.Byextendingknotsandcontrolpoints,theinterpolatedcurveclosedentirelyatclosingpoint.Consideringthecharacterofincompletebondmatrixforcoefficientmatrixofinterpolationsystemofequations,analgorithmforsolvinglinearbandedsystemofequationswithtwocolumnsofnon-zeroelementswasintroducedtoimprovethecomputationalefficiency.Examplesofapplicationshowthattheproposedtechnologyhasthestableperformanceandhighefficiency,canbeusedtointerpolatecomplexclosedcurvewithwhatevershape,anditisespeciallyusefulforproblemswithlargenumberofdatapoints.KeywordsClosedcurve,CubicB-splineinterpolation,Bandmatrix,Periodicextension:2008-10-06:2008-12-17*(JQ200816)(J07YA04):,,,,E-mail:[e-mail],,,,,G1G2[1~2],,,,,,,,,B1BCqh(h=0,1,,N-1;q0=qN-1),B[3],CBC(uj)=mji=mj-3piBi(uj)=qj-3(j=3,4,,N+2)(1)mj=j(j=3,4,,N+1)N+1(j=N+2)piBujqh(h=0,1,,N-1)mjujUBiB(1)B,N,N+2,2,BB[4],B,B11BB,,3[5]N,U=(u0,u0,u0,u3,,ui,,uN+2,uN+5,uN+5,uN+5),,,,B,,Bu0=u3-uN-1u1-u3-uNu2=u3-uN+1(2)uN+3=uN+2+u4uN+4=uN+2+u5uN+5=uN+2+u6(3),,12BB,N,N+2,,3,3p0=pN-1p1=pNp2=pN+1(4)(4)2,(1)p0-pN-1=0mji=mj-3piBi(uj)=qj-3p2-pN+1=0(j=3,4,,N+2)(5)(5),BAP=E(6)P=(p0,p1,,pN+1)TE=(0,q0,q1,,qN-1,0)TPEAA2,i(i=1,2,,N),uiB(ui),B,4(7),,2N-1AD,,c2cN-1,2N-1,AA=[Dc2cN-1](8)D,D=[Gs](9)s=(s0,s1,,sN+1)TG=[gT0,gT1,,gTN+1]TsGsi2522009si=0(i=0)i-1(0iN+1)N-1(i=N+1)(10)gi4,gi=(1,0,0,0)(i=0)B(ui)(0iN+1)(0,0,0,1)(i=N+1)(11)c2cN-1c2=(0,,0-1)N+2cN-1=(-1,0,,0)N+2(12)A,(6),,B2Ax=b(13)A=[ai,j](i,j=0,1,,N-1)x=(x0,x1,,xN-1)Tb=(b0,b1,,bN-1)Txb1Fig.1Bandmatrixwithtwocolumnsofnon-zeroelementsA,,,1,,w;,mnA,22Fig.2Partitionmatrixesofincompletebandmatrix(a)D(b)CAA=D+C(14)C=0,,0m,cm,0,,0n-m-1,cn,0,,0N-nN(15)0=(0,0,,0)TNcm=(c0,m,c1,m,,cN-1,m)Tcn=(c0,n,c1,n,,cN-1,n)Tcmcn0N(14)(15)(13),Dx=b-xmcm-xncn(16)(16)D-1,x=r-xms-xnt(17)r=D-1bs=D-1cmt=D-1cnrtxmxn(17)mn,xm=rm-smxm-tmxnxn=rn-snxm-tnxn(18)(18),xmxn(17),x3B,VisualStudio2005IGES,,,21:C1q0=(-185858,204025,-50000),q1=(-13974,328396,21525),q2=(129961,122974,21525),q3=(-199833,-209615,100189),q4=(336781,-230576,57750),q5=(269704,60089,25724),q6=(-139743,100615,-55660),q7=(-185858,204025,-50000)BU=(-03590,-02297,-00489,0,00965,02062,04087,06410,07703,09511,10000,10965,12062,14087),p0=(-90855,-25637,-61296),p1=(-228365,247173,-54830),p2=(12226,409251,39879),p3=(272933,25312:B13149,69775),p4=(-563221,-234178,123097),p5=(470378,-424273,58772),p6=(292845,223362,35871),p7=(-90855,-25637,-61296),p8=(-228365,247173,-54830),p9=(12226,409251,39879),003ms3aMasterCamX2,,2:C2q0=(-340190,-1997363,-28000),q1=(-208333,-1883966,-24630),q2=(-279536,-1464662,-14928),q3=(18460,-1409283,08313),q4=(-10549,-1894515,17698),q5=(548523,-1923523,13178),q6=(627637,-2158228,09355),q7=(76477,-2189873,-02601),q8=(210970,-2632911,-11814),q9=(39557,-2886076,-14748),q10=(-2373417,-2640823,-17052),q11=(-187236,-2337552,-18648),q12=(-706751,-2258439,-15298),q13=(-949368,-2440401,-04331),q14=(-1010021,-2002637,-04965),q15=(-606540,-1968354,-22428),q16=(-340190,-1997363,-28000)3b,005ms3Fig.3Closedcurveforinterpolation(a)C1(b)C24B,,,,,,k(k3)B1SchaeferS,JuT,WarrenJ.Aunified,integralconstructionforcoordinatesoverclosedcurves[J].ComputerAidedGeometricDesign,2007,24(8~9):481~493.2.C-BezierB-[J].,2002,23(4):303~309.WangChengwei.ClosedC-BeziercurveandB-typesplinecurvewithgiventangentpolygon[J].JournalonNumericalMethodsandComputerApplications,2002,23(4):303~309.(inChinese)3ParkH.ChoosingnodesandknotsinclosedB-splinecurveinterpolationtopointdata[J].Computer-aidedDesign,2001,33(13):967~974.4PieglL,TillerW.Thenurbsbook[M].2nded.Berlin:SpringerVerlag,1997.5DeBoorC.Apracticalguidetosplines[M].Berlin:SpringerVerlag,2001.2542009