三次B样条反求控制点

32320105JOURNALOFTAISHANUNIVERSITYVo.l32NO.3May2010B王凌云(山东服装职业学院科研所,山东泰安271000)[]反求工程作为复杂工业产品设计与制造的重要技术手段之一,深受CAD/CAM领域的广泛重视,特别是自由曲面重构技术作为复杂曲面产品反求工程中的瓶颈问题,是今后一个时期的研究热点.反求工程中最为关键的技术就是曲面重构技术.目前主要有两大重构方法:NURBS曲面重构和三角Bezier曲面重构.本文围绕NURBS曲面重构展开研究,提出一种反算三次B样条曲线曲面控制顶点的简便算法.该算法适用于准均匀和非均匀B样条曲线曲面的反算.算法采用非节点边界条件,不需要由用户提供,从而使反算过程得以简化.[]反求工程;非均匀有理B样条;型值点;控制点[]TP301.6[]A[]1672-2590(2010)03-0040-04[]2010-04-06[](1970-),,,.0CACG,,.NURBS,,,B(NURBS),1991NURBS.NURBS:,.,,.,,,.NURBS,.1B1.1BB,,BB,.B.B:p(u)=ni=1diNi,k(u)i=0,1,!,n(1)di(i=0,1,!,n),,B,Ni,k(u)(i=0,1,!,n)kB,U=[u0,u1,!,un+k+1]Cox-DeBoorkB,:3王凌云:三次B样条反求控制点Ni,0=1Ui∀U∀Ui+10Ni,k(u)=u-uiui+k-uiNi,k-1(u)+ui+k+1-uui+k+1-ui+1Ni+1,k-1(u)00=0,k=3NURBS.B,,,,,B.1.2Ba),.b)B.c).d),.,B.1.2.1kBqi(i=0,1,!,m),,B.qiuk+i(i=0,1,!,m).1.2.2B,pi(i=0,1,!,n)NURBS.NURBSn+3di(i=0,1,!,n+2),U=[u0,u1,!,un+6].ui+3(i=0,1,!,n),.ui:(1):u0=u1=u2=u3=0,ui+3=i/ni=1,2,!!,n-1,un+3=un+4=un+5=un+6=1.(2):u0=u1=u2=u3=0,ui+3=ui+2+|pi-pi-1|/ni=1|pi-pi-1|i=1,2,!,n-1.un+3=un+4=un+5=un+6=1.(3):u0=u1=u2=u3=0,ui+3=ui+2+|pi-pi-1|/ni=1|pi-pi-1|i=1,2,!,n-1.un+3=un+4=un+5=un+6=1.1.2.3Bn+1pi,i=0,1,!,n.p0pnB,p1,p2,!,pn-1B,n.,Bbi,i=0,1,!,n+2n+3.U=[u0,u1,!,un+6],u#[u3,un+3].B,B,B4,B:41泰山学院学报32pi(ui+3)=Bi(ui+3)Bi+1(ui+3)Bi+2(ui+3)Bi+3(ui+3)ViVi+1Vi+2Vi+3ui+3,i[0,m-4],m-3.,,p0=V0;pm-3=Vm-1.:p0=V0pi=[Bi(ui+3)Bi+1(ui+3)Bi+2(ui+3)Bi+3(ui+3)]ViVi+1Vi+2Vi+3i[1,m-4]pn-1=Vm-1mVj,m-2.,:C2B,,q0=qm,,,,dn-2=d0,dn-1=d0,dn=d2;,,,B1,3(u3)B2,3(u3)B0,3(u3)B1,3(u4)B2,3(u3)B3,3(u3)!!Bn-4,3(u3)Bn-3,3(u3)Bn-2,3(u3)Bn-1,3(u3)Bn-3,3(u3)Bn-2,3(u3)V1V2!!Vn-3Vn-4=q0q1!!qn-4qn-3,.Bq0=qmC2BB,n-1n+1,,B.2B2.1BB,Qij(i=0,1,!,m;j=0,1,!,n);Bpij(i=-1,0,1,!,m+1;j=-1,0,1,!,n+1):1)un+1,B,n+1B,,(m+3)*(n+1)Vij(i=-1,0,1,!,m+1;j=-1,0,1,!,n+1).2)Vijv(m+3)B,Bpij(i=-1,0,1,!,m+1;j=-1,0,1,!,n+1).0,pij(i=0,1,!,m+2;j=0,1,!,n+2).2.2,C2,,,.:.423王凌云:三次B样条反求控制点,,,;,,,.,.,.,,,,,.,,.,.,,,.,.3,,,..,..,,.,.BB.:,,.,.[][1].B[J].,1996,(3):84-90.[2].B[M].:,1994.[3],.NURBS[J].,2006,(4):357-359.[4].NURBS[D].:,2003.[5].cad[D].:,2004.[6],.NURBS[J].,2000,32(2).[7].B[D].:,2003.CalculatingControlPointsofNURBSWANGLing-yun(InstituteofScientificResearch,ShandongFashionCollege,Taia'n,271000,China)Abstract:Reverseengineering(RE)isakeytechnologyofadvancedmanufacturing.Asoneofimportanttechnicalwaystodesignandmanufacturecomplexindustrialproducts,reverseengineeringistakenintoaaccountwidelyinthefieldofCAD/CAM.Especially,free-formsurfacereconstructionisachallengingtaskandwillcatchmoreattentioninfuturework.SurfacereconstructionisthemostimportanttechnologyinRE.Therearetwoprevailingsurfacereconstructionmethods.TheyareNURBSsurfacereconstructionmethodandtrianglesurfacereconstructionmethod.ThisthesismakessomeresearchworkonNURBSsurfacereconstruction.OurpaperproposedasimplemethodwhichcancalculatethecontrolpointsinNURBScurvesandsurfaces.Thisarithmeticappliestoquasi-uniformB-splineandnon-uniformB-spline.Thearithmeticusesnon-nodeboundaryconditionsanddoesnotneedinteraction,whichcansimplifytheoperationprocess.Keywords:reverseengineering;non-uniformB-spline;characteristicpoin;tcontrolpoints43