5.Coons曲面

Coons曲面和双三次样条插值主要内容1.Coons曲面2.双三次样条插值3.曲面的基本方法1.Coons曲面1.1Coons的设计思想1.2具有给定边界的Coons曲面1.3具有给定边界及跨界斜率的Coons曲面1.1Coons的设计思想•若干曲面片拼合成要连接的曲面。每个曲面片用四个边界曲线来定义；•设计人员从单张曲面或很少的曲面片开始设计•如果曲面片不满足设计意图，可以修改曲线形状，或添加曲线，将原曲面片划分为更小的曲面片参数曲面的概念r(u,v)=[x(u,v),y(u,v),z(u,v)]0=u,v=1011uw(u,v))0,0(r)1,0(r)0,1(r)1,1(r),(vur)0,(ur),0(vr(1,)rvu和v向切矢：混合偏导矢（扭矢）：u(u,v)r(u,v)ruv(u,v)r(u,v)rv跨界斜率：边界曲线上的v向的偏导矢)(u,0r0vvvvu,ru,0r)()()(00uv,ur)(00vv,ur)(u,0rvvuvu,rvu,r2vu,)()()(00v,ur1.2具有给定边界的Coons曲面•问题：给定两点，以这两点为端点的线段如何表示？0P1P10)1(uPPu]1,0[u)(uP令：uuFuuF)(,1)(101100)()()(PuFPuFuPP(u)有什么特点呢？导函数点点相同：0)('')()()(011100uPPPPuFPuFuP1)()(,0)(,0)(1)1(,0)0(,0)1(,1)0(10101100uFuFuFuFFFFF基函数的特点：两条边界：两条边界的简单曲面片)(,)(v1,rv0,r)()()()()(v1,ruFv0,ruFvu,r101两条边界：)(,)(u,1ru,0r)()()()()(u,0rvFu,0rvFvu,r102)(v0,r)(u,0r)(u,1r)(v1,r四条边界的简单曲面片),(1vur),(2vur),(vur给定四条边界，如何构造曲面呢？？r1,v())(v0,rru,1()ru,0())()()()()()()()(u,0rvFu,0rvFv1,ruFv0,ruFrr101021可是：0)(,1)(010vFvFv时有：1)(,0)(110vFvFv时有：)0,(ur),0(vr0v)1,(ur1v0101v0,F(u)r(0,0)F(u)r(1,0)r(u,0)v1,F(u)r(0,1)F(u)r(1,1)r(u,1)实际边界冗余部分考虑如何去掉“重复信息”构造则])()()()()[(])()()()()[(1,1ruF0,1ruFvF1,0ruF0,0ruFvFr1011003321-rrrr)()()()()()()()()()()()()()()()()()()()()()()()()()()()(vFvF11,1r1,0rv1,r0,1r0,0rv0,ru,1ru,0r0uFuF1vFvF1,1r1,0r0,1r0,0ruFuFu,1ru,0rvFvFv1,rv0,ruFuFr-rrr101010101010321则简单曲面片的性质边界上一点的跨界斜率为该边界两端点跨界斜率的线性组合。即角点的扭矢为零(1,1)r(v)F(1,0)r(v)F(1,v)r(0,1)r(v)F(0,0)r(v)F(0,v)r(1,1)r(u)F(0,1)r(u)F(u,1)r(1,0)r(u)F(0,0)r(u)F(u,0)ru1u0uu1u0uv1v0vv1v0v01,1r1,0r0,1r0,0ruvuvuvuv)()()()()(u,0rv)(0,0ruv跨界连续•曲面片间公共边界自然满足位置连续；•若要满足斜率连续，需公共端点的跨界斜率成正比。(1)(2)(1)(2)vvvv(1)(1)(1)v0v1v(2)(2)0v1v(2)vr(0,1)λr(0,0),r(1,1)λr(1,0)r(u,1)F(u)r(0,1)F(u)r(1,1)λ[F(u)r(0,0)F(u)r(1,0)]λr(u,0))0,()1,()2()1(urur)()()()(1,0rλ1,1rv2v11.3具有给定边界及跨界斜率的Coons曲面)()()()()()()()()(v1,ruGv0,ruGv1,ruFv0,ruFvu,ru1u0101)()()()()()()()()(u,1rvGu,0rvGu,1rvFu,0rvFvu,rv1v0102）（）（）（）（由前面的讨论：vuvuvuvu,r,r,r,r321注意：上面的基函数是Hermit基函数??rrrrr30101vv0vv1uuuvuv0uuuvuv1rFuFuGuGur0,0r0,1r0,0r0,1Fvr1,0r1,1r1,0r1,1Fvr0,0r0,1r0,0r0,1Gvr1,0r1,1r1,0r1,1Gvr-()()()()()()()()()()()()()()()()()()()()()()()()0101vvvvvvuuuuvuuuuv-1FuFuGuGu0ru,0ru,1ru,0ru,1r0,vr0,0r0,1r0,0r0,1r1,vr1,0r1,1r1,0r1,1r0,vr0,0r0,1r0,0rr1,vr1,0r1,1r1,0()()()()()()()()()()()()()()()()()()()()()()()()()()01uv0uv11FvFv0,1Gvr1,1Gv()()()()()()2.双三次样条插值2.1双三次曲面片2.2双三次样条插值2.3曲面插值数据点的参数化2.1双三次曲面片•观察“具有给定边界及跨界斜率的孔斯曲面”中的r3，也是一个曲面函数，并完全由角点的信息定义，而非边界信息。30101vv0vv1uuuvuv0uuuvuv1rFuFuGuGur0,0r0,1r0,0r0,1Fvr1,0r1,1r1,0r1,1Fvr0,0r0,1r0,0r0,1Gvr1,0r1,1r1,0r1,1Gv()()()()()()()()()()()()()()()()()()()()()()()()10101(0)(1)(,)[()()()()](0)(1)uur,vr,vruvFuFuGuGur,vr,v)()()()()1(0r)0(0r)10(r)00(r)0(r1010vGvGvFvF,,,,,vvv若)()()()(])10(r)00(r)10(r)00(r[)0(r1010vGvGvFvF,,,,,vuvuvuuu)()()()(])11(r)01(r)11(r)01(r[)1(r1010uGuGuFuF,,,,,vuvuvuuu)()()()(])11(r)01(r)11(r)01(r[)1(r1010vGvGvFvF,,,,,vvv考察1(,)ruv本质：采用Ferguson参数曲线段定义两边的曲线和切矢量。rrrrr)1(r)0(r)1(r)0(r])()()()([),(r10101,v,v,v,vuGuGuFuFvuuu即)()()()()11(r)01(r)11(r)01(r)10(r)00(r)10(r)00(r)11(r)01(r)11(r)01(r)10(r)00(r)10(r)00(r1010uuuuvGvGvFvF,,,,,,,,,,,,,,,,uvuvuvuvvvvv）（）（有：）中（）（）（所以，vuvuvuvuvuvu,r,r,r,r,r),(r31321）（）（有：）中（）（）（同理，vuvuvuvuvuvu,r,r,r,r,r),(r32321）（所以，vuvu,r),(r3一般实用性文章提到的Coons曲面，多半指上面的这种双三次曲面。由于Ferguson在Coons之前就把参数三次样条曲线推广到这种形式，所以CAGD学术界又把它称为Ferguson双三次曲面片。双三次曲面片的标准形式T2323vvvvuuuvuvuuuvuvrU1uuu,V1vvv1000001033212211r0,0r0,1r0,0r0,1r1,0r1,1r1,0r1,1r0,0r0,1r0,0r0,1r1,0r1,1r1,0r1,1TccUMBMVM()()()()()()()()B()()()()()()()()2.2双三次样条插值目的：将双三次曲面片或Ferguson曲面片合成为光滑曲面。双三次样条函数的概念若函数x(u,v)或r(u,v)满足a.一致通过型值点。b.取值范围单向偏导二阶连续，混合偏导四阶连续。c.每个曲面片为双三次曲面片。则称该函数为双三次样条函数或参数双三次样条函数njcolumn,,1mirow,,1曲面片的坐标变换•对每个曲面片，取则有每个曲面片的方程1ts,0iiiiijjjijkvvvvvvhuuuuuuts111111;)()()()(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),()()()()(),(101011111111111111111010,tGktGktFtFvurvurvurvurvurvurvurvurvurvurvurvurvurvurvurvursGhsGhsFsFvurjjijuvijuvijuijuijuvijuvijuijuijvijvijijijvijvijijiiji???如何计算导矢和扭矢(1)•已知条件如何计算导矢和扭矢(2)Step1计算njmivuriju,,1;,,1),,(Step2计算njmivurijv,,1;,,1),,(向u向v向u向v1,,1],rr[3rr4,r,1,1,1v,v1vmijijijijijiStep1u向导数Step2v向导数1,,1],rr[3rr4,r1,1,1,u,u1unjjijijijijimi,,1nj,,1•Step3在两条边界线，根据,计算如何计算导矢和扭矢(3)),1(),0(vrvr、),1(),0(iuiuvrvr、),1(),0(iuviuvvrvr、1,,1],rr[3rr4,r,1,1,1uv,uv1uvmijuijuijijiji如何计算导矢和扭矢(4)•Step4在各条u向线上，根据，计算),(ivur),1(),0(iuviuvvrvr、),(ijuvvur1,,1],rr[3rr4,r1,1,