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1、扭 带 Twisted Ribbon,设是一条平面或空间曲线。另一条长为h的线段与曲线垂直相交，且交点在曲线上移动。与此同时，该线段又在曲线的法平面上以一定的速度旋转。这条运动的线段形成一条宽度为h的扭带(Twisted Ribbon)。 本课件给出扭带的参数方程，并用数学软件Mathematica绘出图形。,例1,a:=4; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=1;p:=1; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp
2、*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,4,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,3,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h/2,h/2,t,0,2*Pi,Boxed-False, Axes-False
3、,ViewPoint-4,2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,圆,例2,a:=2; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=0;p:=1/2; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=Parametric
4、Plot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h/2,h/2,t,0,2*Pi,Boxed-False, Axes-False,ViewPoint-3,-2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,圆,莫比乌斯带（单侧曲面）,例3,a:=2; rt_:
5、=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=0;p:=3/2; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2
6、,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h/2,h/2,t,0,2*Pi,Boxed-False, Axes-False,ViewPoint-3,-2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,圆,例4,a:=2; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=0;p:=3
7、; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s
8、,-h/2,h/2,t,0,2*Pi,Boxed-False, Axes-False,ViewPoint-3,-2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,圆,例5,a:=2; rt_:=a*Cost,a*Sint,a*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=0;p:=2; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2
9、,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,20,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,0,h,t,0,10,Boxed-False, Axes-False,ViewPoint-3,-2,3,AspectRatio-1.8; ShowQumian,XYZ,Plo
10、tRange-All,螺旋线,例6,a:=1; rt_:=t,0,a*Cost nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=1;k:=0;p:=1/4; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,10,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=Pa
11、rametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h,h,t,-2,10,Boxed-False,Axes-False, ViewPoint-1,4,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,余弦曲线,例7,a:=1; rt_:=t*Cost,t*Sint,a*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Cr
12、ossrt,rt h:=2;k:=0;p:=1; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-4,4,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,4,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,20,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPl
13、ot3Drt+xs,t*mt+ys,t*nt,s,-h,h,t,-2,10,Boxed-False,Axes-False, ViewPoint-4,2,3,AspectRatio-1.2; ShowQumian,XYZ,PlotRange-All,圆锥螺线,例8,a:=2;b:=4; rt_:=a*Cost,b*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=2;k:=0;p:=2; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=Par
14、ametricPlot3Dx,0,0,x,-4,4,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,4,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,0,h,t,0,2*Pi,Boxed-False,Axes-False, ViewPoint-4,2,3,AspectRati
15、o-0.8; ShowQumian,XYZ,PlotRange-All,椭圆,例9,rt_:=t,t,t2 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=1;k:=0;p:=1; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThick
16、ness3; Z=ParametricPlot3D0,0,z,z,-1,6,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h,h,t,-3,3,Boxed-False,Axes-False, ViewPoint-4,-2,3,AspectRatio-1.2; ShowQumian,XYZ,PlotRange-All,例10,rt_:=0,t,t2/10000 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Cross
17、rt,rt h:=1;k:=0;p:=1; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-1,1,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-5,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-1,1,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3D
18、rt+xs,t*mt+ys,t*nt,s,-h,h,t,-5,5,Boxed-False,Axes-False, ViewPoint-2,1,2,AspectRatio-1.2; ShowQumian,XYZ,PlotRange-All,本程序不适合直线（r 的二阶导数是零向量）， 故在z上加一个很小的量，得近似直线。,例10,rt_:=0,t,t2/10000 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt h:=1;k:=0;p:=1; xs_,t_:=Cosp*t*s-Sinp*t*k*s;ys_,t_:=Sinp*t*
19、s+Cosp*t*k*s; X=ParametricPlot3Dx,0,0,x,-1,1,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-5,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-1,1,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs,t*mt+ys,t*nt,s,-h,h,t,-6,8,Boxed-False,Axes-False, ViewPoint-2,1,2,AspectRatio-1.2; ShowQumian,PlotRange-All,