垂直于一空间曲线的曲线构成的曲面.ppt

垂直于一空间曲线的曲线 构成的曲面,2011.6.10,本课件讨论如何作出垂直于一空间曲线的曲线，并用这些曲线构成一个曲面。给出了曲面的参数方程，并用数学软件Mathematica绘制了曲面，附有很多例子和图形。 2011年6月编写，2012年4月修改,设 L 是一条空间曲线，在曲线上任意取一点 P，欲作一个以 P 为圆心且垂直于曲线的圆。 如何处理这个问题？ 我们可以以曲线 L 的单位主法矢和单位副法矢在 L 的法平面上建立直角坐标系，以点 P 为原点，然后就象 xOy 面上那样作出圆的矢量方程。 这个方法也适合其他参数曲线，如椭圆、星形线等，包括极坐标曲线（转化为参数曲线即可）。 如果以空间曲线L的参数为第一参数，以圆的参数为第二参数，则我们就得到动圆形成的参数曲面。 可以用数学软件把这种参数曲面的图形作出来。,以曲线上任一点为圆心，作一个圆垂直于。,例1,曲线是半径为a的圆：,以上的点为中心，作半径为b的圆垂直于曲线。,圆环面,作图的Mathematica程序： a:=4;b:=1; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=b*Coss;ys_:=b*Sins; 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*mt+ys*nt,s,0,2*Pi,t,0,2*Pi, Boxed-False,Axes-False,ViewPoint-4,2,3, AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,a:=4;b:=2; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=Coss;ys_:=b*Sins; 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*mt+ys*nt,s,0,2*Pi,t,0,2*Pi, Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,例2,椭圆L绕圆旋转,例3,圆L绕椭圆旋转,a:=4;b:=2; rt_:=a*Cost,b*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=Coss;ys_:=Sins; 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*mt+ys*nt,s,0,2*Pi,t,0,2*Pi, Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,例4,星形线L绕圆旋转,a:=4;b:=2; rt_:=a*Cost,4*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=Coss3;ys_:=Sins3; 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*mt+ys*nt,s,0,2*Pi,t,0,2*Pi,Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,例5,抛物线L绕圆旋转,a:=4; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=s;ys_:=s2; X=ParametricPlot3Dx,0,0,x,-2,5,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,7,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,-2,1.5,t,0,2*Pi, Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,例6,抛物线L绕螺旋线,a:=4; rt_:=a*Cost,a*Sint,2*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=s;ys_:=s2; X=ParametricPlot3Dx,0,0,x,-2,5,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,20,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,-2,2,t,0,6, Boxed-False,Axes-False,ViewPoint-4,-1,3, AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,例7,圆L绕螺旋线,a:=4; rt_:=a*Cost,a*Sint,2*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=Coss;ys_:=Sins; 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=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,0,10,Boxed-False,Axes-False,AspectRatio-1.5; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,-2,2,例8,曲线L绕螺旋线,a:=3;b:=2 rt_:=a*Cost,a*Sint,b*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=b*Coss3;ys_:=b*Sins; 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=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,0,10,Boxed-False,Axes-False,AspectRatio-1.5; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,-2,2,例9,圆L绕正弦曲线,rt_:=0,t,Sint nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=Coss3;ys_:=Sins; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-6,6,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,-5,5,Boxed-False,Axes-False,AspectRatio-.5; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1,例10,余弦曲线L绕螺旋线,a:=2;b:=2; rt_:=a*Cost,a*Sint,2*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt xs_:=s;ys_:=Cos2s; X=ParametricPlot3Dx,0,0,x,-2,7,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,7,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,12,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,6,t,0,5.8,Boxed-True,Axes-True,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,Boxed-False,Axes-False,ViewPoint-4,2,2,AspectRatio-1,极坐标曲线,例11,心形线L绕圆旋转,a:=4; rt_:=a*Cost,a*Sint,0 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=1-Coss; xs_:=rhos*Coss;ys_:=rhos*Sins; X=ParametricPlot3Dx,0,0,x,-4,5,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,2,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,0,2*Pi,Boxed-True,Axes-True,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8,例12,三叶玫瑰线L绕圆旋转,a:=2.5; rt_:=a*Cost,a*Sint,0; nt_:=NormalizeCrossrt,rt; mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=Cos3*s; xs_:=rhos*Coss;ys_:=rhos*Sins; X=ParametricPlot3Dx,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*mt+ys*nt,s,0,2*Pi,t,0,2*Pi,Boxed-True,Axes-True,AspectRatio-.8; ShowQumian,XYZ,PlotRange-All,Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8,例13,四叶玫瑰线L绕螺旋线,a:=2; rt_:=a*Cost,a*Sint,2*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=Cos2s; xs_:=rhos*Coss;ys_:=rhos*Sins; X=ParametricPlot3Dx,0,0,x,-4,5,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,10,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,-2,5,Boxed-False,Axes-False,AspectRatio-1; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,2,例14,三叶玫瑰线L绕螺旋线,a:=2; rt_:=a*Cost,a*Sint,2*t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=Cos3*s; xs_:=rhos*Coss;ys_:=rhos*Sins; X=ParametricPlot3Dx,0,0,x,-4,5,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-4,5,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,10,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,-2,5,Boxed-False,Axes-False,AspectRatio-1; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,2,例15,8叶玫瑰线L绕螺旋线,a:=2; rt_:=a*Cost,a*Sint,t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=Cos4s; xs_:=rhos*Coss;ys_:=rhos*Sins; 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,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,-1,5,Boxed-False,Axes-False,AspectRatio-1.5; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,2,例16,直线L绕螺旋线,a:=0.01; rt_:=a*Cost,a*Sint,t/2 nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=s;ys_:=0; X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3; Y=ParametricPlot3D0,y,0,y,-2,2,PlotStyle-AbsoluteThickness3; Z=ParametricPlot3D0,0,z,z,-2,8,PlotStyle-AbsoluteThickness3; XYZ=ShowX,Y,Z; Qumian=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2,t,0,14,Boxed-False,Axes-False,AspectRatio-1.5; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,4,例17,抛物线L绕圆锥螺旋线,rt_:=t*Cost,t*Sint,1.5t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=s;ys_:=s2; X=ParametricPlot3Dx,0,0,x,-2,2,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*mt+ys*nt,s,-1.5,1.5,t,0,14,Boxed-False,Axes-False,AspectRatio-1; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1,例18,圆L绕圆锥螺旋线,a:=2; rt_:=t*Cost,t*Sint,1.5t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=2*Coss;ys_:=2*Sins; 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=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,2,12,Boxed-False,Axes-False,AspectRatio-1; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1,例19,椭圆L绕圆锥螺旋线,a:=2;b:=4; rt_:=t*Cost,t*Sint,1.5t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; xs_:=2*Coss;ys_:=b*Sins; 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=ParametricPlot3Drt+xs*mt+ys*nt,s,0,2*Pi,t,0,12,Boxed-False,Axes-False,AspectRatio-1; ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1,例20,玫瑰线L绕圆锥螺旋线,a:=2;b:=4; rt_:=t*Cost,t*Sint,1.5t nt_:=NormalizeCrossrt,rt mt_:=NormalizeCrossrt,Crossrt,rt; rhos_:=a*Cos4s; xs_:=rhos*Coss;ys_:=rhos*Sins; 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=S