垂直于一空间曲线的曲线构成的曲面

null垂直于一空间曲线的曲线 构成的曲面垂直于一空间曲线的曲线 构成的曲面2011.6.10null 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null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。null以曲线Γ上任一点为圆心，作一个圆垂直于Γ。nullnull例1曲线Γ是半径为a的圆：以Γ上的点为中心，作半径为b的圆垂直于曲线Γ。圆环面作图的Mathematica程序： a:=4;b:=1; r[t_]:={a*Cos[t],a*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=b*Cos[s];y[s_]:=b*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,3},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi}, Boxed->False,Axes->False,ViewPoint->{4,2,3}, AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]nulla:=4;b:=2; r[t_]:={a*Cos[t],a*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=Cos[s];y[s_]:=b*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,3},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi}, Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]例2椭圆L绕圆Γ旋转null例3圆L绕椭圆Γ旋转a:=4;b:=2; r[t_]:={a*Cos[t],b*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=Cos[s];y[s_]:=Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,3},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi}, Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]null例4星形线L绕圆Γ旋转a:=4;b:=2; r[t_]:={a*Cos[t],4*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=Cos[s]^3;y[s_]:=Sin[s]^3; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,3},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi},Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]null例5抛物线L绕圆Γ旋转a:=4;; r[t_]:={a*Cos[t],a*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=s;y[s_]:=s^2; X=ParametricPlot3D[{x,0,0},{x,-2,5},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,5},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,7},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,-2,1.5},{t,0,2*Pi}, Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]null例6抛物线L绕螺旋线Γa:=4; r[t_]:={a*Cos[t],a*Sin[t],2*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=s;y[s_]:=s^2; X=ParametricPlot3D[{x,0,0},{x,-2,5},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,5},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,-2,2},{t,0,6}, Boxed->False,Axes->False,ViewPoint->{4,-1,3}, AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All]null例7圆L绕螺旋线Γa:=4; r[t_]:={a*Cos[t],a*Sin[t],2*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=Cos[s];y[s_]:=Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,10},Boxed->False,Axes->False,AspectRatio->1.5]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,-2,2}]null例8曲线L绕螺旋线Γa:=3;b:=2 r[t_]:={a*Cos[t],a*Sin[t],b*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=b*Cos[s]^3;y[s_]:=b*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,10},Boxed->False,Axes->False,AspectRatio->1.5]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,-2,2}]null例9圆L绕正弦曲线Γr[t_]:={0,t,Sin[t]} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=Cos[s]^3;y[s_]:=Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-6,6},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,2},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,-5,5},Boxed->False,Axes->False,AspectRatio->.5]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]null例10余弦曲线L绕螺旋线Γa:=2;b:=2; r[t_]:={a*Cos[t],a*Sin[t],2*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]] x[s_]:=s;y[s_]:=Cos[2s]; X=ParametricPlot3D[{x,0,0},{x,-2,7},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,7},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,12},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,6},{t,0,5.8},Boxed->True,Axes->True,AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All,Boxed->False,Axes->False,ViewPoint->{4,2,2},AspectRatio->1]null极坐标曲线null例11心形线L绕圆Γ旋转a:=4; r[t_]:={a*Cos[t],a*Sin[t],0} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=1-Cos[s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,5},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,5},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,2},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi},Boxed->True,Axes->True,AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All,Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]null例12三叶玫瑰线L绕圆Γ旋转 a:=2.5; r[t_]:={a*Cos[t],a*Sin[t],0}; n[t_]:=Normalize[Cross[r'[t],r''[t]]]; m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=Cos[3*s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,2},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,2*Pi},Boxed->True,Axes->True,AspectRatio->.8]; Show[Qumian,XYZ,PlotRange->All,Boxed->False,Axes->False,ViewPoint->{4,2,3},AspectRatio->.8]null例13四叶玫瑰线L绕螺旋线Γa:=2; r[t_]:={a*Cos[t],a*Sin[t],2*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=Cos[2s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,5},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,5},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,10},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,-2,5},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,2}]null例14三叶玫瑰线L绕螺旋线Γa:=2; r[t_]:={a*Cos[t],a*Sin[t],2*t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=Cos[3*s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,5},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,5},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,10},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,-2,5},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,2}]null例158叶玫瑰线L绕螺旋线Γa:=2; r[t_]:={a*Cos[t],a*Sin[t],t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=Cos[4s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-2,2},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,6},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,-1,5},Boxed->False,Axes->False,AspectRatio->1.5]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,2}]null例16直线L绕螺旋线Γ a:=0.01; r[t_]:={a*Cos[t],a*Sin[t],t/2} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=s;y[s_]:=0; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-2,2},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,8},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2},{t,0,14},Boxed->False,Axes->False,AspectRatio->1.5]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,4}]null例17抛物线L绕圆锥螺旋线Γr[t_]:={t*Cos[t],t*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=s;y[s_]:=s^2; X=ParametricPlot3D[{x,0,0},{x,-2,2},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-2,2},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,-1.5,1.5},{t,0,14},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]null例18圆L绕圆锥螺旋线Γa:=2; r[t_]:={t*Cos[t],t*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=2*Cos[s];y[s_]:=2*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,2,12},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]nullnull例19椭圆L绕圆锥螺旋线Γ a:=2;b:=4; r[t_]:={t*Cos[t],t*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=2*Cos[s];y[s_]:=b*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,12},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]nullnull例20玫瑰线L绕圆锥螺旋线Γa:=2;b:=4; r[t_]:={t*Cos[t],t*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; rho[s_]:=a*Cos[4s]; x[s_]:=rho[s]*Cos[s];y[s_]:=rho[s]*Sin[s]; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,11.8},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]nullnull例21星形线L绕圆锥螺旋线Γa:=2;; r[t_]:={t*Cos[t],t*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=a*Cos[s]^3;y[s_]:=a*Sin[s]^3; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,-6,8},Boxed->False,Axes->False,AspectRatio->1]; Show[Qumian,PlotRange->All,ViewPoint->{4,2,1}]nullnull例22星形线L绕圆柱螺旋线Γa:=2; r[t_]:={a*Cos[t],a*Sin[t],1.5t} n[t_]:=Normalize[Cross[r'[t],r''[t]]] m[t_]:=Normalize[Cross[r'[t],Cross[r'[t],r''[t]]]]; x[s_]:=a*Cos[s]^3;y[s_]:=a*Sin[s]^3; X=ParametricPlot3D[{x,0,0},{x,-4,4},PlotStyle->AbsoluteThickness[3]]; Y=ParametricPlot3D[{0,y,0},{y,-4,4},PlotStyle->AbsoluteThickness[3]]; Z=ParametricPlot3D[{0,0,z},{z,-2,20},PlotStyle->AbsoluteThickness[3]]; XYZ=Show[X,Y,Z]; Qumian=ParametricPlot3D[r[t]+x[s]*m[t]+y[s]*n[t],{s,0,2*Pi},{t,0,12},Boxed->False,Axes->False,AspectRatio->1.4]; Show[Qumian,XYZ,PlotRange->All,ViewPoint->{4,2,1}]null