作出垂直于一空间曲线的曲线构成的曲面研究学习课件.ppt

1、作出垂直于一空间曲线的曲线作出垂直于一空间曲线的曲线构成的曲面构成的曲面2018.9.10 本课件讨论如何作出垂直于一空间曲线的曲线，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，并用数学软件方程，并用数学软件Mathematica绘制了曲面，附绘制了曲面，附有很多例子和图形。有很多例子和图形。设设 L 是一条空间曲线，在曲线上任意取一点是一条空间曲线，在曲线上任意取一点 P，欲作一个，欲作一个以以 P 为圆心且垂直于曲线的圆。为圆心且垂直于曲线的圆。如何处理这个问题？如何处理这个问题？我们可以以曲线我们可以

2、以曲线 L 的单位主法矢和单位副法矢在的单位主法矢和单位副法矢在 L 的法的法平面上建立直角坐标系，以点平面上建立直角坐标系，以点 P 为原点，然后就象为原点，然后就象 xOy 面上那样作出圆的矢量方程。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）。极坐标曲线（转化为参数曲线即可）。如果以空间曲线如果以空间曲线L的参数为第一参数，以圆的参数为第二的参数为第一参数，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。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

3、的图形作出来。可以用数学软件把这种参数曲面的图形作出来。:(),(),()()xx tyy tzz tatb 以曲线以曲线上任一点为圆心，作一个圆垂直于上任一点为圆心，作一个圆垂直于。()(),(),()tx ty tz tr位矢：()(),(),()tx t y t z tr切矢：单位化:()()ttrrn副法矢：单位化:()()()tttrrrm主法矢：单位化:nm()()cossintvrmn圆的矢量方程：(,)()cos()sin()vtr tttmn所有这些圆构成的曲面的矢量方程：,02atb 02nm例例1:()cos,sin,0(02)tat att r曲线曲线是半径为是半径为a

4、的圆：的圆：以以上的点为中心，作半径为上的点为中心，作半径为b的圆垂直于曲的圆垂直于曲线线。圆环面圆环面作图的Mathematica程序：a:=4;b:=1;rt_:=a*Cost,a*Sint,0nt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=b*Coss;ys_:=b*Sins;X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3;Y=ParametricPlot3D0,y,0,y,-4,4,PlotStyle-AbsoluteThickness3;Z

5、=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-Alla:=4;b:=2;rt_:=a*Cost,a*Sint,0nt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_

6、:=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,ViewPoin

7、t-4,2,3,AspectRatio-.8;ShowQumian,XYZ,PlotRange-All例例2:()4cos,4sin,0(02)tttt r:cos,2sin(02)L xy椭圆椭圆L绕圆绕圆旋转旋转例例3:()4cos,2sin,0(02)tttt r圆圆L绕椭圆绕椭圆旋转旋转:cos,sin(02)L xya:=4;b:=2;rt_:=a*Cost,b*Sint,0nt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=Coss;ys_:=Sins;X=ParametricPlot3Dx,0,0,x,-2,2

8、,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-A

9、ll例例4:()4cos,4sin,0(02)tttt r星形线星形线L绕圆绕圆旋转旋转33:cos,sin(02)L xya:=4;b:=2;rt_:=a*Cost,4*Sint,0nt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=Coss3;ys_:=Sins3;X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3;Y=ParametricPlot3D0,y,0,y,-4,4,PlotStyle-AbsoluteThickness3;Z=Parametr

10、icPlot3D0,0,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例例5:()4cos,4sin,0(02)tttt r抛物线抛物线L绕圆绕圆旋转旋转2:,(21.5)L xs yss a:=4;rt_:=a*Cost,a*Sint,0nt_:=Normaliz

11、eCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=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

12、,0,2*Pi,Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8;ShowQumian,XYZ,PlotRange-All例例6:()4cos,4sin,2 (06)ttttt r抛物线抛物线L绕螺旋线绕螺旋线2:,(22)L xs yss a:=4;rt_:=a*Cost,a*Sint,2*tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=s;ys_:=s2;X=ParametricPlot3Dx,0,0,x,-2,5,PlotStyle-AbsoluteThi

13、ckness3;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例例7:()4cos,4sin,2 (010)

14、ttttt r圆圆L绕螺旋线绕螺旋线:cos,sin(02)L xs yssa:=4;rt_:=a*Cost,a*Sint,2*tnt_:=NormalizeCrossrt,rtmt_:=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,PlotSt

15、yle-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例例8:()3cos,3sin,2 (010)ttttt r曲线曲线L绕螺旋线绕螺旋线3:2cos,2sin(02)L xs yssa:=3;b:=2rt_:=a*Cost,a*Sint,b*tnt_:=NormalizeCrossrt,rtmt_:

16、=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,

17、10,Boxed-False,Axes-False,AspectRatio-1.5;ShowQumian,XYZ,PlotRange-All,ViewPoint-4,-2,2例例9:()0,sin (55)tttt r圆圆L绕正弦曲线绕正弦曲线:cos,sin(02)L xs yssrt_:=0,t,Sintnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rt;xs_:=Coss3;ys_:=Sins;X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3;Y=Param

18、etricPlot3D0,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例例10:()2cos,2sin,2 (06)ttttt r余弦曲线余弦曲

19、线L绕螺旋线绕螺旋线:,cos2(06)L xs yssa:=2;b:=2;rt_:=a*Cost,a*Sint,2*tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rtxs_:=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-Abso

20、luteThickness3;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极坐标曲线极坐标曲线例例11:()4cos,4sin,0(02)tttt r心形线心形线L绕圆绕圆旋转旋转:1 cos(02)L a:=4;rt_:=a*Cost,a*Sint,0nt_:=Normali

21、zeCrossrt,rtmt_:=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=Parametric

22、Plot3Drt+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例例12:()2.5cos,2.5sin,0(02)tttt r三叶玫瑰线三叶玫瑰线L绕圆绕圆旋转旋转:cos3(02)L a:=2.5;rt_:=a*Cost,a*Sint,0;nt_:=NormalizeCrossrt,rt;mt_:=NormalizeCrossrt,Crossr

23、t,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,B

24、oxed-True,Axes-True,AspectRatio-.8;ShowQumian,XYZ,PlotRange-All,Boxed-False,Axes-False,ViewPoint-4,2,3,AspectRatio-.8例例13:()2cos,2sin,2 (25)ttttt r四叶玫瑰线四叶玫瑰线L绕螺旋线绕螺旋线:cos2(02)La:=2;rt_:=a*Cost,a*Sint,2*tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rt;rhos_:=Cos2s;xs_:=rhos*Coss;ys_:=rhos*Si

25、ns;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,

26、XYZ,PlotRange-All,ViewPoint-4,2,2例例14:()2cos,2sin,2 (25)ttttt r三叶玫瑰线三叶玫瑰线L绕螺旋线绕螺旋线:cos3(02)La:=2;rt_:=a*Cost,a*Sint,2*tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rt;rhos_:=Cos3*s;xs_:=rhos*Coss;ys_:=rhos*Sins;X=ParametricPlot3Dx,0,0,x,-4,5,PlotStyle-AbsoluteThickness3;Y=ParametricPlot3D0,

27、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例例15:()2cos,2sin,(15)ttt tt r8叶玫瑰线叶玫瑰线L绕螺旋线绕螺旋线:co

28、s4(02)La:=2;rt_:=a*Cost,a*Sint,tnt_:=NormalizeCrossrt,rtmt_:=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-Absolut

29、eThickness3;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例例16:()0.01cos,0.01sin,(014)2ttttt r直线直线L绕螺旋线绕螺旋线:,0(02)L xs ys a:=0.01;rt_:=a*Cost,a*Sint,t/2nt_:=NormalizeCrossrt,rtmt_:=NormalizeCross

30、rt,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,Aspe

31、ctRatio-1.5;ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,4例例17:()cos,sin,1.5 (014)ttt tttt r抛物线抛物线L绕圆锥螺旋线绕圆锥螺旋线2:,(1.51.5)L xs yssrt_:=t*Cost,t*Sint,1.5tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rt;xs_:=s;ys_:=s2;X=ParametricPlot3Dx,0,0,x,-2,2,PlotStyle-AbsoluteThickness3;Y=ParametricPlot3

32、D0,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例例18:()cos,sin,1.5 (212)ttt tttt r圆圆L绕圆锥螺旋线绕圆

33、锥螺旋线:2cos,2sin(02)L xs yssa:=2;rt_:=t*Cost,t*Sint,1.5tnt_:=NormalizeCrossrt,rtmt_:=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-Abs

34、oluteThickness3;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:()cos,sin,1.5 (810)ttt tttt r:2cos,2sin(02)L xs yss例例19:()cos,sin,1.5 (212)ttt tttt r椭圆椭圆L绕圆锥螺旋线绕圆锥螺旋线:2cos,4sin(02)L xs yss a:=2;b:=4

35、;rt_:=t*Cost,t*Sint,1.5tnt_:=NormalizeCrossrt,rtmt_:=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;

36、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:()cos,sin,1.5 (88)ttt tttt r:2cos,4sin(02)L xs yss例例20:()cos,sin,1.5 (012)ttt tttt r玫瑰线玫瑰线L绕圆锥螺旋线绕圆锥螺旋线:2cos4(02)La:=2;b:=4;rt_:=t*Cost,t*Sint,1.5tnt_:=NormalizeCro

37、ssrt,rtmt_:=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=ShowX,Y,Z;Qumian=ParametricPlo

38、t3Drt+xs*mt+ys*nt,s,0,2*Pi,t,0,11.8,Boxed-False,Axes-False,AspectRatio-1;ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1:()cos,sin,1.5 (88)ttt tttt r:2cos4(02)L例例21:()cos,sin,1.5 (012)ttt tttt r星形线星形线L绕圆锥螺旋线绕圆锥螺旋线33:2cos,2sin(02)L xya:=2;rt_:=t*Cost,t*Sint,1.5tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCros

39、srt,Crossrt,rt;xs_:=a*Coss3;ys_:=a*Sins3;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,-6,8,Boxed-Fals

40、e,Axes-False,AspectRatio-1;ShowQumian,PlotRange-All,ViewPoint-4,2,1:()cos,sin,1.5 (012)ttt tttt r33:2cos,2sin(02)L xy例例22:()2cos,2sin,1.5 (012)ttttt r星形线星形线L绕圆柱螺旋线绕圆柱螺旋线33:2cos,2sin(02)L xya:=2;rt_:=a*Cost,a*Sint,1.5tnt_:=NormalizeCrossrt,rtmt_:=NormalizeCrossrt,Crossrt,rt;xs_:=a*Coss3;ys_:=a*Sins3;

41、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-1.4;ShowQumian,XYZ,PlotRange-All,ViewPoint-4,2,1:()2cos,2sin,1.5(66)ttttt r33:2cos,2sin(02)L xy