垂直于一空間曲線的曲線構成的曲面

1、santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011垂直于一空間曲線的曲線垂直于一空間曲線的曲線構成的曲面構成的曲面2011.6.10santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011 本課件討論如何作出垂直于一空間曲線的曲線，本課件討論如何作出垂直于一空間曲線的曲線，并用這些曲線構成一個曲面。給出了曲面的參數(shù)并用這些曲線構成一個曲面。給出了曲面的參數(shù)方程，并用數(shù)學軟件方程，并用數(shù)學軟件mathematica繪制了曲面，附繪制了曲面，附有很多例子和圖形。有很多例子和圖形。2011年年6月編寫，月編寫，2012年年4月修改月修改santa i

2、i, p.135四川大學數(shù)學學院 徐小湛10 june 2011設設 l 是一條空間曲線，在曲線上任意取一點是一條空間曲線，在曲線上任意取一點 p，欲作一個，欲作一個以以 p 為圓心且垂直于曲線的圓。為圓心且垂直于曲線的圓。如何處理這個問題？如何處理這個問題？ 我們可以以曲線我們可以以曲線 l 的單位主法矢和單位副法矢在的單位主法矢和單位副法矢在 l 的法的法平面上建立直角坐標系，以點平面上建立直角坐標系，以點 p 為原點，然后就象為原點，然后就象 xoy 面上那樣作出圓的矢量方程。面上那樣作出圓的矢量方程。這個方法也適合其他參數(shù)曲線，如橢圓、星形線等，包括這個方法也適合其他參數(shù)曲線，如橢圓、

3、星形線等，包括極坐標曲線（轉(zhuǎn)化為參數(shù)曲線即可）。極坐標曲線（轉(zhuǎn)化為參數(shù)曲線即可）。如果以空間曲線如果以空間曲線l的參數(shù)為第一參數(shù)，以圓的參數(shù)為第二的參數(shù)為第一參數(shù)，以圓的參數(shù)為第二參數(shù)，則我們就得到動圓形成的參數(shù)曲面。參數(shù)，則我們就得到動圓形成的參數(shù)曲面?？梢杂脭?shù)學軟件把這種參數(shù)曲面的圖形作出來。可以用數(shù)學軟件把這種參數(shù)曲面的圖形作出來。santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011:( ),( ),( ) ()xx tyy tzz tatb 以曲線以曲線上任一點為圓心，作一個圓垂直于上任一點為圓心，作一個圓垂直于。( ) ( ), ( ), ( )tx ty

4、tz tr位矢：( ) ( ), ( ), ( )tx t y t z tr切矢：單位化:( )( )ttrrn副法矢：單位化:( ) ( ( )( )tttrrrm主法矢：單位化:nmsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011( )( )cossintvrmn圓的矢量方程：( , )( )cos( )sin( )vtr tttmn所有這些圓構成的曲面的矢量方程：, 02atb 02nmsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例1: ( ) cos , sin ,0 (02 )tat att r曲線曲線是半徑為是半徑為

5、a的圓：的圓：以以上的點為中心，作半徑為上的點為中心，作半徑為b的圓垂直于曲的圓垂直于曲線線。圓環(huán)面圓環(huán)面作圖的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;

6、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-allsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011a:=4;b:=2;rt_:=a*cost,a*sint,0nt_:=normalizecro

7、ssrt,rtmt_:=normalizecrossrt,crossrt,rtxs_:=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*p

8、i,t,0,2*pi, boxed-false,axes-false,viewpoint-4,2,3,aspectratio-.8;showqumian,xyz,plotrange-all例例2: ( )4cos ,4sin ,0 (02 )tttt r:cos ,2sin(02 )l xy橢圓橢圓l繞圓繞圓旋轉(zhuǎn)旋轉(zhuǎn)santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例3: ( )4cos ,2sin ,0 (02 )tttt r圓圓l繞橢圓繞橢圓旋轉(zhuǎn)旋轉(zhuǎn):cos ,sin(02 )l xya:=4;b:=2;rt_:=a*cost,b*sint,0nt_:=nor

9、malizecrossrt,rtmt_:=normalizecrossrt,crossrt,rtxs_:=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,

10、s,0,2*pi,t,0,2*pi, boxed-false,axes-false,viewpoint-4,2,3,aspectratio-.8;showqumian,xyz,plotrange-allsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例4: ( )4cos ,4sin ,0 (02 )tttt r星形線星形線l繞圓繞圓旋轉(zhuǎn)旋轉(zhuǎn)33:cos,sin(02 )l xya:=4;b:=2;rt_:=a*cost,4*sint,0nt_:=normalizecrossrt,rtmt_:=normalizecrossrt,crossrt,rtxs_:=co

11、ss3;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,

12、2,3,aspectratio-.8;showqumian,xyz,plotrange-allsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例5: ( )4cos ,4sin ,0 (02 )tttt r拋物線拋物線l繞圓繞圓旋轉(zhuǎn)旋轉(zhuǎn)2:,( 21.5)l xs yss a:=4;rt_:=a*cost,a*sint,0nt_:=normalizecrossrt,rtmt_:=normalizecrossrt,crossrt,rtxs_:=s;ys_:=s2;x=parametricplot3dx,0,0,x,-2,5,plotstyle-absoluteth

13、ickness3;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-allsanta ii, p.135四川大

14、學數(shù)學學院 徐小湛10 june 2011例例6: ( )4cos ,4sin ,2 (06)ttttt r拋物線拋物線l繞螺旋線繞螺旋線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-absolutethickness3;y=parametricplot3d0,y,0,y,-4,5,plotstyle-absolutethickne

15、ss3;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-allsanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例7: ( )4cos ,4sin ,2 (010)ttttt r圓圓l繞螺旋線繞螺旋線:

16、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,plotstyle-absolutethick

17、ness3;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,-2,2santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例8: ( )3cos ,3sin ,2 (010)ttttt r曲線曲線l繞螺旋線繞螺旋線3:2cos,2sin(02 )l xs yssa:=3;b:=2rt_:=a*cost,a*sint,b*tn

18、t_:=normalizecrossrt,rtmt_:=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=parametricplot3dr

19、t+xs*mt+ys*nt,s,0,2*pi,t,0,10,boxed-false,axes-false,aspectratio-1.5;showqumian,xyz,plotrange-all,viewpoint-4,-2,2santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例9: ( )0, ,sin ( 55)tttt r圓圓l繞正弦曲線繞正弦曲線:cos ,sin(02 )l xs yssrt_:=0,t,sintnt_:=normalizecrossrt,rtmt_:=normalizecrossrt,crossrt,rt;xs_:=coss3;ys_:

20、=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;showqumi

21、an,xyz,plotrange-all,viewpoint-4,2,1santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例10: ( )2cos ,2sin ,2 (06)ttttt r余弦曲線余弦曲線l繞螺旋線繞螺旋線:,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-absolute

22、thickness3;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,

23、2,aspectratio-1santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011極坐標曲線極坐標曲線santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例11: ( )4cos ,4sin ,0 (02 )tttt r心形線心形線l繞圓繞圓旋轉(zhuǎn)旋轉(zhuǎn):1 cos(02 )l a:=4;rt_:=a*cost,a*sint,0nt_:=normalizecrossrt,rtmt_:=normalizecrossrt,crossrt,rt;rhos_:=1-coss;xs_:=rhos*coss;ys_:=rhos*sins;x=paramet

24、ricplot3dx,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,plotrang

25、e-all,boxed-false,axes-false,viewpoint-4,2,3,aspectratio-.8santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例12: ( )2.5cos ,2.5sin ,0 (02 )tttt r三葉玫瑰線三葉玫瑰線l繞圓繞圓旋轉(zhuǎn)旋轉(zhuǎn):cos3 (02 )l 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=

26、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,p

27、lotrange-all,boxed-false,axes-false,viewpoint-4,2,3,aspectratio-.8santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例13: ( )2cos ,2sin ,2 ( 25)ttttt r四葉玫瑰線四葉玫瑰線l繞螺旋線繞螺旋線:cos2 (02 )la:=2;rt_:=a*cost,a*sint,2*tnt_:=normalizecrossrt,rtmt_:=normalizecrossrt,crossrt,rt;rhos_:=cos2s;xs_:=rhos*coss;ys_:=rhos*sins;x=

28、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,p

29、lotrange-all,viewpoint-4,2,2santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例14: ( )2cos ,2sin ,2 ( 25)ttttt r三葉玫瑰線三葉玫瑰線l繞螺旋線繞螺旋線: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,plotstyl

30、e-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-4,2,2santa ii,

31、 p.135四川大學數(shù)學學院 徐小湛10 june 2011例例15: ( )2cos ,2sin , ( 15)ttt tt r8葉玫瑰線葉玫瑰線l繞螺旋線繞螺旋線:cos4 (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,

32、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-4,2,2santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例16: ( )0

33、.01cos ,0.01sin , (014)2ttttt r直線直線l繞螺旋線繞螺旋線:,0 (02)l xs ys a:=0.01;rt_:=a*cost,a*sint,t/2nt_:=normalizecrossrt,rtmt_:=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

34、,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-4,2,4santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例17: ( ) cos , sin ,1.5 (014)ttt tttt r拋物線拋物線l繞圓錐螺旋線繞圓錐螺旋線2:,( 1.51.5)l

35、 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=parametricplot3d0,y,0,y,-2,2,plotstyle-absolutethickness3;z=parametricplot3d0,0,z,z,-2,20,plotstyle-absolutethickness3;xyz=showx,y,z;qum

36、ian=parametricplot3drt+xs*mt+ys*nt,s,-1.5,1.5,t,0,14,boxed-false,axes-false,aspectratio-1;showqumian,xyz,plotrange-all,viewpoint-4,2,1santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例18: ( ) cos , sin ,1.5 (212)ttt tttt r圓圓l繞圓錐螺旋線繞圓錐螺旋線:2cos ,2sin(02 )l xs yssa:=2;rt_:=t*cost,t*sint,1.5tnt_:=normalizecrossr

37、t,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-absolutethickness3;xyz=showx,y,z;qumian=parametricplot3drt+xs*mt+ys*nt,s,0,2*

38、pi,t,2,12,boxed-false,axes-false,aspectratio-1;showqumian,xyz,plotrange-all,viewpoint-4,2,1santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011: ( ) cos , sin ,1.5 ( 810)ttt tttt r:2cos ,2sin(02 )l xs ysssanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例19: ( ) cos , sin ,1.5 (212)ttt tttt r橢圓橢圓l繞圓錐螺旋線繞圓錐螺旋線:2cos ,4sin(0

39、2 )l xs yss a:=2;b:=4;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-absolutethic

40、kness3;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-4,2,1santa ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011: ( ) cos , sin ,1.5 ( 88)ttt tttt r:2cos ,4sin(02 )l xs ysssanta ii, p.135四川大學數(shù)學學院 徐小湛10 june 2011例例20:

41、( ) cos , sin ,1.5 (012)ttt tttt r玫瑰線玫瑰線l繞圓錐螺旋線繞圓錐螺旋線:2cos4(02 )la:=2;b:=4;rt_:=t*cost,t*sint,1.5tnt_:=normalizecrossrt,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=parametricplot3drt+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,