垂直于一空間曲線的曲線構(gòu)成的曲面

1、santa ii, p.135四川大學(xué)數(shù)學(xué)學(xué)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(gòu)成的曲面構(gòu)成的曲面2011.6.10santa ii, p.135四川大學(xué)數(shù)學(xué)學(xué)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(gòu)成一個(gè)曲面。給出了曲面的參數(shù)并用這些曲線構(gòu)成一個(gè)曲面。給出了曲面的參數(shù)方程，并用數(shù)學(xué)軟件方程，并用數(shù)學(xué)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(xué)數(shù)學(xué)學(xué)院 徐小湛10 june 2011設(shè)設(shè) l 是一條空間曲線，在曲線上任意取一點(diǎn)是一條空間曲線，在曲線上任意取一點(diǎn) p，欲作一個(gè)，欲作一個(gè)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(gè)問(wèn)題？如何處理這個(gè)問(wèn)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(biāo)系，以點(diǎn)平面上建立直角坐標(biāo)系，以點(diǎn) p 為原點(diǎn)，然后就象為原點(diǎn)，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(gè)方法也適合其他參數(shù)曲線，如橢圓、星形線等，包括這個(gè)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(biāo)曲線（轉(zhuǎn)化為參數(shù)曲線即可）。極坐標(biāo)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(dòng)圓形成的參數(shù)曲面。參數(shù)，則我們就得到動(dòng)圓形成的參數(shù)曲面?？梢杂脭?shù)學(xué)軟件把這種參數(shù)曲面的圖形作出來(lái)?？梢杂脭?shù)學(xué)軟件把這種參數(shù)曲面的圖形作出來(lái)。santa ii, p.135四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛10 june 2011:( ),( ),( ) ()xx tyy tzz tatb 以曲線以曲線上任一點(diǎn)為圓心，作一個(gè)圓垂直于上任一點(diǎn)為圓心，作一個(gè)圓垂直于。( ) ( ), ( ), ( )tx ty

4、tz tr位矢：( ) ( ), ( ), ( )tx t y t z tr切矢：?jiǎn)挝换?( )( )ttrrn副法矢：?jiǎn)挝换?( ) ( ( )( )tttrrrm主法矢：?jiǎn)挝换?nmsanta ii, p.135四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛10 june 2011( )( )cossintvrmn圓的矢量方程：( , )( )cos( )sin( )vtr tttmn所有這些圓構(gòu)成的曲面的矢量方程：, 02atb 02nmsanta ii, p.135四川大學(xué)數(shù)學(xué)學(xué)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(diǎn)為中心，作半徑為上的點(diǎn)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)院 徐小湛10 june 2011極坐標(biāo)曲線極坐標(biāo)曲線santa ii, p.135四川大學(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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(xué)數(shù)學(xué)學(xué)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,