扭带的动画下载_PPT模板_22

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扭带的动画nullnull扭带 Maple 动画null设Γ是一条平面或空间曲线。另一条长为h的线段与曲线垂直相交，且交点在曲线上移动。与此同时，该线段又在曲线Γ的法平面上以一定的速度旋转。这条运动的线段形成一条宽度为 h 的扭带(Twisted Ribbon)。本课件给出扭带的参数方程，并用数学软件Maple绘出图形和动画。nullnull例1with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; ...

nullnull扭带 Maple 动画null设Γ是一条平面或空间曲线。另一条长为h的线段与曲线垂直相交，且交点在曲线上移动。与此同时，该线段又在曲线Γ的法平面上以一定的速度旋转。这条运动的线段形成一条宽度为 h 的扭带(Twisted Ribbon)。本课件给出扭带的参数方程，并用数学软件Maple绘出图形和动画。nullnull例1with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; Q(t):=; n(t):=normalize(crossprod(T(t),Q(t))); m(t):=normalize(crossprod(T(t),n(t))); h:=2;k:=1;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s; Y(s,t):=sin(p*t)*s+cos(p*t)*k*s; x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]; y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]; z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]; qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..2*Pi,style=patch,lightmodel=light2,grid=[10,66]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-150,50]);圆nullwith(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=2;k:=1;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=60: for i from 1 to K do ti:=i*2.2*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patch,lightmodel=light2,grid=[10,60]): quxian1[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2-0.1..-h/2,t=0..ti,color=red,grid=[2,60]): quxian2[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=h/2..h/2+0.1,t=0..ti,color=red,grid=[2,60]):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): quxian1:=display(seq(quxian1[i],i=1..K),insequence=true): quxian2:=display(seq(quxian2[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(quxian1,quxian2,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23null例2 with(plots): with(linalg): a:=2: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; Q(t):=; n(t):=normalize(crossprod(T(t),Q(t))); m(t):=normalize(crossprod(T(t),n(t))); h:=2;k:=1;p:=1/2; X(s,t):=cos(p*t)*s-sin(p*t)*k*s; Y(s,t):=sin(p*t)*s+cos(p*t)*k*s; x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]; y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]; z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]; qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..2*Pi,style=patch,lightmodel=light2,grid=[10,66]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-100,50]);圆莫比乌斯带（单侧曲面）nullwith(plots): with(linalg): a:=2: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=1;p:=1/2: X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=60: for i from 1 to K do ti:=i*2.2*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patch,lightmodel=light2,grid=[10,60]): quxian1[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2-0.1..-h/2,t=0..ti,color=red,grid=[2,60]): quxian2[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=h/2..h/2+0.1,t=0..ti,color=red,grid=[2,60]):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): quxian1:=display(seq(quxian1[i],i=1..K),insequence=true): quxian2:=display(seq(quxian2[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(quxian1,quxian2,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23nullwith(plots): with(linalg): a:=2: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=1;p:=1/2: X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=60: for i from 1 to K do ti:=i*3*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patchnogrid,lightmodel=light2): quxian1[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2-0.1..-h/2,t=0..ti,color=red,style=patchnogrid,lightmodel=light2): quxian2[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=h/2..h/2+0.1,t=0..ti,color=red,style=patchnogrid,lightmodel=light2):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): quxian1:=display(seq(quxian1[i],i=1..K),insequence=true): quxian2:=display(seq(quxian2[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(quxian1,quxian2,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23nullwith(plots): with(linalg): a:=2: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=1;p:=1/2: X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=120: for i from 1 to K do ti:=i*3*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patch,lightmodel=light2,grid=[10,60]): zhixian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h..h,t=ti-0.01..ti+0.01,color=blue):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): zhixian:=display(seq(zhixian[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(zhixian,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23null例3with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; Q(t):=; n(t):=normalize(crossprod(T(t),Q(t))); m(t):=normalize(crossprod(T(t),n(t))); h:=2;k:=1;p:=1.5; X(s,t):=cos(p*t)*s-sin(p*t)*k*s; Y(s,t):=sin(p*t)*s+cos(p*t)*k*s; x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]; y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]; z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]; qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..2*Pi,style=patch,lightmodel=light2,grid=[10,66]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-100,50]);圆null> with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=2;k:=1;p:=3/2: X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=120: for i from 1 to K do ti:=i*3*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patchnogrid,lightmodel=light2): quxian1[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2-0.1..-h/2,t=0..ti,color=red,style=patchnogrid,lightmodel=light2): quxian2[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=h/2..h/2+0.1,t=0..ti,color=red,style=patchnogrid,lightmodel=light2):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): quxian1:=display(seq(quxian1[i],i=1..K),insequence=true): quxian2:=display(seq(quxian2[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(quxian1,quxian2,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23nullwith(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=2;k:=1;p:=3/2: X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: K:=120: for i from 1 to K do ti:=i*3*Pi/K: qumian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..ti,style=patch,lightmodel=light2,grid=[10,60]): zhixian[i]:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h..h,t=ti-0.01..ti+0.01,color=blue):od: qumian:=display(seq(qumian[i],i=1..K),insequence=true): zhixian:=display(seq(zhixian[i],i=1..K),insequence=true): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(zhixian,qumian,xyz,scaling=constrained,orientation=[-150,50]);动画 2012.4.23null例4with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; Q(t):=; n(t):=normalize(crossprod(T(t),Q(t))); m(t):=normalize(crossprod(T(t),n(t))); h:=2;k:=1;p:=3; X(s,t):=cos(p*t)*s-sin(p*t)*k*s; Y(s,t):=sin(p*t)*s+cos(p*t)*k*s; x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]; y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]; z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]; qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..2*Pi,style=patch,lightmodel=light2,grid=[10,66]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[130,50]);圆null例5with(plots): with(linalg): a:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=; Q(t):=; n(t):=normalize(crossprod(T(t),Q(t))); m(t):=normalize(crossprod(T(t),n(t))); h:=2;k:=0;p:=2; X(s,t):=cos(p*t)*s-sin(p*t)*k*s; Y(s,t):=sin(p*t)*s+cos(p*t)*k*s; x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]; y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]; z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]; qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=0..h,t=0..10,style=patch,lightmodel=light1,grid=[10,100]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..40,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=unconstrained,orientation=[30,50]);螺旋线null例7with(plots): with(linalg): a:=1: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=3;k:=0;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..15,style=patch,lightmodel=light1,grid=[6,100]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..20,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-50,70]); 圆锥螺线null例8with(plots): with(linalg): a:=2:b:=4: r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=2;k:=0;p:=2; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h/2..h/2,t=0..2*Pi,style=patch,lightmodel=light1,grid=[6,100]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-50,70]);椭圆null例9with(plots): with(linalg): r:=t-> ; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=0;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h..h,t=-3..3,style=patch,lightmodel=light1,grid=[6,100]): x_axis:=plot3d([u,0,0],u=-3..3,v=0..0.01): y_axis:=plot3d([0,u,0],u=-3..3,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-55,70]);nullnullnullnull例10with(plots): with(linalg): r:=t-> <0,t,t^2/10000>; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=0;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h..h,t=-5..5,style=patch,lightmodel=light1,grid=[6,100]): x_axis:=plot3d([u,0,0],u=-1..1,v=0..0.01): y_axis:=plot3d([0,u,0],u=-6..6,v=0..0.01): z_axis:=plot3d([0,0,u],u=-2..2,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[-55,70]);本程序不适合直线（r 的二阶导数是零向量），故在z上加一个很小的量，得近似直线。null例10with(plots): with(linalg): r:=t-> <0,t,t^2/10000>; x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]: T(t):=: Q(t):=: n(t):=normalize(crossprod(T(t),Q(t))): m(t):=normalize(crossprod(T(t),n(t))): h:=1;k:=0;p:=1; X(s,t):=cos(p*t)*s-sin(p*t)*k*s: Y(s,t):=sin(p*t)*s+cos(p*t)*k*s: x(s,t):=r(t)[1]+X(s,t)*m(t)[1]+Y(s,t)*n(t)[1]: y(s,t):=r(t)[2]+X(s,t)*m(t)[2]+Y(s,t)*n(t)[2]: z(s,t):=r(t)[3]+X(s,t)*m(t)[3]+Y(s,t)*n(t)[3]: qumian:=plot3d([x(s,t),y(s,t),z(s,t)],s=-h..h,t=-6..8,style=patch,lightmodel=light3,grid=[6,100]): x_axis:=plot3d([u,0,0],u=-1..1,v=0..0.01): y_axis:=plot3d([0,u,0],u=-6..6,v=0..0.01): z_axis:=plot3d([0,0,u],u=-1..1,v=0..0.01): xyz:=display(x_axis,y_axis,z_axis,thickness=3): display(qumian,xyz,scaling=constrained,orientation=[50,50]);

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