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徐小湛的博客

高等数学课件、函数图形动画、数学软件应用

 
 
 

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扭带 (Maple动画)  

2012-04-29 21:09:49|  分类: 高等数学图形动画 |  标签: |举报 |字号 订阅

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本文转载自calculus《扭带》

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。
     
 

扭带 - calculus - 高等数学博客

 
 
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 

 
with(plots): with(linalg):
a:=2:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>;
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>;
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]);
扭带 (Maple动画) - calculus - 徐小湛的博客
with(plots): with(linalg):
a:=4:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=2:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>;
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>;
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=2:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=2:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=4:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>;
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>;
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=4:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=2:
r:=t-> <a*cos(t),a*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
n(t):=normalize(crossprod(T(t),Q(t))):
m(t):=normalize(crossprod(T(t),n(t))):
h:=1;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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
 
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
with(plots): with(linalg):
a:=4:
r:=t-> <a*cos(t),a*sin(t),a*t>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>;
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>;
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
with(plots): with(linalg):
a:=1:
r:=t-> <t*cos(t),t*sin(t),a*t>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 扭带 (Maple动画) - calculus - 徐小湛的博客
 
    
    
with(plots): with(linalg):
a:=2:b:=4:
r:=t-> <a*cos(t),b*sin(t),0>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);

 

扭带 (Maple动画) - calculus - 徐小湛的博客

with(plots): with(linalg):
r:=t-> <t,t,t^2>;
x(t):=r(t)[1]:y(t):=r(t)[2]:z(t):=r(t)[3]:
T(t):=<diff(x(t),t),diff(y(t),t),diff(z(t),t)>:
Q(t):=<diff(x(t),t$2),diff(y(t),t$2),diff(z(t),t$2)>:
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]);
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 
 
扭带 (Maple动画) - calculus - 徐小湛的博客
 

   
 
 
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