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投影

2009-09-06 23:37:26| 分类：高等数学图形动画 | 标签： |举报 |字号大中小订阅

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投影

在多元微积分学中，我们经常需要将空间的立体、曲面和曲线投影到坐标面上，以便于进行一些运算。以下展示一些几何图形在坐标面上的投影。

1. 旋转抛物面 z=x^2+y^2+1 在三个坐标面上的投影：

with(plots):with(plottools):
qumian:=plot3d([u*sin(theta), u*cos(theta), u^2+1],
u=-.5..1.5, theta=0..2*Pi, style=patch):
A1:=[0,0,0]: B1:=[1,0,0]: C1:=[0,1,0]: A2:=[-3,1,0]: B2:=[-3,0,0]: C2:=[-3,0,1]:
A3:=[1,-3,0]: B3:=[0,-3,0]: C3:=[0,-3,1]:
touying1:=project(qumian, [A1, B1, C1]): touying2:=project(qumian, [A2, B2, C2]):
touying3:=project(qumian, [A3, B3, C3]):
display(qumian, touying1,touying2,touying3, orientation=[40,70], lightmodel=light2, axes=boxed, scaling=constrained);

2. 旋转抛物面 z=x^2+y^2+1（2<z<6）在三个坐标面上的投影：

with(plots):with(plottools):
qumian:=plot3d([u*sin(theta), u*cos(theta), u^2+1],
u=1..2, theta=0..2*Pi, style=patch):
A1:=[0,0,0]: B1:=[1,0,0]: C1:=[0,1,0]: A2:=[-6,1,0]: B2:=[-6,0,0]: C2:=[-6,0,1]:
A3:=[1,-6,0]: B3:=[0,-6,0]: C3:=[0,-6,1]:
touying1:=project(qumian, [A1, B1, C1]): touying2:=project(qumian, [A2, B2, C2]):
touying3:=project(qumian, [A3, B3, C3]):
display(qumian, touying1,touying2,touying3, orientation=[40,70], lightmodel=light2, axes=boxed, scaling=constrained);

3. 椭球体在三个坐标面上的投影：

with(plots):with(plottools):a:=2:b:=3:c:=1:
qumian:=plot3d([a*sin(phi)*cos(t),b*sin(phi)*sin(t),c*cos(phi)],
phi=0..Pi, t=0..2*Pi, style=patch):
A1:=[0,0,-3]: B1:=[1,0,-3]: C1:=[0,1,-3]: A2:=[-6,1,0]: B2:=[-6,0,0]: C2:=[-6,0,1]:
A3:=[1,-6,0]: B3:=[0,-6,0]: C3:=[0,-6,1]:
touying1:=project(qumian, [A1, B1, C1]): touying2:=project(qumian, [A2, B2, C2]):
touying3:=project(qumian, [A3, B3, C3]):
display(qumian, touying1,touying2,touying3, orientation=[40,70], lightmodel=light2, axes=boxed, scaling=constrained);

4. 冰淇淋在三个坐标面上的投影:

5. 曲面在三个坐标面上的投影:

6. 立体在三个坐标面上的投影:

7. 单叶双曲面在三个坐标面上的投影:

8. 螺旋线在三个坐标面上的投影:

with(plots): with(plottools):
A:=-3:B:=-3:C:=0:
quxian1:=spacecurve([cos(t), sin(t), t], t=0..6*Pi, color=red, thickness=3):
quxian:=spacecurve([cos(t), sin(t), t], t=0..6*Pi, color=blue, thickness=3):
A1:=[0,0,C]: B1:=[1,0,C]: C1:=[0,1,C]:
A2:=[A,1,0]: B2:=[A,0,0]: C2:=[A,0,1]:
A3:=[1,B,0]: B3:=[0,B,0]: C3:=[0,B,1]:
touying1:=project(quxian, [A1, B1, C1]):
touying2:=project(quxian, [A2, B2, C2]):
touying3:=project(quxian, [A3, B3, C3]):
display(quxian1, touying1,touying2,touying3,axes=boxed,tickmarks=[0,0,0]);

9. 圆锥螺线在三个坐标面上的投影:

with(plots):
x:=t*cos(t):y:=t*sin(t):z:=t:
a:=0:b:=20:
A:=-30:B:=-50:C:=0:
quxian:=spacecurve([x(t),y(t),z(t)], t=a..b,color=red):
touying1:=spacecurve([x(t),y(t),C], t=a..b,color=blue):
touying2:=spacecurve([x(t),B,z(t)], t=a..b,color=green):
touying3:=spacecurve([A,y(t),z(t)], t=a..b,color=brown):
display(quxian, touying1,touying2,touying3,thickness=3,axes=boxed,tickmarks=[0,0,0]);

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