matlab画直交圆柱面,三个直交圆柱面所围成的立体

谁借莪１个温暖的怀抱￠ 2023-01-17 10:53 76阅读 0赞

with(plots): R:=1:

zhumian1:=plot3d(\[R\*cos(t),R\*sin(t),z\],z=-2\*R..2\*R, t=0..2\*Pi,color=yellow):

zhumian2:=plot3d(\[R\*cos(t),y,R\*sin(t)\],y=-2\*R..2\*R, t=0..2\*Pi,color=green):

zhumian3:=plot3d(\[x,R\*cos(t),R\*sin(t)\],x=-2\*R..2\*R, t=0..2\*Pi,color=red):

display(zhumian1,zhumian2,zhumian3,scaling=constrained,orientation=\[40,65\]);

三个圆柱面所围成的立体：

![71c80abb3b1ef07dbe9775496af5ebb2.png][]![02d32427dacc10410a237e5a0b0234d2.png][]

with(plots): R:=1:

zhumian\_a1:=plot3d(\[R\*cos(t),R\*sin(t),z\],z=-R\*sin(t)..R\*sin(t),t=-Pi/4..Pi/4,color=yellow):

zhumian\_a2:=plot3d(\[R\*cos(t),R\*sin(t),z\],z=-R\*cos(t)..R\*cos(t),t=Pi/4..3\*Pi/4,color=yellow):

zhumian\_a3:=plot3d(\[R\*cos(t),R\*sin(t),z\],z=-R\*sin(t)..R\*sin(t),t=3\*Pi/4..5\*Pi/4,color=yellow):

zhumian\_a4:=plot3d(\[R\*cos(t),R\*sin(t),z\],z=-R\*cos(t)..R\*cos(t),t=5\*Pi/4..7\*Pi/4,color=yellow):

zhumian\_b1:=plot3d(\[R\*cos(t),y,R\*sin(t)\],y=-R\*sin(t)..R\*sin(t),t=-Pi/4..Pi/4,color=green):

zhumian\_b2:=plot3d(\[R\*cos(t),y,R\*sin(t)\],y=-R\*cos(t)..R\*cos(t),t=Pi/4..3\*Pi/4,color=green):

zhumian\_b3:=plot3d(\[R\*cos(t),y,R\*sin(t)\],y=-R\*sin(t)..R\*sin(t),t=3\*Pi/4..5\*Pi/4,color=green):

zhumian\_b4:=plot3d(\[R\*cos(t),y,R\*sin(t)\],y=-R\*cos(t)..R\*cos(t),t=5\*Pi/4..7\*Pi/4,color=green):

zhumian\_c1:=plot3d(\[x,R\*cos(t),R\*sin(t)\],x=-R\*sin(t)..R\*sin(t),t=-Pi/4..Pi/4,color=red):

zhumian\_c2:=plot3d(\[x,R\*cos(t),R\*sin(t)\],x=-R\*cos(t)..R\*cos(t),t=Pi/4..3\*Pi/4,color=red):

zhumian\_c3:=plot3d(\[x,R\*cos(t),R\*sin(t)\],x=-R\*sin(t)..R\*sin(t),t=3\*Pi/4..5\*Pi/4,color=red):

zhumian\_c4:=plot3d(\[x,R\*cos(t),R\*sin(t)\],x=-R\*cos(t)..R\*cos(t),t=5\*Pi/4..7\*Pi/4,color=red):

a:=display(zhumian\_a1,zhumian\_a2,zhumian\_a3,zhumian\_a4):

b:=display(zhumian\_b1,zhumian\_b2,zhumian\_b3,zhumian\_b4):

c:=display(zhumian\_c1,zhumian\_c2,zhumian\_c3,zhumian\_c4):

display(a,b,c,scaling=constrained,orientation=\[40,65\],style=patch);

立体的体积是：

![2537215440086786618.png][]

推导如下：

![2824319916331657375.png][]

用Mathematica求体积(利用Boole函数)：

Integrate\[Boole\[x^2 + y^2 < R^2 && x^2 + z^2 < R^2 &&  y^2 + z^2 < R^2\], \{x, -Infinity, Infinity\}, \{y, -Infinity,  Infinity\}, \{z, -Infinity, Infinity\}\]

![603482350084580303.png][]

选择R>0的结果：

![2537215440086786618.png][]

![609674799572214992.png][]

![2504564342788347536.png][]

[71c80abb3b1ef07dbe9775496af5ebb2.png]: https://img-blog.csdnimg.cn/img_convert/71c80abb3b1ef07dbe9775496af5ebb2.png
[02d32427dacc10410a237e5a0b0234d2.png]: /images/20221021/ee69a77341674352ae02c47aab6d1abb.png
[2537215440086786618.png]: http://img.ph.126.net/OZz7RtSKrkZx0zNPojb9Vg==/2537215440086786618.png
[2824319916331657375.png]: http://img.ph.126.net/ouVuYIfQOklEtUFGGKml0g==/2824319916331657375.png
[603482350084580303.png]: http://img.ph.126.net/Gq7noFfKxQHjeW-kSQpIEg==/603482350084580303.png
[609674799572214992.png]: http://img.ph.126.net/0YsHhINOJGWX0TCXPWufhQ==/609674799572214992.png
[2504564342788347536.png]: http://img.ph.126.net/3Z6bK6gwypwz4IBWYdO_-Q==/2504564342788347536.png