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Discovery Project

--- The Intersection of Three Cylinders

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1.

The solid enclosed by the three cylinders , , and  is shown above; where the green surface is the cylinder   , the blue surface is the cylinder   , and the yellow surface is the cylinder  .

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2.

To find the volume, we can split the solid into sixteen congruent pieces, one of which lies in the part of the first octant with  in [0, ],as shown below.

This piece is described by  

{  (,,)  |  and ,  and , and }.

So the volume of the solid is

=

=                                   .

we can also split the solid into forty eight congruent pieces as follows :

One of the piece which lies in the part of the first octant is shown below.

This piece is described by  

{ (,,) | and , and ,and }.

So the volume of the solid is

 =           

=   = .

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3.

One possible set of parametric equations ( with all sign choices allowed ) of the edges of the solid is

,  or ,  or for all   in [,] .

or , or , for all in [,] .

or  ,  ,  or   for all in [,] .

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4. Let the three cylinder be ,  , and   .

(a) If , the four faces in problem 1 collapse into a single face, as in the graph below.

(b) If <<, then each pair of vertical opposite faces, defined by one of the other two cylinders, collapse into a single face. The graph of the case for is shown below.

(c) If , then the solid of intersection coincides with the solid of intersection of the two cylinder , and   , as in the graph below, where the blue surface is the cylinder , and the green surface is the cylinder .

5.  

(a) If , we split the solid into sixteen congruent pieces, one of which can be described as the solid above the polar region

{ (,) | and , and  }

in the xy-plane and below the surface , as shown below.

Thus, the total volume is

      .

(b) If <<, we split the solid into sixteen congruent pieces, one of which is the solid above the regions ,  and below the surface , where is the right triangular region

{ (,) | and ,  and  } ,

and  is the circular polar region  

{ (,) |  and ,  and  } .

( as the figure below).

Using rectangular coordinates for the region and polar coordinates for the region  , we find the total volume of the solid to be

+ .

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(c) If , we split the solid into sixteen congruent pieces, one of which can be described as the solid above the region { (,) | and ,  and  } in the xy -plane and below the surface  ,as shown below.

Thus, the total volume is

.