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Here's a shot ( the idea is good, the manipulation should probably
be checked ) :
Problem:
What is the volume of the solid resulting from the
intersection of a cube and a cylinder such that the
axis of the cylinder coincides with the long diagonal
of the cube.
Assumptions:
Unit cube
Cylinder radius (r) from 0 to 1
Cube vertex labling:
F -------------------- B
/| /|
/ | / |
/ | / |
/ | / |
A /-------------------- O |
| | | |
| | | |
| | | |
| D ---------------|----
| / | /
| / | /
| / | /
| / |/
---------------------
The clinder's axis will coincide with OD.
Method:
Consisder cylinder of height Len(OD). Find volume of clinder not
in intersection and subtract from total volume.
Create coordinate system such that cylinder axis coincides with
Z-axis with O at origin and point F when projected along Z-axis
falls on positive X-axis. Equation of this plane is
(1) Z=Sqr(2)*X.
Vector to A is
(2) ( 1/Sqr(6), 1/Sqr(2), 1/Sqr(3) )
Vector to B is
(3) ( 1/Sqr(6), -1/Sqr(2), 1/Sqr(3) )
The projection of the (half) circle X^2 + Y^2 = r^2 onto plane OAFB is
given by
(4) X = Sqr( r^2-y^2)
Z = Sqr(2) * X
Proposition: By symmetry arguments, the volume of the part of the
cylinder not in the intersection is equal to six times the volume
of the object bounded by:
. Plane Z=0
. Plane OAFB
. Cylinder X^2 + Y^2 = r^2
. Plane Y = Sqr(3) * X
. Plane Y = - Sqr(3) * X
Inspection of the resulting volume integral reveals that this is
twice the volume of the same object with either of the last two planes
replaced by the plane Y = 0. Thus, the boundary conditions on the
volume integral derived from (1), (2), (3), and (4) are:
Y = [ -r / Sqr(2), 0 ]
X = [ -Y / Sqr(3), Sqr( r^2 - Y^2 ) ]
Z = [ 0, Sqr(2) * X ]
so
/ / /
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V = 12 | | | dz dx dy (with the above boundaries)
| | |
/ / /
this gives V = 46 * Sqr( 6 ) * r^3 / 27
so Vint = ( Sqr(3) * pi * r^2 ) - ( 46 * Sqr( 6 ) * r^3 / 27 )
Monty
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