[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

503.0. "Drilling a Cube" by COMET::ROBERTS (Dwayne Roberts) Sun Jun 08 1986 22:48

    
    Consider the unit cube which is intersected with a circular cylinder of
    radius r such that the axis of the cylinder passes through two opposite
    vertices of the cube.  (Picture a wooden cube with a hole drilled in it
    from one corner, through the center, to it's opposite corner.) 
    
    What expression describes the volume of the cube that is within the
    cylinder?  (How much sawdust is there?)

    [If you can solve this, my hat's off to you.  I've been working on this
    one off and on for over ten years.  (Not that that means much; it's
    just been out of my reach.)]
T.RTitleUserPersonal
Name		DateLines
503.2pass large cub THROUGH small oneSIERRA::OSMANand silos to fill before I feep, and silos to fill before I feepMon Jun 09 1986 17:126
    This reminds me of a related novelty, which is that by drilling
    a square hole from corner to diagonal corner of a solid cube, you
    can actually pass a LARGER cube through a SMALLER one.
    
    /Eric
503.3part of cylinder not in cubeSTAR::BRANDENBERGCivilization is the progress toward a society of privacy.Wed Jun 11 1986 11:16101
    
    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
			/ / /
		       | | |
		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