For background please use the two links which are the bold and blue words.
A solid whose plan view and front and side elevations resembled a multiplication symbol (x) bounded by a circle may be viewed here.
It is the third on that page.
The object can be created by imposing 3 cylinders on a cube in each of the x, y and z dimensions.
If the edges of the cube are of unit length, What is the volume of this object?
How mundane (as being simple) a solution can we get?
Two ideas, nothing definite.
(1) If the cube has unit length, then the cylindars all have unit diameter, or radius = 1/2 = r
In each plane, a relationship holds:
x^2 + y^2 <= r^2
y^2 + z^2 <= r^2
x^2 + z^2 <= r^2
I can imagine a computer solution in which a sector of the cube is meshed, and each point in the mesh is tested, then accepted or rejected as being part of the solid, subject to the above 3 equations.
(2) Other thought is to first determine what percent of a cylindar's volume (r=1/2, h=1) is also included in a similar cylindar at 90 degrees. Say it turns out to be 90%. Since all 3 cylindars are orthogonal to each other I suspect it can then be deduced what the overlap is for all 3.
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Posted by Larry
on 2006-11-04 14:36:36 |
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