In article <3df273e6@news.povray.org>, Warp <war### [at] tagpovrayorg> 
wrote:

>   Have you actually made the math, which shape "wastes" more space when
> bounding (optimally) a regular tetrahedron ("regular" meaning all sides
> have the same length), the sphere or the box?

Ok, for a tetrahedron with these corners:
<-1, 1,-1>
< 1, 1, 1>
<-1,-1, 1>
< 1,-1,-1>
(regular tetrahedron, every point equidistant from the other three)
The radius of a perfect bounding sphere is sqrt(3). Volume is pi*3/4*r^3 
= 21.763 cubic units. The volume of the perfect axis-aligned bounding 
box is 8 cubic units, though it will increase for other orientations (I 
think the worst case makes for a bounding box sqrt(8)xsqrt(8)x2, 16 
cubic units.

I think the "average visible area" would make a better measurement than 
volume, though much harder to compute. And the difference is bigger than 
I expected, but I can't find anything wrong with my math, please 
re-check it.


>   Also I think that a ray-sphere intersection is faster than a ray-box
> intersection, so the sphere is in that sense a better bounding object.

Hmm...the bounding box calculation is very well optimized, I think it 
only requires 3 intersections with axis-aligned planes and some code for 
clipping that. A sphere is fast, but I think a bounding box is faster. 
However, I'm not too sure how the bounded_by keyword fits in with the 
bounding box heirarchy...it looks like a bounding box is always used, 
which bounds the bounding shapes (multiple?), which bound the object. 
Much of the bounding code seems to be very per-object...it's pretty 
disorganized, and probably not as fast as it could be.

-- 
Christopher James Huff <cja### [at] earthlinknet>
http://home.earthlink.net/~cjameshuff/
POV-Ray TAG: chr### [at] tagpovrayorg
http://tag.povray.org/

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