RE: [Algorithms] Extracting a Matrix from a Tetrahedron (was: Preventing Verlet Rigid Bodies From Going Inside-Out)

[Tyson J. <twj...@sa...>]

I don't see how either method would "bite" you, but I would probably
just for my own mental sanity and debugging convenience define one of
the points as "up" and another as "forward".  I'd put the origin of the
frame at the base of the tetrahedron opposite "up."*.  From there
construction is trivial**.  But I'm doing extra work to get there and
only so that I have a nice convenient local space.  OTOH, nice
convenient coordinate frames have saved me hours of debugging that I can
then use to speed up really slow parts of the game.

* Average the three bottom points
** up - base = up_vector, forward - base = forward_vector, up_vector X
forward_vector = right_vector

-----Original Message-----
From: gda...@li...
[mailto:gda...@li...] On Behalf Of
Steve Ranck
Sent: Tuesday, December 10, 2002 12:26 AM
To: gda...@li...
Subject: [Algorithms] Extracting a Matrix from a Tetrahedron (was:
Preventing Verlet Rigid Bodies From Going Inside-Out)

Hi Charles,

This is another great question. I'm currently using only 3 of the points
and completely ignoring the 4th. Though I'm getting pretty decent
results, I wonder if it's going to bite me under certain circumstances.

In fact, the method I use to extract the matrix is even worst. I
implemented the most straightforward approach, where the line between
the first two vertices is one of my axis, and the other two are derived
via cross products. This seems like two of the points would have direct
influence over the matrix generation, followed by the 3rd, and ignoring
the 4th. The ideal solution would generate a matrix considering all four
points.

It seems there should be a way to determine the matrix by somehow
"averaging" the 3 axis formed by the particles of a right tetrahedron.
Or if we measured how far off two axis were (using dot product, for
example) from each other and called that the error, maybe we could
select a matrix whose 3 axis form a minimum error when summed together.

I wonder what Jakobsen did.

Steve



> -----Original Message-----
> From: gda...@li... 
> [mailto:gda...@li...] On 
> Behalf Of Charles Bloom
> Sent: Monday, December 09, 2002 10:28 AM
> To: gda...@li...
> Subject: Re: [Algorithms] Preventing Verlet Rigid Bodies From 
> Going Inside-Out
> 
> 
> 
> At 12:43 AM 12/9/2002 -0800, Steve Ranck wrote:
> >The "Rigid Bodies" section speaks about using a tetrahedron 
> >configuration of 4 particles with distance constraints 
> between them to 
> >represent a rigid body. From the particle positions, a matrix can be 
> >extracted and the graphics geo (or whatever) can be oriented 
> by using 
> >that matrix.
> 
> BTW how are people extracting a frame from a tetrahedron?  
> The problem is over-constrained, since you only need 3 points 
> to define a coordinate frame.
> 
> -------------------------------------------------------
> Charles Bloom    cb...@cb...    http://www.cbloom.com
> 
> 
> 
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