RE: [Algorithms] Bounding cones.

[Johnson, J. <Jam...@si...>]

You're correct. I should have tested the obvious first.

James


-----Original Message-----
From: gda...@li...
[mailto:gda...@li...] On Behalf Of Nick
Carter
Sent: Tuesday, January 27, 2004 4:45 PM
To: 'gda...@li...'
Subject: RE: [Algorithms] Bounding cones.


I don't think that this iterative technique will always generate the
smallest cone -- it doesn't seem to work in the case of 2D circles.
Consider three small circles of equal radius whose centers are mutually
equidistant (i.e., in the pattern of an equilateral triangle).  Then the
minimum bounding circle of any two of the small circles will not be
fully contained in the minimum bounding circle of all three of the small
circles. So there's no way to "build up" a minimum bounding circle in
the iterative way that James suggests.

This counterexample is easily extended from circles to cones: consider
three cones, of small theta, whose centerlines, pairwise, are at equal
angles to one another.

I'm very curious about bounds on optimality of like iterative
techniques, in either the 2D-circle or 3D-cone (which is still 2 dof)
cases.  Does anyone know?

 - nick carter


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