Yes, of course, you're right about my wording.  I see what you are talking
about WRT to the smallest sphere, too.  Nearly coplanar points would produce
an enormous sphere if you only allowed the points to be on the surface of
the sphere.

"Tor Olav Kristensen" <tor### [at] hotmailcom> wrote in message
news:web.3dfe7e1187ba9bcf38149fba0@news.povray.org...
>
> Rick Gutleber wrote:
> >If the points are _tangential_ to the sphere, wouldn't there only be one
> >solution?
>
> Rick,
> I don't think one can say about points
> that they can be tangential to anything.
>
> You probably mean that they are on the
> surface of the sphere.
>
> If you have four points in 3D space that
> are not coplanar (*), then yes; there
> are only one specific sphere that have
> them all on its surface. (And the macro
> I mentioned in my other post finds that
> very sphere.)
>
> This sphere will, of coarse, enclose a
> tetrahedron that has these four points
> as its vertices.
>
> But as Peter and I pointed out, this is
> not always the most optimal sphere to
> choose for bounding of such a tetrahedron.
>
> It will in some cases be possible to
> find spheres with smaller radii, that
> encloses the tetrahedron.
>
> And in these cases only 2 or 3 of the
> 4 points (vertices) will be on the surface
> of the sphere. The other 2 or 1 will be
> inside it.
>
> The problem is now to find the bounding
> sphere with the smallest radius.
>
>
> (*) More precisely I mean:
> vdot(p1 - p0, vcross(p2 - p0, p3 - p0)) != 0
>
>
> Tor Olav
>
>
> >"Peter Popov" <pet### [at] vipbg> wrote in message
> >news:6ldgvu8m3haqih3b2cud5narcqn8p82pvi[at]4ax.com...
> >> On Wed, 11 Dec 2002 11:45:10 -0500, "Rick Gutleber" <ric### [at] hiscom>
> >> wrote:
> >>
> >> >I bet the Graphics Gems books have code that will allow you to
determine
> >a
> >> >sphere tangential to 4 points in 3-space.  The code for those books
can
> >be
> >> >found on-line.
> >>
> >> There sure is, but keep in mind that the circumscribed sphere is
> >> usually not the smallest sphere containing four points.
>
>

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