[help-3dldf] Re: Polyhedra and ellipses

[Martijn van Manen]

Hi Laurence,

I'll try to answer some of your questions in another
installment. What I would like to say though is that the
number of fun and exciting polyhedra is rather endless.
I'll make some selection. Ofcourse there are "smart" methods
of finding the vertices and the edges and so on. 
Those mathematicians in the nineteenth century produced
formula after formula. There are endless and sometimes
pointless calculations.
What you should maybe figure out is how to construct a nice
data structure for them, so that afterwards you can traverse it
and find intersections with lines and planes. The people in
computational geometry love the "doubly connected edge list".
You might want to implement it. 
For the tangencies of two ellipses, or two quadrics, I'll
give you some methods too.

GRTZ,

Martijn 

----- Original Message -----
From: "Laurence Finston" <address@hidden>
To: address@hidden
Subject: Polyhedra and ellipses
Date: Mon, 06 Dec 2004 23:17:34 +0100

> 
> Hello Martijn et al,
> 
> I've taken another look at what you've written.
> I can see there's a lot to say about the Platonic polyhedra, and I shouldn't
> be in such a hurry to get to the more "exciting" ones.  However, I'd still
> like to know whether it's possible to find the vertices of the Archimedean
> polyhedra, their duals, and the Kepler-Poinsot polyhedra the way you do for
> the Platonic ones.
> 
> Someone recommended the book _An Introduction to Solid Modeling_ by Martti
> Maentylae, and I ordered it through inter-library loan.
> It came today, and while it discusses polyhedra in general, it doesn't go into
> these kinds of polyhedra.  I'm not really surprised, because _none_ of the
> books on computer graphics I've looked at do (and I've looked at quite a few
> and even bought a couple).  The people who program screen savers seem to know
> how to do it, though, unless they just fold up virtual cardboard models, too.
> 
> 
> I may actually try to implement your method for finding ellipse intersections
> for the coming release.  The math doesn't seem too difficult, and if you can
> give me a formula I can just go ahead and program it without learning linear
> algebra and analytic geometry first.  I don't mind if the formula's a bit
> complicated.
> It would certainly be worth the effort.
> 
> I think it should be possible to pretest whether the ellipses are tangent.
> For this, however, it will make a difference whether the axes of the ellipses
> are parallel and perpendicular, since this case will be easy to test.  I think
> it might be possible to test whether they're too far apart to be tangent, or
> whether one lies completely inside another with no tangents, even if their
> axes are non-parallel and non-perpendicular.  Do you know whether there's a
> way of finding the point on an ellipse that's closest to another ellipse, or
> would this be just as hard as finding the tangent points?
> 
> Thanks again for all your help.
> 
> Laurence
> 

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