[help-3dldf] Re: Ellipsoids

[Laurence Finston]

Hans Aberg wrote:

> There is also a big step between doing one the one hand linear algebra,
> splines and graphics, and on the other hand, formal mathematical geometry.
> So it can be wise to stay with the former.

Yes, this is the crux of the problem.  The only computer graphics book I've
really used so far is Huw Jones, _Computer Graphics through Key Mathematics_. 
 e justifies his results but doesn't give formal proofs.  However, his
treatment of some topics, such as NURBS, isn't detailed enough for me to base
an implementation on it.  In addition, most computer graphics programs don't
do the kind of traditional geometry that 3DLDF is intended to do, so computer
graphics texts don't treat the topics I'm interested in.  And since these
topics, e.g., polyhedra and intersection theory, are non-trivial, it's hard to
find explanations in words of one syllable.  So I'll just have to try to learn
them.

Months ago I worked out a formula for the intersections of an ellipse and a
circle in a plane using substitution, but it's quite long, so I'm not sure
it's really correct.  I've never gotten around to testing it.  If it works, I
believe I can use  it to find the intersections of two coplanar ellipses,
because I could just transform one to a circle.  I have never seen this
problem covered in a math book.  Currently, I use an iterative routine that
traverses one of the ellipses, but it's quite slow--up to thirty seconds, at a
guess.  

Laurence