[help-3dldf] Ellipsoid-Plane intersection

[Laurence Finston]

Hi Martijn,

I've had  a chance to think about what you wrote.  I've done substitution to
get intersections before.  It's just been awhile, and I got rattled by the
three different radii for the ellipsoid.  Obviously, they're known, so it's
not a problem.  

I'd also completely forgotten about the implicit equation for a plane, and how
it relates to the standard form.  I've now looked this up, and feel pretty
confident about implementing a solution.

It wouldn't be possible to account for the transformations that have been
applied to the the ellipsoid e and the plane p, since 3DLDF doesn't store this
information.  However, neither is it  necessary.
I can put e into "standard position" using Transform::align_with_axis() twice
to get Transform t, then multiply a plane lying in p by t and get its plane
using Path::get_plane().  I haven't implemented Plane::operator*() yet, but it
would amount to the same thing.  So I can just get the coefficients for the
plane equation from the data members of Plane directly.

Unless I'm mistaken, using substitution to solve for x, y, and z, I get two
values for each, unless they're zero, in which case they're the same.  So I
would get a maximum of 8 points.   Is anything known about these points?  For
example, can they be used to get a pair of conjugate axes?  Otherwise, it's a
bit involved finding the main axes, as described on my "Ellipsoid" webpage,
unless you know a better way.  
By the way, those pages are updated hourly, so if you took a look soon after
my post, you  might not have seen the updated version.  However,  it might be
too elementary to be of interest to you.

> An ellipse in the plane is determined by five points.

Okay,  but how do I get the main axes?  As far as I could tell from the
description of the Braikenridge-Maclaurin construction, it requires six points
to start with.  Otherwise, there's no way to find the point Z.
Most of the constructions I've found require either the main axes or a
tangent, and those are things I ain't got.  I found the method for finding the
center and the main axes in a book for draughtsmen, not a math book.

Thanks again.  

Laurence