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[help-3dldf] Ellipsoid-Plane intersection
From: |
Laurence Finston |
Subject: |
[help-3dldf] Ellipsoid-Plane intersection |
Date: |
Sun, 06 Nov 2005 16:13:30 +0100 |
User-agent: |
IMHO/0.98.3+G (Webmail for Roxen) |
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
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