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[help-3dldf] Ellipsoids
From: |
Laurence Finston |
Subject: |
[help-3dldf] Ellipsoids |
Date: |
Wed, 17 Nov 2004 00:33:38 +0100 |
User-agent: |
IMHO/0.98.3+G (Webmail for Roxen) |
Hello,
I've been thinking about how to implement `class Ellipsoid'.
This is how I'd like to do it:
class Ellipsoid
{
Point center;
real axis_x;
real axis_y;
real axis_z;
Point x_axis_pt;
Point y_axis_pt;
Point z_axis_pt;
Transform transform;
// ...
};
If an `Ellipsoid' is constructed with no arguments for
shifting, and not subsequently transformed, it
will be centered about the origin, and `x_axis_pt',
`y_axis_pt', and `z_axis_pt' will each lie on the
corresponding axis.
My question is this:
If after one or more transformations, I want to test
whether the `Ellipsoid' is still ellipsoidal,
and assuming that the vectors (x_axis_pt - center),
(y_axis_pt - center), and (z_axis_pt - center) are all
still orthogonal, and that, for purposes of the test and
subsequent to the transformations in question,
the `Ellipsoid' has been rotated and shifted such that
`center' == (0, 0, 0) and `axis_x_pt', `axis_y_pt', and
`axis_z_pt' all lie on the corresponding axes,
will it suffice to test the Point p, such that
p == (sqrt(a^2/3), sqrt(b^2/3), sqrt(c^2/3)), where
a, b, and c are the lengths of the x, y, and z- axes of the
`Ellipsoid'?
Or is it possible
that a transformation could make the `Ellipsoid'
non-ellipsoidal while p still satifies the relation
({x_p}^2/a^2 + {y_p}^2/b^2 + {z_p}^2/b^c == 1)?
There is no guarantee that the transformations that may be
applied to an `Ellipsoid' will be affine.
Any help would be much appreciated.
Laurence Finston
- [help-3dldf] Ellipsoids,
Laurence Finston <=