[help-3dldf] Ellipsoids

[Laurence Finston]

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