[metapost] Constructing ellipse from 4 points

Laurence Finston lfinsto1 at gwdg.de
Wed Nov 2 19:36:38 CET 2005


I'm a bit stuck and hoping someone here will be able to help me.
The attached PostScript file contains illustrations of my attempts
to find the intersection of an ellipsoid with a plane.

The ellipsoid e:  length along the x-axis: 10cm
                  length along the y-axis: 5cm
                  length along the z-axis: 7cm

                  center at the origin, rotated 15° about the y-axis,
                  then shifted -1cm along the x-axis (center (-1, 0, 0)).

The square s:    center at origin, lying in x-z plane
                 side length: 15cm
                 Then, rotated 15° about the y-axis and 45° about the z-axis,
                 and shifted 1cm along the x-axis (center (-1, 0, 0)).

The points h0, h1, and v0 ... v5 all lie in the plane of s.
The points h0 and h1 are on one of the ellipses on e,
and the points v0 and v1 are on another one, perpendicular to the first.
These four points therefore lie on the ellipse that represents the
intersection of the ellipsoid and the plane.

In more nicely behaved cases, the polygon "v5, v4, v3, v2" is
rectangular, and it seemed that its enclosed ellipse was the one I
was looking for.  However, in this case (and others), it is not, as
can be seen from the projections onto the x-z and z-y planes.

It seems to me that it ought to be possible to construct an ellipse
from the four points v0, v1, h0, and h1.  Does anyone here know how, or
know where I could look to find out?

Any help would be much appreciated.

Laurence Finston
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