[metapost] Constructing ellipse from 4 points

[Brooks Moses]

At 11:10 AM 11/2/2005, Laurence Finston wrote:
>On Wed, 2 Nov 2005, Brooks Moses wrote:
> > A Google search on "ellipse construct 'four points'" finds the 
> following links:
> >
> >    http://steiner.math.nthu.edu.tw/ne01/whw/Ellipse1/part2.htm
> >    http://mathworld.wolfram.com/ConicSection.html
> >    http://mathworld.wolfram.com/Ellipse.html
>
>Aha!  I just searched for "ellipse constructions" and found a lot of sites
>that didn't look too promising.  Thanks.

You're welcome.

(I added the "four points" string because it seemed like it ought to be 
possible with either four or five, and would have tried "five points" if I 
hadn't found anything.)

> > The short form is that, to construct an ellipse, you need either five
> > points, four points and a tangent line, or four points and knowledge of the
> > direction of the major and minor axes.
>
>I can get more points by finding the intersections of the plane with more
>of the ellipses on the ellipsoid, and I can get as many of those as I
>want.

Ah, indeed.  Now I'm wondering if the best option might not be to do that 
in a general way, then take the derivative of the distance between the 
points, and thus find the maximum distance -- and thereby find the major 
axis of the ellipse, assuming the equations are tractable.

- Brooks