[help-3dldf] Re: [metapost] Constructing ellipse from 4 points

[Laurence Finston]

On Wed, 2 Nov 2005, Brooks Moses wrote:

> > >    http://steiner.math.nthu.edu.tw/ne01/whw/Ellipse1/part2.htm

This page has a little movie that demonstrates the ruler-and-compass
construction, but there's no explanation.  The library system here
doesn't list the reference, T.H. Eagle, _Constructive Geometry For
Plane Curves_, but I've ordered a couple of other books in the field.

> > >    http://mathworld.wolfram.com/ConicSection.html
> > >    http://mathworld.wolfram.com/Ellipse.html

There's a lot of information on these pages, but no constructions
are given, unless I missed them.

> >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,

I've attached a PostScript file with an illustration of an ellipsoid
and a square, with red dots where a selection of ellipses intersects
the plane of the square.  I've also attached the 3DLDF code.

> then take the derivative of the distance between the
> points,

Could you explain this?  I'm only familiar with derivatives of functions,
but I haven't taken a derivative since my college days.

> and thus find the maximum distance -- and thereby find the major
> axis of the ellipse, assuming the equations are tractable.

You mean algebraically, don't you?  I'm afraid I don't have a clue
of how to go about doing this.

I thought about looking for the maximum distance from a selection of
points, and then using binary search to find the maximum in the
neighborhood of my first approximation.  The problem is, I would have to
generate an ellipse and transform it at least once for every try.
I do something similar, but which doesn't require quite so much work,
to find the intersection points of two coplanar ellipses.  Someone
explained a better way to do this to me, but it involves mathematical
concepts that I (still) don't understand.  Given a choice, I'd almost
always rather solve problems like this algebraically.

Thanks again for your help.

Laurence

ellpln01.ps.gz
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ellpln01.ldf
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