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Feb 27, 1998, 3:00:00 AM2/27/98

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Given an ellipse on a paper, is it a geometrical way to find the focal

points or the long and short axis?

Alf

Feb 28, 1998, 3:00:00 AM2/28/98

to

In article <34F70BC3...@online.no>,

Alf Jacob Munthe <alfj...@online.no> wrote:

>Given an ellipse on a paper, is it a geometrical way to find the focal

>points or the long and short axis?

We had this problem once before. Check out

http://www.math.niu.edu/~rusin/known-math/index/51M04.html

and scan for "ellipse". I think you'll want to skip over the responses

by someone named "rusin"; the others are much clearer.

dave

Mar 1, 1998, 3:00:00 AM3/1/98

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In article <6d9fhj$2g7$1...@gannett.math.niu.edu>,

In the last sentence of that document you wrote:

> The more general problem, of constructing the foci from 5 _arbitrary_

> points on the ellipse, seems even more formidable.

If I remember well this was a standard problem in a course about

projective geometry that I took some (well, many) years ago. The

solution was more or less as follows:

1) Given the points M, N, P, Q and R use Pascal's theorem (limit case)

to construct the tangent m to the ellipse at M (the rule suffices).

2) Draw a circle c tangent to m at M. This circle is homologous to the

ellipse, with M as center of homology (this last word is used here with

the meaning it has in the context of projective geometry, of course).

3) Let N', P' and Q' be the (second) points where MN, MP and MQ meet

the circle c, respectively. The axis e of the homology is determined by

the points NP.N'P' and PQ.P'Q'.

4) Let S be the point where P'Q' intersect a parallel to PQ by M. The

parallel i' to the axis e by S' is the "limit line", i.e. the homologous

of the line i at infinity. Let X' be the intersection of m and i'. With

center X' and radius X'M intersect i' at points Y' and Z'. Since the angle

Y'MZ' is right, MY' and MZ' give the directions of the principal diameters.

5) Let O' be the pole of the line i' with respect to c. Let the line O'Y'

meet c at points A' and B', and let O'Z' meet c at C'and D'. The points A,

B, C and D homologous of A', B', C' and D' are the vertices of the ellipse.

The point O = AB.CD (homologous of O') is the center. The foci are easily

found: if AB is the major axis draw a circle with center C and radius OA

and intersect it with AB.

In other post cited in the same document you say:

>Back in the century in which emperors and presidents gave mathematical

>proofs there was so much collective understanding of Euclidean constructions

>that someone would have leapt in to demonstrate the elegant geometrical

>construction

It seems that I'm older than I thought :-(

JosÃ© H. Nieto

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Mar 8, 1998, 3:00:00 AM3/8/98

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In article <34F70BC3...@online.no>,

Alf Jacob Munthe <alfj...@online.no> wrote:

>

> Given an ellipse on a paper, is it a geometrical way to find the focal

> points or the long and short axis?

>

> I saw this in an old geometry textbook. Totally impractical, but it seems

correct:

Take a hollow cone and cut it to appropriate angle to get the desired

ellipse. Place a sphere between the vertex and the ellipse touching the

ellipse at point S. Now place another sphere under the ellipse this time

touching it at point H. (Of course spheres snugly fit inside the cone). The

points S and H are the focii.

Mar 12, 1998, 3:00:00 AM3/12/98

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Practically do the following:

Centre: Draw two arbitrary parallel chords of

the given ellipse.

Find the midpoints of the two chords.

Draw the chord of the ellipse passing

through the two midpoints.This chord

is a diameter of the ellipse.

Find the midpoint of the last chord

drawn,this is the centre of the ellipse.

Axes: Draw a circumference centred on the

centre of the ellipse so as to

intersect it at four points.

Choose three of these points and draw

the two mutually perpendicular chords

passing through them.

Draw two chords passing through the

centre of the ellipse and

parallel to the above two chords.

They are the major and minor axes of

the ellipse.

Foci: Draw a circumference centering on one

of the vertices of

the minor axis and using half lenght

of the major axis as the radius.

The two intersections with the major

axis are the foci.

Angelo

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