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Topic: ellipse, mean distance from point on perimeter to a focus ?
Replies: 8   Last Post: Feb 12, 2010 5:56 PM

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David W. Cantrell

Posts: 3,395
Registered: 12/3/04
Re: ellipse, mean distance from point on perimeter to a focus ?
Posted: Feb 11, 2010 11:22 AM
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rob@trash.whim.org (Rob Johnson) wrote:
> In article
> <a36f3aaf-77d5-499b-bd30-f603613dcc8d@j6g2000vbd.googlegroups.com>, G
> Patel <gaya.patel@gmail.com> wrote:

> >Is the mean distance from a point on perimeter to a focus equal to the
> >semi major axis length?

>
> Yes.


Well, as Achava correctly pointed out, it depends on the variable with
respect to which the mean is calculated.

> Recall the "string and pins" method of drawing an ellipse, which
> depends on the fact that the sum of the distances from a point on the
> ellipse to the two foci of that ellipse is the major axis of that
> ellipse. Since the sum of those two distances is always the major
> axis, the sum of their means is the major axis. By symmetry, the
> means of these distances are the same. Therefore, each is equal to
> the semi-major axis.


That's a simple and excellent argument that _a_ mean is the semi-major axis
length. Perhaps in some sense, that is the most useful mean.

But I initially (before seeing Achava's post) calculated the mean distance
with respect to theta, thinking of the ellipse in polar coordinates as
given by

r = a (1 - e^2) / (1 - e cos(theta))

Doing that, we find instead

mean distance = a sqrt(1 - e^2) = b, the semi-minor axis length.

Best regards,
David



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