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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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Rob Johnson

Posts: 1,771
Registered: 12/6/04
Re: ellipse, mean distance from point on perimeter to a focus ?
Posted: Feb 12, 2010 5:56 PM
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In article <>,
Rob Johnson <> wrote:
>In article <>,
>Cary <cary@domain.invalid> wrote:

>>On Fri, 12 Feb 2010 10:22:33 +0000 (UTC), (Rob
>>Johnson) wrote:

>>>In article <20100211111323.200$>,
>>>David W. Cantrell <> wrote:

>>>> (Rob Johnson) wrote:
>>>>> In article
>>>>> <>, G
>>>>> Patel <> wrote:

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

>>>>> Yes.


>>>>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.

>>>I get that as well. Furthermore, averaging with respect to time,
>>>using equal area in equal time, I get a mean of a(1+e^2/2).

>>This paper may be of interest to those following this thread.

>Thanks for the reference; that paper goes the distance. It covers
>the means that David Cantrell (True Anomaly) and I (Time) discussed.
>I note that they also discuss average over arc-length, and since arc
>length is symmetric with respect to the foci, that average is a.
>It may not be obvious at first, but eccentric anomaly is also
>symmetric with respect to the foci. The eccentric anomaly is the
>angle from the center of the ellipse after the ellipse has been
>scaled in the direction of either the major or the minor axis to a
>circle. Thus, the mean with respect to the eccentric anomaly is
>also a.

Another mean with possible physical meaning is the mean distance
weighted by the amount of energy received from the star. The
intensity of the energy is k/r^2 while the time spent receiving
that energy is 1/2 r^2 d(theta). Thus, the average distance
weighted by energy would be identical to the average distance with
respect to the True Anomaly, which, as David has pointed out, is the
semi-minor axis.

Rob Johnson <>
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