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Re: ellipse, mean distance from point on perimeter to a focus ?
Posted:
Feb 11, 2010 11:22 AM


rob@trash.whim.org (Rob Johnson) wrote: > In article > <a36f3aaf77d5499bbd30f603613dcc8d@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 semimajor axis.
That's a simple and excellent argument that _a_ mean is the semimajor 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 semiminor axis length.
Best regards, David



