email@example.com (Rob Johnson) wrote: > In article > <firstname.lastname@example.org>, G > Patel <email@example.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.