Re: how to calculate the expected distance from a point to the center in square grid?

From: Leonardo B. Oliveira (leob_at_dcc.ufmg.br)
Date: 10/08/04 

Date: Fri, 8 Oct 2004 08:23:52 -0300
To: "David W. Cantrell" <DWCantrell@sigmaxi.org>

Ok, David, many thanks. And do you know any place (e.g.
reference, link, etc) where I can see the demonstration of
the result below?

> (sqrt(2) + log(1 + sqrt(2)))*R .

I am developing a work and it is very important to me present
how i achieved this result.

About circle, in fact, i also has found 2/3*R - which is very similar
result to R sqrt(2)/2 (both 0.7, approximately). The latter
was found, by calculating the radius r of the inner circle
that has the half of the area of the circle of radius R.
Namely,

pR**2 = 2pr**2, so r = R sqrt(2)/2

Thanks again!
    
Regards,

     Leonardo

On Fri, 8 Oct 2004, David W. Cantrell wrote:

> "Leonardo B. Oliveira" <leob@dcc.ufmg.br> wrote:
>
> > The expected distance between a random point
> > (i.e., a point picked at random) in a circle of
> > radius R and the center of this circle
> > is R sqrt(2)/2 - as shown in "Estimating Hop=20
> > Counts in Position Based Routing Schemes for Ad Hoc"
>
> I'm not familiar with that article.
> And I must wonder if you're presenting their result accurately. I seems to
> me that what you have described as "expected distance" should be merely
> the average distance from the center of the circle to points within it,
> and that distance is 2/3*R, not sqrt(2)/2*R.
>
> > But and what about the expected distance from random
> > point to the center of a square grid, say n x n?
>
> To make this case easily comparable to that of the circle, let's say
> instead that we're dealing with a square of side length 2*R. The average
> distance from the center of the square to points within it is then
>
> (sqrt(2) + log(1 + sqrt(2)))*R .
>
> David Cantrell
>

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