On Thu, Mar 31, 2016 at 11:38 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies for my lack of imagination, but I'm not understanding the problem. Except the part about ignoring cases where 2 or more distances are equal.
1) Are the points labeled?
I don't know how to make sense of "counting the number of orderings" if the points are unlabeled; The order is "A point, then another point, then another point..."
2) Are you taking N labeled points, (selected in advance from whatever space and then asking what are their different orders from a variable additional point?
Yes, I think that's correct/
3) How are they selected?
((( The first 2 paragraphs seem to assume that in at least 1D, however they be selected there is going to be one answer, just a function of N. Possible, but not immediately clear to me.
But then in paragraphs 3 and 4, you are considering some probability distribution and mentioning a simulation, indicating that you agree that (in general) the answer is not necessarily just a function of N. )))
He was speculating that the answer was not constant, but I believe that I've proved that there's a maximum number of orderings, which occurs except on a measure-0 subset of possible N-tuples of points. For example, if 3 of the points are colinear, or if there are 4 points A, B, C, and D such that AB is parallel to CD, there will be fewer possible orderings.
The number of possible orderings depends on both N and the dimension of the space, but is otherwise generically constant. A previously posted an incorrect derivation of an incorrect formula as a find-the-error puzzle, but I'll follow up with what I think is a correct proof of the correct formula sometime this weekend. I think my derivation can be expanded to find the formula for larger numbers of dimensions, but it gets trickier, and I don't yet have a formula
Keith, can you please express the problem rigorously so that the answers to my 3 questions above are included either explicitly or implicitly?
Keith, apologies if the problem you were thinking of is not the one I'm talking about above; if I've gotten it wrong, please post the problem you were thinking about, and we'll have two interesting problems to talk about! Andy
On Mar 30, 2016, at 8:35 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[I]f you have N random points on a line, and you measure their distances from a point, there are always (except, again, in the case of measure-zero coincidences) exactly N(N-1)/2 + 1 possible different sort orders. Which sort order you get depends on where the point you measure the distance from is. That point doesn't have to be on the line.
Also, if you have N random points in M-dimensional space, there are once again N(N-1)/2 + 1 possible different sort orders, *if* the points you measure the distances from are all constrained to a line.
But now I'm wondering about the more general case. For instance the number of sort orders of distances to N random points in a plane, as measured from any point in the plane. I think it would not be a constant. Maybe I should write a simulation. For definiteness, I'll say the N points are distributed, with equal probability density per unit area, within a circle. The point the distances are measured from need not be within the circle.
Would I get a different answer if, instead of a region of a plane, the distribution was over the whole of a sphere, a torus, or a Klein bottle? (Distances of course to be measured along the surface in each case.) What about instead of a Euclidian metric (sqrt(x^2+y^2)), using a Minkowski metric (sqrt(x^2-y^2))? But then how would I sort the imaginary distances?
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