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Topic: Distances in Bounded Regions
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Kevin Brown

Posts: 31
Registered: 12/12/04
Distances in Bounded Regions
Posted: Jul 10, 1996 8:43 PM
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One of the standard problems in introductory calculus courses is
to find the average distance between two randomly selected points
inside a unit sphere. (This problem also comes up now and then in
sci.math and rec.puzzles.) The popularity of this particular problem
is probably due to the fact that it happens to lead to an integral
that can be evaluated in "closed form" to give a nice explicit answer
(36/35). However, for shapes other than a sphere the solution is not
always so simple, although sometimes a parametric formulation of the
distance density can be found to simplify the analysis.

Anyway, I've been trying to compile a catalog of the average
distances (and powers of distances) within various shapes of various
dimensions, and also the "distance densities" for these shapes. I'd
appreciate any information that anyone can provide on this subject.
So far I've looked mainly at circles (spheres) and squares (cubes)
of different dimensions, starting with the classic problem of the
solid unit sphere in 3D space. For completeness, I'll start with
this well-known case:

Let R and r be the radial distances from the origin to two randomly
chosen points in a unit sphere, and let w be the angle between these
vectors. The distance between the two points is

sqrt[ R^2 - 2Rrcos(w) + r^2 ]

Covering just the case R>r, we need to triple-integrate this
quantity over r, R, and w, and we need to weight the quantity
in proportion to the fraction of the two-point state-space
corresponding to each set of parameters.

For given values of r and R, the angle w defines a circle whose
circumference is proportional to sin(w), so this is the weight for
the w integration. Similarly each value of r and R defines a sphere
of surface area proportional to r^2 and R^2, respectively, so these
are the weights for the r and R integrations.

We can restrict our analysis to just the case R>r because the other
case (R<r) is symmetrical and has the same distribution of distances.
Therefore, we just need to integrate the distance function with the
appropriate weights for the ranges r=[0,1], R=[r,1], and w=[0,pi].
Then we divide the result by the triple integral of just the weights
(rR)^2 sin(w), which is 1/9. Thus, the problem reduces to the triple

1 1 pi
/ / / ______________________
9 | r^2 | R^2 | sin(w) / R^2 + r^2 - 2Rrcos(w) dw dR dr (1)
/ / /
0 r 0

which is easily evaluated to give the familiar result 36/35. In
fact, we can evaluate this integral with the distance function
raised to any integral power, to give the average of the nth
powers of the distances as (72*2^n)/(n+3)(n+4)(n+6).

A couple of comments can be made here. First, notice that the
weight for w is very fortuitous, because the factor of sin(w)
enables us to evaluate the integral in closed-form (using the
easy integral for sin(x)*sqrt(A-Bcos(x)) ). In contrast, the
seemingly simpler case of a unit DISK is actually more difficult
because it lacks this convenient weight factor.

Second, you might notice that if you try to cover both the cases
R>r and R<r at once by integrating over r=[0,1] and R=[0,1] you
may get a result like 21/20 instead of 36/35. The problem is that
the integral of sin(w) * sqrt[R^2 - 2Rrcos(w) + r^2] over the
range w=[0,pi] (divided by the integral of the weight) is

3/2 3/2
[ (R+r)^2 ] - [ (R-r)^2 ]
------------------------------------ (2)

which is formally symmetrical in R and r. Now, since R+r is always
non-negative there's not much ambiguity in evaluating the left hand
term in the numerator; we just take the positive value (R+r)^3,
glossing over the fact that there's really a square root there
in the 3/2 power, so we could have taken the negative root. However,
the right hand term is a bit tricky: do we evaluate this as (R-r)^3
or (r-R)^3 ? The answer depends on whether R>r or R<r. In these
two cases the above expression reduces to

R + (r^2)/3R if R > r

r + (R^2)/3r if R < r

so it's necessary (not just convenient) to treat the case separately.
Fortunately, due to symmetry, the cases give the same distribution
of distances, so we only need to treat one of them.

So much for the easy case. Now let's consider the distances on
a unit disk. This case can be formulated in essentially the same
way as with the unit sphere, except that the weights are different.
Each angle w now represents only a single point, so it's weight is
just 1. Each radius r and R now represents a circle with length
proportional to r and R respectively, so these are the weights.
The triple integral of these weights is pi/8, so the average
distance on a unit disk can be expressed as

1 1 pi
8 / / / ______________________
-- | r | R | / R^2 + r^2 - 2Rrcos(w) dw dR dr (3)
pi / / /
0 r 0

Unfortunately, this integral isn't as easy to evaluate as the
one for the sphere, because it lacks the sin(w) factor. However,
by messing around with some gamma functions we can show that the
result is 128/(45pi). We can also evaluate the above integral
for any integer power of the distance function. For odd powers
of the form 2k-1 the general result is

4k+5 3
2 (2k-1)! (k!) 1
------------------------ ----
[(2k+1)!] (k-1)! (2k+3) pi

whereas for even powers of the form 2k the general result is

[(k+1)!] (k+1)(k+2)

Incidentally, this last formula shows that if c[n] denotes the nth
Catalan number, then the average (2k)th power of distances on a unit
disk is just c[k+1]/(k+1).

Of course, as an alternative to (3) we could just integrate the unit
disk by scanning two points (x,y) and (X,Y) orthogonally across the
disk. The weight factors are all 1 in this case, and they integrate
to the squared area of the disk (pi^2), so we have the quadruple

1 a 1 b
1 / / / / _________________
-- | | | | /(x-X)^2 + (y-Y)^2 dY dX dy dx (4)
pi^2 / / / /
-1 -a -1 -b

where a = sqrt(1-x^2) and b = sqrt(1-X^2). This gives the same
results as (3), but it's slightly more laborious to integrate.

Now let's consider the distribution of distances on a 1x1 unit
square. We could use the approach of equation (4) and just
integrate the distances between two points (x,y) and (X,Y) as
each parameter ranges from 0 to 1, but the resulting quadruple
integral is not very easy to evaluate. A much more efficient
approach is to notice that the distances on a unit square are
distributed according to the parametric formulas
s(u,v) = / u^2 + v^2 dens(u,v) = 4(1-u)(1-v) (5)

Therefore, we just need to integrate dens(u,v)*s(u,v) as u and v
range from 0 to 1. The orthogonal way of formulating this double-
integral is

1 1
/ / __________
4 | (1-u) | (1-v) / u^2 + v^2 dv du (6)
/ /
0 0

This is a big improvement over the quadruple integral, but it still
is not easily evaluated in closed form. Let's try polar coordinates
in this parametric space by setting u = r cos(w) and v = r sin(w)
and integrating over the region u<v be letting r range from 0 to
1/cos(w) at each w from 0 to pi/4. For any incremental slice of w
the weight at r is proportional to r, and of course the integral
of r over this region u<v is just 1/2, so we have the integral

pi/4 1/cos(w)
/ /
8 | | (1-r sin(w))(1-rcos(w)) r^2 dr dw (7)
/ /
0 0

This value of this double-integral is

2 + sqrt(2) + 5 ln(1+sqrt(2))
----------------------------- = 0.521405433...

By incrementing the exponent of r in (6) we can evaluate the average
of the nth powers of distances on the unit square. In general the
results are of the form

A + B sqrt(2) + C ln(1+sqrt(2)
------------------------------ (8)

where the values of A,B,C,D are as shown below

n A B C D
--- --- ---- ---- -----
1 2 1 5 15
2 1 0 0 3
3 8 17 21 210
4 17 0 0 90
5 16 73 45 1008
6 29 0 0 210
7 384 3239 1155 47520
8 187 0 0 1575

Now, what about the unit cube, or the unit 4D hyper-cube, etc.?
The nice thing about the parametric distance density equations (5)
is that they immediately generalize to higher dimensions. In
general the parametric equations for the distance density of a
d-dimensional unit cube are

s(x1,x2,..,xd) = sqrt[ x1^2 + x2^2 + ... + xd^2 ]

dens(x1,x2,..,xd) = 2^d (1-x1)(1-x2)...(1-xd)

For even powers of the distance we can immediately evaluate the
d-dimensional analogs of equation (6). We find that the average
squared distance in a d-dimensional unit cube is simply d/6.
(This gives a nice trivia question: In what dimensional space
is the average squared distance in a unit cube equal to unity?)
The average nth powers of distances in a d-dimensional unit cube
are given by the following formulas for the first few even values
of n:

n average nth power of distance in d-cube
--- ----------------------------------------------
2 (d)/(1*6)
4 (5d^2 + 7d)/(2*90)
6 (35d^3 + 147d^2 + 88d)/(6*1260)
8 (175d^4 + 1470d^3 + 2789d^2 + 606d)/(24*9450)

What most intrigues me about the above is the parametric density
formulas for the distances inside a d-dimensional cube. Notice
that in each case the density of the vector V = [x1,x2,..xd] within
a bounded region B is proportional to the intersection of B with a
copy of B shifted by the vector V. It seems plausible to me that
this may be true in general. For example, take an arbitrary
tetrahedron T and a vector V. Is it true that the density of V in
T is proportional to the intersection of T with a copy of T
offset (without rotation) by V?

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