Suppose I have a square, let's say the sides have length 1. I will then partition the square into $N^2$ sub-squares, where $N \in \mathbb{N}$ and the sub-squares are all the same size. Now, suppose we place two points $A$ and $B$ randomly within the large square a distance of $D$ apart from each other, and then draw the line segment $AB$. The question is, what is the expected number of small squares that $AB$ will pass through?

I have no idea how to solve this analytically, and was thinking of attempting to get a sense using simulation, but was curious what I could learn about it here. (This isn't a homework question; it's related to a problem I'm working on in economics research.)

  • $\begingroup$ So basically you have $N$ copies of the partitioned square (one for each $x$) and you pick the same points $A$ and $B$ in each copy and add up the total squares intersected in the copies? Also, is each partition "uniform"? i.e. the sub-squares are equal sized, like a chessboard grid etc? $\endgroup$ – Aryabhata Mar 22 '12 at 22:40
  • $\begingroup$ All of the sub-squares are the same size. But we only pick one point $A$ and $B$ in the big square, not in each sub-square. Maybe it's clearer if you think about the points first, and then do the partition. Thanks. $\endgroup$ – itzy Mar 22 '12 at 23:48
  • $\begingroup$ It is still not clear. What does "partition into $x^2$ sub-squares, where $x \in \{1,2,\dots, N\}$" exactly mean? Perhaps you meant to say partition into $N^2$ sub-squares? Please clarify. $\endgroup$ – Aryabhata Mar 23 '12 at 0:01
  • $\begingroup$ I mean partition it into some prespecified number of squares. Now that I think about it, $x=1$ doesn't make any sense -- that wouldn't be a partition. But we could partition it into 4 squares, or 9 squares, etc. So I meant, choose an integer greater than one, square it, and that's how many sub-squares there will be. I could have specified the number of partitions, but I want the answer to depend on $x$. $\endgroup$ – itzy Mar 23 '12 at 0:05
  • $\begingroup$ If you just say $N^2$ sub-squares, the answer will depend on $N$ and that would be good enough, won't it? $\endgroup$ – Aryabhata Mar 23 '12 at 0:07

The number of of the cells touched by the line is (following Byron's answer) $N (|X_1-X_2| + |Y_1-Y_2| +1) $ . Calling $X =|X_1 - X_2|$ and $Y = |Y_1 - X_2|$ and leaving apart the $N$ factor, this is approximately equivalent, for large $N$, to $d_M=X+Y$, the Manhattan distance.

Hence, I'll attack the following problem: We throw two points at random on the unit square; we want to compute $$E(d_M | d)$$

where $d=\sqrt{X^2+Y^2}$ is the euclidean distance.

The expected number of touched cells, in this approximation, would be $N E(d_M | d) + 1$.

It's not difficult to see that $X$ (absolute value of the difference of two uniform v.a) has a triangular density : $f_X(x)=2(1-x)$ The same for $Y$, and both are independent, hence: $$f_{XY}(x y)=4(1-x)(1-y)$$ on the unit square.

UPDATE: Below I found a much simpler solution

--------------- begin ignore -----

Conditioning on a fixed value of $d$ implies that we must restrict to an arc, the intersection of $x^2+y^2=d^2$ and the unit square.

Let's assume first that $d<1$

Note that $d\le d_M \le \sqrt{2} d$

To compute $P(d_M \le \xi | d)$, we must integrate over two pieces of arcs begining on the axis (because of symmetry, we compute just one and multiply by two). (blue arcs in the figure) The first limit point is given by $$x_1=\frac{\xi - \sqrt{2 d^2-\xi^2}}{2}$$ enter image description here Further to change the integration over the arc for an integral over $x$, we note that $ds=\sqrt{1+\frac{x^2}{y^2}}{dx} = \frac{d}{y}dx$. So, leaving out the constants (that disappear on the division), we get that the cumulative distribution function (conditioned on $d$) is given by

$$F_{d_M|d}(\xi;d)=P(d_M \le \xi | d) = \frac{2 \int_{0}^{x_1}\left( 1-x\right) \,\left( \frac{1}{\sqrt{{d}^{2}-{x}^{2}}}-1\right) dx}{\int_{0}^{d}\left( 1-x\right) \,\left( \frac{1}{\sqrt{{d}^{2}-{x}^{2}}}-1\right) dx}$$

Finally, we must integrate to get the expected value

$$E(d_M |d) = d+ \int_{d}^{\sqrt{2} d} \left[1- F_{d_M|d}(\xi;d) \right] \; d\xi $$

This is all.. but it seems pretty complicated to get in close form. Let's see Maxima:

integrate((1-x)*(1/sqrt(d^2-x^2)-1), x,0,x1);

$$\frac{2\,\mathrm{asin}\left( \frac{x1}{d}\right) +2\,\sqrt{{d}^{2}-{x1}^{2}}+{x1}^{2}-2\,x1-2\,d}{2}$$

assume(dm>d); assume(dm<sqrt(2)*d);

This gets a little messy. Let's try at least some numerical values:

dd+quad_qags(1-fdist(dm,dd), dm, dd, sqrt(2)*dd)


Let's simulate to check: Matlab/Octave:

xy = [abs(t(:,1)-t(:,3)),abs(t(:,2)-t(:,4))];
>>>ans =  1.0174

Not bad, I'd say. Some other values:

  d   Maxima      Octave (simul)
 0.9  1.15847     1.1569
 0.8  1.01798     1.0174
 0.7  0.88740     0.8863
 0.6  0.75983     0.7579
 0.5  0.63328     0.6331
 0.4  0.50698     0.5063
 0.3  0.38062     0.3808

The case $d>1$ should be solved in a similar way.

--------------- end ignore -----

A much simpler solution:

Let's change to polar coordinates: $x = r \cos(t)$, $y = r \sin(t)$. The joint density is

$$f_{r,t}(r,t) = 4 r (1- r \cos(t))(1- r \sin(t))$$

in the domain $0 \le r \le \sqrt{2}$, $0 \le t \le \pi/2$ for $r\le 1$, $ arccos(1/r) \le t \le arcsin(1/r)$ for $r > 1$

The conditional density is then

$$f_{t|r}(t|r) = \frac{1}{g(r)} (1- r \cos(t))(1- r \sin(t))$$

where $g(r)$ is the normalization constant (indepent of $t$).

Assuming first that $r<1$, then

$$g(r) = \int_0^{\pi/2} (1- r \cos(t))(1- r \sin(t)) dt = \frac{{r}^{2}}{2}-2\,r+\frac{\pi }{2}$$

Further, $d_M= x + y= r (\cos(t)+\sin(t))$, so the conditional expectation is given by

$$E[d_M | r] = r \frac{1}{g(r)} \int_0^{\pi/2} (1- r \cos(t))(1- r \sin(t)) (\cos(t)+\sin(t)) dt$$

which gives

$$ E[d_M | r] = r \frac{4\,{r}^{2}-3\,\left( \pi +2\right) \,r+12}{3\,{r}^{2}-12\,r+3\,\pi }$$

For $r>1$ it's more complicated. The result is

$$ E[d_M | r] = r \frac{\sqrt{{r}^{2}-1}\,\left( 4\,{r}^{2}+8\right) +\left( 6\,\mathrm{asin}\left( \frac{1}{r}\right) -6\,\mathrm{acos}\left( \frac{1}{r}\right) -6\right) \,{r}^{2}-4}{3\,{r}^{3}-12\,r\,\sqrt{{r}^{2}-1}-\left( 6\,\mathrm{asin}\left( \frac{1}{r}\right) -6\,\mathrm{acos}\left( \frac{1}{r}\right) -6\right) \,r} $$

The figure shows $E[d_M | r] / r$, ie. the factor by which the expected Manhattan distance exceeds the euclidean distance (we already knew that this must be in the $[1,\sqrt{2}]$ range).

enter image description here

(BTW: sorry if the formatting is not optimal, but I got sick of my Chorme crashing on the edition, lots of times, sometimes losing changes - am I the only one?)

Added: Notice that for small distances ($r \to 0$) the expected number of squares results $r N \frac{4}{\pi} +1 $, which agrees with Ronald's answer.

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Imagine the subdivided square as an $N\times N$ chessboard of subsquares. Every point on the board belongs to one subsquare with coordinates $(r,c)$, where $r$ (the row index) ranges from $1$ to $N$ and similarly for the $c$ (the column index).

Now the number of subsquares that the line segment $\overline{AB}$ between two uniformly selected points $A$ and $B$ touches equals $d(A,B)=|A(r)-B(r)|+|A(c)-B(c)|+1$. Here $A(r)$ means row index for $A$, and similarly for the other notation. The points $A$ and $B$ are in "general position" because of the randomness, so the segment always passes through the maximum number of subsquares. The segment doesn't hit any corners.

Since the random variables $A(r),B(r),A(c),B(c)$ are independent and uniformly distributed on $\{1,\dots, N\}$, it is not hard to calculate that the expected number is $$\mathbb{E}(d(A,B))=1+{2(N^2-1)\over3N}.$$

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  • $\begingroup$ I think your answer is correct if we don't condition on $D$. But I meant for the answer to depend on $D$. So for example, if $N$=2 and $D$ is small enough, then I would think $E(d(a,B))$ would be close to 1. But if $D$ is large $E(\cdot)$ must be at least 2. Does that make sense? $\endgroup$ – itzy Mar 23 '12 at 13:35
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    $\begingroup$ @itzy Yes, I could only solve the simplest case of this problem. $\endgroup$ – user940 Mar 23 '12 at 15:47

Suppose that you divide your square into $N^2$ sub-squares, where $N\in\mathbb{N}$. Then suppose that $A=(x_{1},y_{1})$ and $B=(x_{2},y_{2})$ where $x_{1},x_{2},y_{1}$ and $y_{2}$ are real numbers in $[0,1]$. Then the segment $AB$ must pass through at least $N|y_{2}-y_{1}|$ squares in the $y$ (or vertical) direction and at least $N|x_{2}-x_{1}|$ squares in the $x$ (or horizontal) direction. So, we would expect the segment $AB$ to pass through $N|x_{2}-x_{1}|+ N|y_{2}-y_{1}|$ squares.

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  • $\begingroup$ Isn't this double-counting some squares? A square that is passed through in the $x$ direction may be the same square that is passed through in the $y$ direction. $\endgroup$ – itzy Mar 23 '12 at 13:27
  • $\begingroup$ It ends up that this answer is an undercount by 1, because we should include the squares at the end and beginning of each direction (contrary to usual counting). The $x$ and $y$ directions are kind of independent, because you can either pass up into the next square, or left into the next square - there isn't really a risk of double-count. However, we are assuming that the line doesn't pass exactly through a corner, which would complicate things a bit. Which is fine, because it's almost sure that it won't pass through a corner. $\endgroup$ – Ronald Mar 31 '12 at 20:11

I broadly agree with @Byron Schmuland's answer, which seems correct, but I have some additional suggestions.

I am assuming that the size of a small square is 1, for simplicity, compared to a distance of D.

If the question is to place A and B "randomly" with a given distance of D apart, this may fall foul of Bertrand's paradox: that this concept of randomness is not clearly enough defined.

However, it seems to make sense to place A first, suggesting we should place B on an arc with centre A and radius D. The arc will be restricted depending on how close it is to the edge of the square.

Let's imagine that we choose the point A, and then the angle $\theta$ is chosen randomly. Each direction $\theta$ is equally likely (by assumption here and by symmetry of the square).

Then the number of squares passed through will be $D|\sin(\theta)|+D|\cos(\theta)|+1$ from Byron's work and using a right-angled triangle.

$$E[D|\sin(\theta)|+D|\cos(\theta)|+1]$$ $$= D \cdot E[|\sin(\theta)|+|\cos(\theta)|] + 1$$ $$= 4D/\pi + 1$$

Or, considering $(1/N)$ to be the length of a little square (according to the question as given) $$= 4ND/\pi + 1$$

In the case where $N = 2$ ($2$ little squares per side) and $D = 0.5$ ($D$ is half the length of a long side), we expect to pass through $2.27$ squares. Seems right. Sometimes it will pass through only $1$, but more often $3$.

This will break down badly when D gets long compared to the side of the big square, I imagine ($D>1$?), when the line no longer has the option of being horizontal or vertical.

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  • $\begingroup$ +1. I have tried to make your formulae a little prettier. When you say "Then the number of squares passed though", I think you mean "Then the expected number of squares passed though" $\endgroup$ – Henry Mar 31 '12 at 2:44
  • $\begingroup$ @Henry Thanks, and you're right. Any thoughts on the N=1 case? Have I made an error? $\endgroup$ – Ronald Mar 31 '12 at 2:46
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    $\begingroup$ I am not interested in the original finite square: I think the question would be better posed as an infinite integer grid. $\endgroup$ – Henry Mar 31 '12 at 2:51
  • $\begingroup$ agree; at least, it's much easier in an infinite grid :) $\endgroup$ – Ronald Mar 31 '12 at 9:50
  • $\begingroup$ +1 for mentioning Bertrand's paradox. In fact that is what I had in mind when I asked itzy to clarify how the points are selected. Alas, I might have been too terse, and that comment was ignored. Now we have a bounty! on a possibly ill-formed question. $\endgroup$ – Aryabhata Apr 1 '12 at 20:06

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