Ave length of intersection of all lines through a unit cube

[bahamagreen]

I'm having trouble even beginning to figure out how to approach solutions for this. I begin with a unit cube, and imagine all the possible lines that intersect the cube. I am assuming there must be an average length of these intersections; I want to find that average length.

Another way to express it is, for every point on and within the unit cube, what is the average length of all possible lines through those points, length meaning on and within the cube?

Here is where I am so far...

There is a group that intersect the cube through one perimeter point:
- those that pass through one of the vertices
- those that pass through one point of an edge

There is a group that intersect the cube through two perimeter points:
- those that enter and exit through adjacent vertices and intersect all points of the connecting edge (12 of these)
- those that enter and exit through opposite vertices of a face and intersect all points of a face diagonal (24 of these of length SQRT(2)
- those that enter and exit adjacent edges of a face and pass entirely through the face
- those that enter and exit opposite edges of a face and pass entirely through the face
- those that enter and exit through opposite faces
- those that enter and exit through adjacent faces
- those that enter and exit through opposite vertices of the cube - 4 of these and they are the longest at SQRT(3)

I think that is all...? It is not clear to me how to distinguish some instances of some groups - when or whether to include vertices as parts of edges, or when or whether to include edges as parts of faces. Is there a convention for this?

I know there are infinitely many intersection lengths here, but the shortest are zero where the intersection is a point, and the longest are SQRT(3) between most distal vertices.

For a while I was wondering if it would be mathematically clearer to use a unit sphere rather than a unit cube, but I have no proof the average intersection length would be the same... but that just got me wondering what radius sphere would have the same average intersection length as the unit cube.

Maybe I should back up a dimension and work with the average intersection length through a unit square?

The average intersection length through a point looks like zero.
The average intersection length through a unit line looks like zero, too; one instance of length 1 and an infinite number of length zero
The average intersection length through a unit square looks like it is greater than zero and is less than SQRT(2).

I'm thinking that using calculus on trigonometric functions is the way forward, but it looks like the problem may need to be broken up into parts corresponding to the different groups of intersections?

Anyone have any clues to figuring out an approach (maybe to just the unit square for now)?

 

[Nidum]

Place an N by N grid of node points on each face of the cube . Devise an algorithm that calculates distance from any nodal point on each face to any nodal point on the other faces . For any N find the average distance . Determine whether this average distance converges to some fixed value as N increases .

 

[mfb]

Didn't we have that thread a while ago? You have to specify what you want to average over. The cube surface area (->Nidum)? The solid angle (looks more natural I think)?
Just simulating it is probably faster than attempts to find analytic solutions.

 

[Stephen Tashi]

I am assuming there must be an average length of these intersections;

Problems like the one you pose are studied in the branch of mathematics called "geometric probability".

The average length isn't defined until you state a method for assigning a "measure" to the sets of lines used in doing the integration that produces the average. Different ways of assigning the measure can produce different averages.

We can try various probability measures.

For example (for line segments inside the cube):

1) Pick a face $S_A$ of the cube at random. Using a uniform distribution on that face, pick point $A$ from that face. Pick a different face $S_B$ at random. Using a uniform distribution on that face, pick a point $S_B$ from that face at random. Let the randomly selected line segment be $AB$. Let $L_A$ be the average length of such randomly selected line segments.

or:

2) Pick a point $C$ inside the cube from a uniform distribution over the volume of the cube. Draw a unit circle with center at $C$. Pick a point $U$ on the surface of the unit circle at random using a uniform distribution over the surface of the sphere. Let the line be $CU$ and let the segment of $CU$ that is inside the cube be the randomly selected line segment. Let $L_C$ be the average length of such randomly selected line segments.Whether $L_A = L_C$ is a complicated question and I don't know the answer. The point is that we can't be sure that $L_A = L_C$ without doing some analysis.

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Consider a deterministic situation. The average mass density of an object is its total mass divided by its total volume. Applying that analogy to lengths of line segments, we would need a number for the total length of all the line segments and another number for the "volume" of all the line segments. That approach won't work. So consider an alternate method for approximating the mass density of an object as $\frac{ \sum_i s_i dv_i}{\sum_i dv_i}$ where each $dv_i$ is a small sub-volume of the object and $s_i$ is a mass density representative of the density at each point in the sub-volume. If we can define a sub-volume $dv_i$ of the possible line segments in such a manner that each line segment in the subvolume has approximately the same length $l_i$ then we can approximate the average length of a line segment as $\frac{ \sum_i l_i dv_i}{\sum_i dv_i}$. However, the numerical outcome of this approach depends on how we define the subvolumes $dv_i$ of the totality of line segments.

 

[bahamagreen]

Thanks everyone... mfb mentioned "You have to specify what you want to average over.", and I was not sure how that would matter, but Stephen Tashi made it clear.

Let me explain where the question is coming from.

I'm thinking about a bound space of, for example, 10x10x10, so 1000 unit cubes of interest. I am imagining a point moving and bouncing off the inner bound space walls elastically and indefinitley so that within the bound space that the point is passing through these various cubes. For simplicity of subsequent thinking, I was wanting to assign the velocity of the point so that it might have a convenient form like "average number of cubes passed through per second"... so you see why my question. I was hoping there might even be an old proof that the average path length through the cubes might be something interesting like 1/SQRT(2)...

The method of using pairs of face points is interesting because I can turn reflections of the point into a continuing line path projection into an adjacent bound space sharing that face, by flipping the angles of incidence as inversions/rotations of the second bound space, and can repeat this indefinitely to determine what is the location of the nth reflection on the original bound space walls. That is, the path of the point could be imagined as a continuous straight line through a series of outside bound spaces, each inverted/rotated as needed to keep the path straight. I would need an indexing method to keep track of these projected bound spaces, axes, and rotations.
But the thing, this would provide a path to estimating the length of average intersection for the primary bound space, scalable to unit cubes or other.

The method of using the circle... it's nice but my intuition suggests that might not be the way, here. In my toy, the point is moving to make the lines, so its motion as it encounters the various cube faces seems to be better matched to a face distribution oriented production of the line... ? It also works better if I use the adjustments to the outer bound spaces to host the reflection history as a straight line projection through a series of inverted/rotated spaces.