The shortest path between two points on a cuboid on its surface

Question

I can't find a universal solution to "The Spider and the Fly Problem" (the shortest path between two points on a cuboid on its surface). Everybody solves a one specific case but what when two points can be anywhere?

My idea was to create an algorithm that considers various nets of a cuboid, calculates shortest paths on 2D and then returns the minimum but I have no idea for the algorithm to generate these grids (I guess hardcoding all combinations is not the best way).

Answer 1

Simplistic approach (only works where the points are on the same or adjacent faces)

Flatten the cube structure to 2d as follows...

1. Start with a face containing one of the two points. If this also contains the other point, you can stop there and the solution is trivial.
2. There are only 4 neighbouring faces. If any of them contain the other point, you can place that face adjoining the first, and plot the straight line.
3. Otherwise, then the points are on opposing faces. You need to try placing the final face adjoining each of the 4 neighbouring faces, and choose the shortest of the 4 alternatives. This will not always give the best solution, but it's not far off, and is cheap.

Generic approach

Jim Propp's surface distance conjecture is that For a centrally symmetric convex compact body, the greatest surface distance between two points is achieved only for pairs which are opposites through the centre. My conjecture based on that would be that the shortest distance is approximately where the plane made by the two points and the centre of the body meets the surface. So you simply need to find where that plane intersects the faces using 3d geometry, and use the faces that are crossed by the shorter of the two alternatives when looking at possible routes. If the plane runs along an edge of the cube (e.g. if the points are on opposite faces and are both between the centre of the face and the corner of the face, and those corners are linked by an edge) then routes through both faces should be considered, although I speculate they will be equivalent lengths.

This solution is more generic, and also satisfies scenarios where the points are on the same face, connected faces and opposite faces.

The only problem with this approach arises where the line between the two points passes through the centre of the body, which by definition means that the two points are exactly opposite each other, because that means the 3 points are in a straight line, so there isn't a plane...

Answer 2

I think this is a good question, for which the answer is not at all obvious. In the smooth realm, it is an extraordinarily difficult problem. Geodesics (shortest paths) on a sphere (which is a smooth analog of a cube) are easy to find. Geodesics on a biaxial ellipsoid (an ellipsoid of revolution; one cross section is a circle) are much harder to find. Finding geodesics on a triaxial ellipsoid (a smooth analog of a general cuboid) was a challenging unsolved problem in the first half of the 19th century. See the Wikipedia page.

On the other hand, geodesics on cuboid are made from straight line segments so are much simpler. But some of the difficulty of the problem remains.

You may be able to find some literature on the subject if you search for the term "net". A polyhedron cut along some edges so that it can be flattened is often called a "net". I was able to quickly find a site that claims (without proof) that there are just 11 different nets for a cub(oid). But I agree with you that hard coding all the variations is not the best way.

It's not even obvious to me that the approach using nets will work for all polyhedra. I think I see an argument that will work for cuboids, but for general polyhedra, even convex polyhedra, it is not known whether they must have even one net. See the Wikipedia page. I think a satisfying solution to the problem on cuboids should work more generally on polyhedra, and the net idea seems to be insufficiently general, in my view.

What I'm thinking might work is a dynamic programming solution, where you look at the different edges your path can pass through between the initial and final points. There is a hierarchy of edges (those on the starting face; those containing a vertex on the starting face; those on the faces adjacent to the starting face; etc.). For each point on each of those edges you can find the minimum distance to the start point, culminating in the minimum distance from the end point to the start point.

Another way to think about this is to use something akin to the reflection principle, except instead of reflections, we use rotations in space which rotate the polyhedron about one of its edges so that the other face adjacent to the edge becomes coplanar with the starting face. Then we don't have to worry about whether we have a good net or not. You just pick a sequence of edges so that the final point is eventually rotated onto the plane of the initial face. The sequence of edges is finite because any loop is not part of a minimal path. I'll think about how I might be able to communicate this idea better.