Spider and fly problem

Question

I am a novice in computational geometry and I would like to address some questions about the spider - fly problem as stated in wiki: https://en.wikipedia.org/wiki/The_spider_and_the_fly_problem

1. I know that there is at least a shortest arc on the cuboid, connecting the point "fly" and the point "spider" but I do not know why this is called geodesic? Geodesic is locally a shortsest arc but not via versa...

2. In order to find a shortest arc (according to wiki), we have to:

   a. Check in all the 11 possible nets (edge unfolding, without overlapping) of the cuboid if the line segment "fly""spider" belongs to the corresponding net.

   b. Compare the lengths of these line segments (at most 11) and choose the shortest. This is a shortest arc. But why to check only the edge, not overlapping unfoldings and not all the possible general, not overlapping unfoldings? Maybe there is a shortest arc also there...

Thanks.

EDIT

When I say general unfolding, I mean such as: Star Unfolding, Source Unfolding, Aleksandrov Unfolding, etc

Answer 1

According to user 2661923 helpfull comments, it suffices to examine all the possible "edge paths" that lead spider "A" to fly point "B". So the possible paths are:

1. A-left wall-floor-B

2. A-left wall-B

3. A-left wall-ceiling-B

4. A-floor-B

5. A-floor-left wall-B

6. A-floor-left wall-ceiling-B

7. A-celling-B

8. A-ceilling-left wall-B

9. A-ceilling-left wall-floor-B

Notice we do not have to examine right wall, bcs of symmetry and of course not to take the same edge again as it would make the path greater.

According to each of these edge paths we can produce an edge unfolding of the cuboid.

Now because in the above 9 considerations the segment $AB$ lie inside the appropriate planar net (I made some paper modeling and found it out) we can easily find the distance accordingly. This is not always the case if we have a general cuboid....