Twisting prisms : Do all polygon prisms behave in the same manner?
First, some observations:
If we consider the cylinder containing all the vertices of the prism (including the circumcircles of the polygons), polygon $B$ is moving back and forth in this cylinder like a rotating piston.
Given a radius $r$ and a side length $l$, consider the point $B_1$.
It is always at a distance $l$ from $A_1$.
So it is always on the sphere of radius $l$ centered at $A_1$.So the point $B_1$ moves along the path which is the intersection of the cylinder and the sphere. So this path depends only on $r$ and $l$.
Now, some answers:
Your hypothesis is correct. You can see that the path of $B_1$ does not depend on how many points are in the polygon. It depends only on $r$ and $l$. The same is true for the motion of B towards and away from A.
This question is really about visualizing a hyperboloid of one sheet. The best way to visualize it, however, is just to make one yourself. Cut two circles of cardboard, make little cuts around their edges to hold the string, and thread some string between the two. Then you can see it in 3D in your own hands.
After you play with it, it will be obvious why the strings touch only at 180°.If you are talking about the angle $\angle A_2 A_1 B_1$, then this question does not have a simple answer. Unlike the previous two questions, this angle will depend on $A_2$, whose position relative to $A_1$ depends on the number of sides of the polygon, as well as on $r$ and $l$. You can extend your equation for the location of $B_1$ with some trigonometry to calculate this angle, but unfortunately the formula will just be a big mess.