Flipping objects in a bounded region

Shape and space, angle and circle properties, ...
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Flipping objects in a bounded region

Post by Bubbler » Mon Feb 23, 2015 12:03 pm

(Sorry, this problem is kind of computational geometry, not math-oriented...)

A convex polygonal region R with k vertices is given on a 2D plane, and there are n convex polygons Pi, with k vertices each, which are pairwise disjoint and completely inside R. These polygons do not sum to R; in other words, there are some gaps inside R that are not part of any of Pi.

Now you want to flip each Pi - place the mirror image Pi' of Pi so that the position of its center of mass remains constant. Each Pi' must still lie inside R and be pairwise disjoint. Rotation of Pi' around the center of mass is freely allowed.

Is it possible to computationally determine whether each polygon can be successfully flipped or not?

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Re: Flipping objects in a bounded region

Post by jaap » Mon Feb 23, 2015 3:00 pm

I don't doubt that it can be done, but it will be very difficult. You will have to reduce it to a combinatorial problem.
That means that the n degrees of freedom, which are the mirror angles for the n pieces, each have to be restricted to a finite set of values in such a way that a solution is present whenever the original problem has a solution.

However even if this reduction can be done, it may be the case that this class of combinatorial problem is difficult. It would not surprise me if it is a PSPACE-complete problem, even if all the pieces are only allowed to be mirrored vertically or horizontally.
I have written a page about PSPACE completeness for a different kind of puzzle here: http://www.jaapsch.net/puzzles/pspace.htm
It feels to me that there will probably be a way to define wires, AND and OR widgets, etcetera using only horizontal/vertical mirroring of pieces. If that is the case, then it means that if you were to actually find a program that can solve it quickly (in polynomial time) then you will win a million dollars.

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