Carpenter's ruler problem

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The carpenter's ruler problem is a discrete geometry problem, which can be stated in the following manner: Can a simple planar polygon be moved continuously to a position where all its vertices are in convex position, so that the edge lengths and simplicity are preserved along the way? A closely related problem is to show that any polygon can be convexified, that is, continuously transformed, preserving edge distances and avoiding crossings, into a convex polygon.

Both problems were successfully solved by Robert Connelly, Erik Demaine and Günter Rote in 2000.

Subsequently to their work, Ileana Streinu provided a simplified combinatorial proof. Both the original proof and Streinu's proof work by finding non-expansive motions of the input, continuous transformations such that no two points ever move towards each other. Streinu's version of the proof adds edges to the input to form a pointed pseudotriangulation, removes one added convex hull edge from this graph, and shows that the remaining graph has a one-parameter family of motions in which all distances are nondecreasing. By repeatedly applying such motions, one eventually reaches a state in which no further expansive motions are possible, which can only happen when the input has been straightened or convexified.

Streinu and Whitely (2005) provide an application of this result to the mathematics of paper folding: they describe how to fold any single-vertex origami shape using only simple non-self-intersecting motions of the paper. Essentially, this folding process is a time-reversed version of the problem of convexifying a polygon, but on the surface of a sphere rather than in the Euclidean plane.

[edit] References

  • Streinu, Ileana; Whiteley, Walter (2005). "Single-vertex origami and spherical expansive motions". Discrete and Computational Geometry: 161–173, Springer-Verlag, Lecture Notes in Computer Science 3742. MR2212105.