Carpenter's rule problem

The carpenter's rule 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 Connelly, Demaine & Rote (2003).

Combinatorial proof

Subsequently to their work, Ileana Streinu provided a simplified combinatorial proof formulated in the terminology of robot arm motion planning. 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 & Whiteley (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 of length smaller than π, but on the surface of a sphere rather than in the Euclidean plane. This result was extended by Panina & Streinu (2010) for spherical polygons of edge length smaller than 2π.

Generalization

John Pardon (2009) generalized the Carpenter's rule problem to rectifiable curves. He showed that every rectifiable Jordan curve can be made convex, without increasing its length and without decreasing the distance between any pair of points. This research, performed while he was still a high school student, won the second-place prize for Pardon in the 2007 Intel Science Talent Search (Cunningham 2007).

See also

• Curve-shortening flow, a continuous transformation of a closed curve in the plane that eventually convexifies it

References

• Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete and Computational Geometry 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840. Preliminary version appeared at 41st Annual Symposium on Foundations of Computer Science, 2000.
• Cunningham, Aimee (17 March 2007), "The Next Generation", Science News: 166.
• Streinu, Ileana (2000), "A combinatorial approach to planar non-colliding robot arm motion planning", Proceedings of the 41st Annual Symposium on Foundations of Computer Science, IEEE Computer Society, pp. 443–453, doi:10.1109/SFCS.2000.892132, MR 1931841
• Panina, Gaiane; Streinu, Ileana (2010), "Flattening single-vertex origami: The non-expansive case", Computational Geometry: Theory and Applications 43 (8): 678–687, arXiv:1003.3490, doi:10.1016/j.comgeo.2010.04.002, MR 1931841
• Pardon, John (2009), "On the unfolding of simple closed curves", Transactions of the American Mathematical Society 361 (4): 1749–1764, doi:10.1090/S0002-9947-08-04781-8, MR 2465815.
• Streinu, Ileana; Whiteley, Walter (2005), "Single-vertex origami and spherical expansive motions", Discrete and Computational Geometry: Japanese Conference, JCDCG 2004, Tokyo, Japan, October 8-11, 2004, Revised Selected Papers, Lecture Notes in Computer Science 3742, Springer-Verlag, pp. 161–173, MR 2212105