Carpenter's Rule Problem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. Bot...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 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.
