Threadable Curves
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We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. Our main result is a linear-time algorithm for deciding whether a polygonal curve is threadable, and if so, finding a sequence of rigid motions to thread it through a hole. In addition, we sketch arguments that show that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve, and that threading a 3D polygonal curve through a point-hole in a plane can be decided in quadratic time. Finally, we connect threadable curves to the problem known as "moving a chair through a doorway."
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