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Convexi cation of Planar Polygons in R 3 Boris Aronov ,

Summary: Convexi cation of Planar Polygons in R 3
Boris Aronov  ,
Jacob E. Goodman y ,
Richard Pollack z
In memory of Paul Erd}os
We prove that a planar polygon can be convexi ed, in R 3 , by O(n) simple moves. This
bound is already known [Bea99], but we believe our argument is simpler.
We consider a variant of the problem rst proposed by Erd}os in [Erd35], concerning a simple
linkage, i.e., a family x 1 ; x 2 ; : : : ; x n ; x n+1 = x 1 of points, whose links (or edges) x i x i+1 , i = 1; : : : ; n
are xed in length, and whose joints (or vertices) x i are free to move continuously, subject to the
condition that, except for x i 1 x i and x i x i+1 , which meet at their common endpoint x i , no two
edges meet. We are interested in how e∆ciently such a linkage can be moved into convex position
if it is permitted to move in three-dimensional space. For a complete history of the problem, see
[Bea99] and [Tou99].
We call a continuous motion of a linkage simple when only O(1) joint angles or angles between
the planes of consecutive joints change. Consider a planar quadrilateral with distinguished edge e
and edge lengths e, a, b, c, in counterclockwise order around its perimeter. We call the quadrilateral
special if e + a = b + c and e + c < a + b, or if e + c = a + b and e + a < b + c.
Main Theorem. A linkage in the form of a simple planar n-gon can be unfolded to a convex planar


Source: Aronov, Boris - Department of Computer Science and Engineering, Polytechnic Institute of New York University


Collections: Computer Technologies and Information Sciences