Conical Existence of Closed Curves on Convex Polyhedra

Details Conical Existence of Closed Curves on Convex Polyhedra Conical Existence of Closed Curves on Convex Polyhedra

2011-02-04

arxiv.org/abs/1102.0823v2

Computer Science

Discrete Mathematics

24 pages, 15 figures, 6 references. Version 2 includes a solution to one of the open problems posed in Version 1, concerning q

Scientific paper

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of "source unfolding" of the entire surface of P to one non-overlapping piece, as reported in a companion paper.

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