Document Type
Article
Publication Date
2-4-2011
Abstract
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.
Recommended Citation
O'Rourke, Joseph and Vîlcu, Costin, "Conical Existence of Closed Curves on Convex Polyhedra" (2011). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/29
Included in
Computer Sciences Commons, Discrete Mathematics and Combinatorics Commons, Geometry and Topology Commons
Comments
Author’s submitted manuscript.