Conical Existence of Closed Curves on Convex Polyhedra

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 isomet-rically embedded on the surface of a cone Λ, with the apex a of Λ 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.