  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.
