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Non-uniform corner cutting
The convergence of a non-uniform corner cutting process is
investigated. It is shown that the limit curve will be differentiable provided the proportions of the corner cuts are kept within appropriate
constraints.
Keywords Subdivision, corner cuttin
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 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.Comment: 24 pages, 15 figures, 6 references. Version 2 includes a solution to
one of the open problems posed in Version 1, concerning quasigeodesic loop
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