A notion of "radially monotone" cut paths is introduced as an effective
choice for finding a non-overlapping edge-unfolding of a convex polyhedron.
These paths have the property that the two sides of the cut avoid overlap
locally as the cut is infinitesimally opened by the curvature at the vertices
along the path. It is shown that a class of planar, triangulated convex domains
always have a radially monotone spanning forest, a forest that can be found by
an essentially greedy algorithm. This algorithm can be mimicked in 3D and
applied to polyhedra inscribed in a sphere. Although the algorithm does not
provably find a radially monotone cut tree, it in fact does find such a tree
with high frequency, and after cutting unfolds without overlap. This
performance of a greedy algorithm leads to the conjecture that spherical
polyhedra always have a radially monotone cut tree and unfold without overlap.Comment: 41 pages, 39 figures. V2 updated to cite in an addendum work on
"self-approaching curves.
