Abstract
There are two known ways to unfold a convex polyhedron without overlap: the star unfolding and the source unfolding, both of which use shortest paths from vertices to a source point on the surface of the polyhedron. Non-overlap of the source unfolding is straightforward; non-overlap of the star unfolding was proved by Aronov and O’Rourke (Discrete Comput Geom 8(3):219–250, 1992). Our first contribution is a simpler proof of non-overlap of the star unfolding. Both the source and star unfolding can be generalized to use a simple geodesic curve instead of a source point. The star unfolding from a geodesic curve cuts the geodesic curve and a shortest path from each vertex to the geodesic curve. Demaine and Lubiw conjectured that the star unfolding from a geodesic curve does not overlap. We prove a special case of the conjecture. Our special case includes the previously known case of unfolding from a geodesic loop. For the general case we prove that the star unfolding from a geodesic curve can be separated into at most two non-overlapping pieces.
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Notes
Private communication from J. O’Rourke
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Acknowledgments
We thank Timothy Chan for suggesting Lemma 8. We thank Joseph O’Rourke, Costin Vîlcu, and anonymous referees for helpful comments. Partially supported by NSERC
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Kiazyk, S., Lubiw, A. Star Unfolding from a Geodesic Curve. Discrete Comput Geom 56, 1018–1036 (2016). https://doi.org/10.1007/s00454-016-9795-1
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DOI: https://doi.org/10.1007/s00454-016-9795-1