Unfolding Face-Neighborhood Convex Patches: Counterexamples and Positive Results
Joseph O’Rourke · Canadian Conference on Computational Geometry · 2013
We address unsolved problems of polyhedra in a new context, focusing on special convex patches|disklike polyhedral subsets of the surface of a convex polyhedron. One long-unsolved problem is edge-unfolding prismatoids. We show that several natural strategies for a prismatoid can fail, but obtain a positive result for \petal unfolding topless prismatoids, which can be viewed as particular convex patches. We also show that the natural extension of an earlier result on face-neighborhood convex patches fails, but we obtain a positive result for nonobtusely triangulated faceneighborhoods.