15 Cut edges: strengthening Lemma: The cut edges of an edge unfolding of a convex polyhedron to a single, connected piece form a spanning tree of the 1-skeleton of the polyhedron.[Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, Snoeyink 01]
16 What is an unfolding?Cut surface and unfold to a single nonoverlapping piece in the plane.
44 Open: Fewest NetsFor a convex polyhedron of n vertices and F faces, what is the fewest number of nets (simple, nonoverlapping polygons) into which it may be cut along edges?≤ F≤ (2/3)F[Spriggs]≤ (1/2)F[Dujmenovic, Moran, Wood]≤ (3/8)F[Pincu, 2007]
45 Outline What is an unfolding? Why study? Main questions: status Convex: edge unfoldingConvex: general unfoldingNonconvex: edge unfoldingNonconvex: general unfoldingOrthogonal polyhedra
46 Status of main questions ShapesEdge Unfolding?General Unfolding?Convex polyhedra???Yes: always possibleNonconvex polyhedraNo: not always possible
47 General Unfoldings of Convex Polyhedra Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net).Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]Star unfolding [Aronov & JOR ’92]Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007][Poincare?]
55 Geodesics & Closed Geodesics Geodesic: locally shortest path; straightest lines on surfaceSimple geodesic: non-self-intersectingSimple, closed geodesic:Closed geodesic: returns to start w/o cornerGeodesic loop: returns to start at corner(closed geodesic = simple, closed geodesic)
56 Lyusternick-Schnirelmann Theorem Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics.Birkoff 1927: at least one closed geodesicLS 1929: at least three“gaps” filled in 1978 [BTZ83]Pogorelov 1949: extended to polyhedral surfaces
57 Quasigeodesic Aleksandrov 1948 left(p) = total incident face angle from leftquasigeodesic: curve s.t.left(p) ≤ right(p) ≤ at each point p of curve.
101 ResultArbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece,which may be viewed as a 2n x 2n grid unfolding(and so is, in places, 1/2n thin).Arbitrary (non-orthogonal) polyhedra: still open.
102 Status of main questions ShapesEdge Unfolding?General Unfolding?Convex polyhedra???Yes: always possibleNonconvex polyhedraNo: not always possible