This note shows that the hope expressed in [ADL+07]--that the new algorithm
for edge-unfolding any polyhedral band without overlap might lead to an
algorithm for unfolding any prismatoid without overlap--cannot be realized. A
prismatoid is constructed whose sides constitute a nested polyhedral band, with
the property that every placement of the prismatoid top face overlaps with the
band unfolding.Comment: 5 pages, 3 figures. v2 replaced Fig.1(b) and Fig.3 to illustrate the
  angles delta=(1/2)epsilon (rather than delta=epsilon

O'Rourke, Joseph

English

arXiv

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding

O\u27Rourke, Joseph

Smith College: Smith ScholarWorks

Smith ScholarWorksComputer Science: Faculty Publications Computer Science10-4-2007Band Unfoldings and Prismatoids: ACounterexampleJoseph O'RourkeSmith College, jorourke@smith.eduFollow this and additional works at: https://scholarworks.smith.edu/csc_facpubsPart of the Computer Sciences Commons, and the Geometry and Topology CommonsThis Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. Formore information, please contact scholarworks@smith.eduRecommended CitationO'Rourke, Joseph, "Band Unfoldings and Prismatoids: A Counterexample" (2007). Computer Science: Faculty Publications. 57.https://scholarworks.smith.edu/csc_facpubs/57Band Unfoldings and Prismatoids:A CounterexampleJoseph O’Rourke∗March 6, 2008AbstractThis note shows that the hope expressed in [ADL+07]—that the newalgorithm for edge-unfolding any polyhedral band without overlap mightlead to an algorithm for unfolding any prismatoid without overlap—cannotbe realized. A prismatoid is constructed whose sides constitute a nestedpolyhedral band, with the property that every placement of the prisma-toid top face overlaps with the band unfolding.1 IntroductionAn edge unfolding of a polyhedron is a cutting of its surface along edges sothat it unfolds flat to a single, non-overlapping piece in the plane. Few classesof convex polyhedra are known to have edge unfoldings. In particular, it isunknown if the class of prismatoids have edge unfoldings. A prismatoid is theconvex hull of two convex polygons A and B lying in parallel planes. See [DO07,Chap. 22] for background on this problem.Recently an edge unfolding result has been obtained for bands. A polyhedralband is the intersection of the surface of a convex polyhedron with the spacebetween two parallel planes, as long as this space does not contain any verticesof the polyhedron. In [ADL+04] [ADL+07], it was established that nested poly-hedral bands, where the top polygon projects orthogonally inside the bottomor vice versa, have an edge unfolding. This result was extended in [Alo05] toremove the nesting restriction, and to permit vertices along the top and bottomrims.A prismatoid can be viewed as a polyhedral band closed by top and bottomconvex polygons A and B. As there has been no success achieving a “volcanounfolding” of a prismatoid [DO07, pp. 321–3], the suggestion made in [ADL+07]is natural:∗Dept. Comput. Sci., Smith College, Northampton, MA 01063, USA. orourke@cs.smith.edu.1arXiv:0710.0811v1  [cs.CG]  3 Oct 2007“with it established that arbitrary bands can be unfolded withoutoverlap, it remains interesting to see whether this can lead to a non-overlapping unfolding of prismatoids, including the top and bottomfaces. It is natural to hope that these faces could be nestled onopposite sides of the unfolded band, but we do not know how toensure non-overlap.”The purpose of this note is to dash this hope by constructing a prismatoid,whose side faces constitute a nested polyhedral band, such that, for every edgeunfolding of the band, the top A cannot be attached to the band without overlap.2 Prismatoid DescriptionThere are two key aspects to the design of the prismatoid counterexample. First,the top face A has three acute angles. It is a hexagon formed by replacing eachside of an equilateral triangle with two nearly collinear edges. Figure 1(a) illus-trates the basic structure, exaggerating the 3D height. Second, the prismatoidis nearly flat, as in (b) of the figure.(b)(a)Figure 1: (a) The prismatoid with z/y = 1/2; (b) The prismatoid with z/y =0.18 leading to δ = ε ≈ 1◦.Figure 2 shows an overhead view of the prismatoid, and establishes notation.The six vertices of A are (a0, . . . , a5), and each is connected to its counterpart bion the bottom face B. The side faces (ai, bi, bi+1, ai+1) are each parallelograms.Three parameters determine the dimensions of the shape:1. z, the distance between the planes containing A and B.2. y, the y-distance between a0 and b0, as illustrated in the figure.3. h, the offset of a0 with respect to the edge of the equilateral triangleinscribed in A (again illustrated in the figure).Let δ be the curvature, i.e., the angle deficit, at a0, and let ε be the curvatureat a1. Cutting and flattening a vertex opens it by an amount equal to thecurvature. By choosing z/y small, we can make these curvatures small. And—although this is not essential—adjustment of the parameters permits achievingδ = ε (e.g., as in Figure 1(b)).2The counterexample works for a variety of combinations of the three pa-rameters. No attempt has been made to delimit the precise set of parametercombinations that lead to overlap, as choosing δ and ε sufficiently small suffices.a0a5a4a1a3a2Ab0b5b4b1b3b2hεδyFigure 2: Overhead view of prismatoid. If an origin is set at the midpointof the bottom side of the equilateral triangle on A, then a0 = (0,−h, 0) andb0 = (0,−(h+ y),−z).3 UnfoldingBecause of the symmetry of A, there are only two distinct band edge-cuts thatneed be considered: cutting at the apex of the equilateral triangle, say, edgea3b3, or cutting at a side vertex, say, edge a0b0. Because we choose δ and ε tobe small, the band opens only a little. Figure 3(a-c) illustrates the opening ata3, and (d-f) the opening at a0. For the opening at a3, there are only threedistinct edges at which to attach A: along edge a0a1, or a1a2, or a2a3. Allthree attachments lead to overlaps. Similarly, cutting at a0 leaves three distinctattachment edges, a3a4, a4a5, and a5a0, each of which leads to overlap.The key here is the three acute angles of A, for it is acute angles which leadto overlap. One might be cut (e.g., a3), but that still leaves two to thwart allplacements. The example can be viewed as an extension of the (non-nested)prismatoid described in Figs. 5, 6, and 7 of [BO07], which has two acute angles,and so thwarted one particular unfolding but not all unfoldings.3(e) (f)(d)(c)(b)(a)a0a3a0a1 a1a2a3a2a4a3a5a4a0a5Figure 3: Placements of A when a3 is cut (top row) and when a0 is cut (bot-tom row). The band rim is shown red; the band itself lies outside the rim(cf. Fig. 1(b)). The attachment edge of the band to A is blue. Circles indicateoverlap. In this figure, δ = ε = 1◦ .4References[ADL+04] Greg Aloupis, Erik D. Demaine, Stefan Langermann, Pat Morin,Joseph O’Rourke, Ileana Streinu, and Godfried Toussaint. Unfoldingpolyhedral bands. In Proc. 16th Canad. Conf. Comp. Geom., pages60–63, 2004.[ADL+07] Greg Aloupis, Erik D. Demaine, Stefan Langermann, Pat Morin,Joseph O’Rourke, Ileana Streinu, and Godfried Toussaint. Edge-unfolding nested polyhedral bands. Comput. Geom. Theory Appl.,39(1):30–42, 2007.[Alo05] Greg Aloupis. Reconfigurations of Polygonal Structures. PhD thesis,McGill Univ., School Comput. Sci., 2005.[BO07] Alex Benton and Joseph O’Rourke. Unfolding polyhedra via cut-tree truncation. In Proc. 19th Canad. Conf. Comput. Geom., pages77–80, 2007.[DO07] Erik D. Demaine and Joseph O’Rourke. Geometric Folding Algo-rithms: Linkages, Origami, Polyhedra. Cambridge University Press,July 2007. http://www.gfalop.org.5

Band Unfoldings and Prismatoids: A Counterexample

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Band Unfoldings and Prismatoids: A Counterexample

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