The concept of pointed pseudo-triangulations is defined and a few of its
applications described.Comment: 3 pages, 1 figur

O'Rourke, Joseph

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The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed

O\u27Rourke, Joseph

Smith College: Smith ScholarWorks

Masthead Logo Smith ScholarWorksComputer Science: Faculty Publications Computer Science2003Computational Geometry Column 44Joseph 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, "Computational Geometry Column 44" (2003). Computer Science: Faculty Publications, Smith College,Northampton, MA.https://scholarworks.smith.edu/csc_facpubs/44arXiv:cs/0304025v1  [cs.CG]  18 Apr 2003Computational Geometry Column 44Joseph O’Rourke∗AbstractThe open problem of whether or not every pair of equal-area polygonshas a hinged dissection is discussed.A dissection of one polygon A to another B is a partition of A into a finitenumber of pieces that may be reassembled to form B. A and B are then saidto be equidecomposable. A staple of puzzle collections for centuries, the fieldhas been given new life through Frederickson’s 1997 monograph [Fre97]. Hequickly followed that with a second book [Fre02] on hinged dissections, themost famous of which is the elegant 1902 Dudeney-McElroy hinged dissectionbetween a square and an equilateral triangle illustrated in Fig. 1.Figure 1: A four-piece hinged dissection between a square and an equilateraltriangle.It has been known since the early 19th century that any two polygons ofequal area are equidecomposable.1 One method [Fre97, p. 221] triangulatespolygon A, dissects each triangle to a rectangle, converts each rectangle to oneof fixed length, and stacks all the rectangles. Applying the same process to Band overlaying the stacked rectangles yields a mutual dissection. No comparablemethod is known for hinged dissections:∗Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. orourke@cs.smith.edu. Supported by NSF Distinguished Teaching Scholar Grant DUE-0123154.1It has been known since the early 20th century that not all equal-volume polyhedra areequidecomposable.1question: Does every pair of equal-area polygons have ahinged dissection?The usual interpretation of this problem ignores possible intersections betweenthe pieces as they hinge, following what Frederickson calls the “wobbly-hinged”model. Removing this freedom is perhaps not difficult, but seems not to besettled in the literature.Here I report on two advances on this problem. The first is a proof thatany two polyominoes of the same area have a hinged dissection [DDE+03]. Apolyomino is a polygon formed by joining unit squares at their edges [Gol65,Kla97]. One composed of n squares is called an n-omino. The authors establish atype of universal dissection by proving that a cycle of 2n right isosceles triangles,hinged at their base vertices, can fold to any n-omino. See Fig. 2. Although thecycle may seem a special case, they prove that any hinged graph structure maybe converted into another dissection hinged in a cycle. The polyomino result62 4351877156843271568432Figure 2: Eight right isosceles triangles hinging to a tetramino.generalizes to hinged dissections of all edge-to-corresponding-edge gluings ofcongruent copies of any polygon. This leads, among other curiosities, to a 128-piece hinged chain of isosceles right triangles that folds to shapes representingall the letters of the alphabet and the ten digits [DD03].The second advance is a proof by Eppstein [Epp01] that any asymmetricpolygon has a hinged dissection to its mirror image. His dissection partitionsthe polygon into a chain of kites, quadrilaterals with reflection symmetry acrossa diagonal. With this partition in hand, it is easy to rotate the kites so alltheir symmetry diagonals align. The mirror image of the unfolding process thenfolds the shape to its mirror image. This procedure is illustrated for a simpleexample in Fig. 3. The kite partition relies on circle packing [BE00], and mustbe supplemented by a refined partition, and threading around the boundary ofa spanning tree to obtain a hinged chain. This work leads to a possible routetoward solving the general problem for hinged dissections, for it reduces it toO(n) equal-area triangle hinge-dissection problems with particular constraints2143214321432Figure 3: A dissection into a chain of kites, hinging to the mirror image. Herethe kites are all squares.on the hinges.References[BE00] Marshall W. Bern and David Eppstein. Quadrilateral meshing by circlepacking. Int. J. Computational Geometry & Applications, 10(4):347–360,August 2000.[DD03] Erik D. Demaine and Martin L. Demaine. Hinged dissection of the alpha-bet. Journal of Recreational Mathematics, 31(3):204–207, 2003.[DDE+03] Erik D. Demaine, Martin L. Demaine, David Eppstein, Greg N. Freder-ickson, and Erich Friedman. Hinged dissection of polyominoes and poly-forms. Computational Geometry: Theory and Applications, 2003. To ap-pear. arXiv:cs.CG/9907018. Revised version of paper in Proc. 11th Canad.Conf. Comput. Geom., 1999, 15–18.[Epp01] David Eppstein. Hinged kite mirror dissection. ACM Computing ResearchRepository, June 2001. arXiv:cs.CG/0106032.[Fre97] Greg Frederickson. Dissections: Plane and Fancy. Cambridge UniversityPress, 1997.[Fre02] Greg Frederickson. Hinged Dissections: Swinging & Twisting. CambridgeUniversity Press, 2002.[Gol65] Solomon W. Golomb. Polyominoes. Charles Scribner’s Sons, New York,1965.[Kla97] David A. Klarner. Polyominoes. In Jacob E. Goodman and JosephO’Rourke, editors, Handbook of Discrete and Computational Geometry,chapter 12, pages 225–242. CRC Press LLC, Boca Raton, FL, 1997.3

Computational Geometry Column 44

The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed. A dissection of one polygon A to another B is a partition of A into a finite number of pieces that may be reassembled to form B. A and B are then said to be equidecomposable. A staple of puzzle collections for centuries, the field has been given new life through Frederickson’s 1997 monograph [Fre97]. He quickly followed that with a second book [Fre02] on hinged dissections, the most famous of which is the elegant 1902 Dudeney-McElroy hinged dissection between a square and an equilateral triangle illustrated in Fig. 1. Figure 1: A four-piece hinged dissection between a square and an equilateral triangle. It has been known since the early 19th century that any two polygons of equal area are equidecomposable. 1 One method [Fre97, p. 221] triangulates polygon A, dissects each triangle to a rectangle, converts each rectangle to one of fixed length, and stacks all the rectangles. Applying the same process to B and overlaying the stacked rectangles yields a mutual dissection. No comparable method is known for hinged dissections

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The concept of pointed pseudo-triangulations is defined and a few of its applications described

Masthead Logo Smith ScholarWorksComputer Science: Faculty Publications Computer Science6-2002Computational Geometry Column 43Joseph 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, "Computational Geometry Column 43" (2002). Computer Science: Faculty Publications, Smith College,Northampton, MA.https://scholarworks.smith.edu/csc_facpubs/62arXiv:cs/0203008v1  [cs.CG]  6 Mar 2002Computational Geometry Column 43Joseph O’Rourke∗AbstractThe concept of pointed pseudo-triangulations is defined and a few ofits applications described.A pseudo-triangle is a planar polygon with exactly three convex vertices.Each pair of convex vertices is connected by a reflex chain, which may be justone segment. Thus, a triangle is a pseudo-triangle. A pseudo-triangulation of aset S of n points in the plane is a partition of the convex hull of S into pseudo-triangles using S as a vertex set. Under the name of “geodesic triangulations,”these found use in ray shooting [CEG+94], and then for visibility algorithmsby Pocchiola and Vegter [PV95], who named them (after a dual relationshipto pseudoline arrangements) and studied many of their properties [PV96]. Re-cently the identification of minimum pseudo-triangulations by Streinu [Str00]has generated a flurry of new applications that we selectively sample here.Minimum pseudo-triangulations have the fewest possible number of edgesfor a given set S of points. The term pointed pseudo-triangulation is gainingprominence because of the following theorem1 (see Fig. 1):Theorem 1 A pseudo-triangulation of S is minimum iff every point p ∈ S ispointed in the sense that its incident edges span less than π (i.e., they fall withina cone apexed at p with aperture angle < π).Figure 1: Two pointed pseudo-triangulations of the same set of n = 10 points.Pointed pseudo-triangulations have remarkable regularity in that each con-tains exactly n−2 pseudo-triangles. In this they parallel triangulations of simple∗Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. orourke@cs.smith.edu. Supported by NSF Distinguished Teaching Scholar Grant DUE-0123154.1 Thm. 3.1(2) of [Str00] (where the property is called “acyclicity”); Lem. 1 in [KKM+01].1polygons and are unlike triangulations of point sets, which may have from n−2to 3(n− 2) triangles. Another indication of their well-behavedness is the recentresult [KKM+01] that every set of n points in general position has a pointedpseudo-triangulation with maximum vertex degree of 5, a property not enjoyedby triangulations (of either polygons or point sets). This bounded degree mayprove useful for designing data structures employing pseudo-triangulations, suchas those used for collision detection [KSS02].We mention two disparate applications of pointed pseudo-triangulations.Two years ago a long-standing open problem was settled by Connelly, Demaine,and Rote when they proved that polygonal linkages cannot lock in the plane:open polygonal chains can be straightened, and closed polygonal chains canbe convexified [CDR00] [O’R00]. Their unlocking motions are expansive: nodistance between any pair of vertices decreases. Their proof relies on provingthe existence of a solution to differential equations describing this motion. Al-though not difficult to solve numerically in practice,2 theirs does not constitutean algebraic proof. This was provided by Streinu [Str00]. She proved that apointed pseudo-triangulation is rigid when viewed as a bar and joint framework,and that removal of a convex hull edge yields a mechanism with one degree offreedom (1-dof) that executes an expansive motion. Adding bars to a givencollection of polygonal chains to produce a pointed pseudo-triangulation, andusing a useful “flipping” property of pseudo-triangulations, leads ultimately toa piecewise-algebraic straightening/convexifying motion. These 1-dof expansivemotions have a rich structure that is just now being elucidated [RSS01].Second, pointed pseudo-triangulations were used to make an advance onan open problem posed by Urrutia [Urr00]: Can a simple polygon of n ver-tices be illuminated by cn interior π-floodlights placed at vertices, with c < 1(for sufficiently large n)? For several years the lower bound on c was 3/5,but no one could prove that c was smaller than 1. Starting from a pointedpseudo-triangulation of the polygon, Speckmann and Tóth have recently estab-lished [ST01] that (2/3)n lights suffice, finally breaking the c = 1 barrier.3We close with an open problem, posed in [RRSS01]. For a given set S ofpoints, let #T be the number of triangulations of S, and #PPT be the numberof pointed pseudo-triangulations of S. They conjecture that #T ≤ #PPT forall S, with equality only when the points of S are in convex position. This hasbeen established for all sets of at most 9 points [BKPS01].References[BKPS01] H. Brönnimann, L. Kettner, M. Pocchiola, and J. Snoeyink. Counting andenumerating pseudo-triangulations with the greedy flip algorithm. Septem-ber 2001. http://www.cs.unc.edu/Research/compgeom/pseudoT/.2 http://db.uwaterloo.ca/~eddemain/linkage/animations/.3 See http://cs.smith.edu/~orourke/TOPP/, Problem 23, for further developments.2[CDR00] R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcsand convexifying polygonal cycles. In Proc. 41st Annu. IEEE Sympos.Found. Comput. Sci., pages 432–442. IEEE, November 2000.[CEG+94] B. Chazelle, H. Edelsbrunner, M. Grigni, Leonidas J. Guibas, J. Her-shberger, M. Sharir, and J. Snoeyink. Ray shooting in polygons usinggeodesic triangulations. Algorithmica, 12:54–68, 1994.[KKM+01] L. Kettner, D. Kirkpatrick, A. Mantler, J. Snoeyink, B. Speckmann, andF. Takeuchi. Tight degree bounds for pseudo-triangulations of points.Comput. Geom. Th. Appl., 2001. To appear. Revision of abstract by L.Kettner, D. Kirkpatrick, and B. Speckmann, in Proc. 13th Canad. Conf.Comput. Geom., pp. 117-120, 2001.[KSS02] D. Kirkpatrick, J. Snoeyink, and B. Speckmann. Kinetic collision detectionfor simple polygons. Internat. J. Comput. Geom. Appl., 2002. To appear.Revised version of Proc. 16th Annu. ACM Sympos. Comput. Geom., pp.322-330, 2000.[O’R00] J. O’Rourke. Computational geometry column 39. Internat. J. Comput.Geom. Appl., 10(4):441–444, 2000. Also in SIGACT News, 31(3):47–49(2000), Issue 116.[PV95] M. Pocchiola and G. Vegter. Computing the visibility graph via pseudo-triangulations. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages248–257, 1995.[PV96] M. Pocchiola and G. Vegter. Pseudo-triangulations: Theory and applica-tions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 291–300,1996.[RRSS01] D. Randall, G. Rote, F. Santos, and J. Snoeyink. Counting triangulationsand pseudotriangulations of wheels. In Proc. 13th Canad. Conf. Comp.Geom., pages 149–152, 2001.[RSS01] G. Rote, F. Santos, and I. Streinu. Expansive motions and the polytope ofpointed pseudo-triangulations. http://cs.smith.edu/~streinu/Papers/polytope.ps, September 2001.[ST01] B. Speckmann and C. D. Tóth. Vertex π-guards in simple polygons. De-cember 2001.[Str00] I. Streinu. A combinatorial approach to planar non-colliding robot armmotion planning. In Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci.IEEE, November 2000. 443–453.[Urr00] J. Urrutia. Art gallery and illumination problems. In J.-R. Sack and J.Urrutia, editors, Handbook of Computational Geometry, pages 973–1027.North-Holland, 2000.3

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