The problem of representing triangulations has been widely studied to obtain convenient encodings and space efficient data structures. In this paper we propose a new practical approach to reduce the amount of space needed to represent in main memory an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometry information (vertex coordinates), since the combinatorial data represents the main storage part of the structure. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define stable catalogs of patches that are close under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results for the quadrilateral-triangle catalog

Castelli Aleardi, Luca

Devillers, Olivier

Mebarki, Abdelkrim

HAL-Polytechnique

2D Triangulation Representation Using Stable CatalogsOlivier Devillers, Abdelkrim Mebarki, Luca Castelli AleardiTo cite this version:Olivier Devillers, Abdelkrim Mebarki, Luca Castelli Aleardi. 2D Triangulation RepresentationUsing Stable Catalogs. Proc. 18th Canadian Conference on Computational Geometry, Aug2006, Kingston, Canada, 2006. <inria-00090631>HAL Id: inria-00090631https://hal.inria.fr/inria-00090631Submitted on 1 Sep 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.CCCG 2006, Kingston, Ontario, August 14–16, 20062D Triangulation Representation Using Stable Catalogs.∗Luca Castelli Aleardi† Olivier Devillers‡ Abdelkrim Mebarki§AbstractThe problem of representing triangulations has beenwidely studied to obtain convenient encodings and spaceefficient data structures. In this paper we propose anew practical approach to reduce the amount of spaceneeded to represent in main memory an arbitrary trian-gulation, while maintaining constant time for some basicqueries. This work focuses on the connectivity informa-tion of the triangulation, rather than the geometry in-formation (vertex coordinates), since the combinatorialdata represents the main storage part of the structure.The main idea is to gather triangles into patches, to re-duce the number of pointers by eliminating the internalpointers in the patches and reducing the multiple refer-ences to vertices. To accomplish this, we define stablecatalogs of patches that are close under basic standardupdate operations such as insertion and deletion of ver-tices, and edge flips. We present some bounds and re-sults concerning special catalogs, and some experimen-tal results for the quadrilateral-triangle catalog.1 IntroductionThe triangulation is the basic data structure in a largespectrum of application domains, ranging from geo-metric applications, to finite element applications andinterpolation schemes. This data structure has beenwidely studied from different points of view, and sev-eral schemes have been recently proposed for represent-ing triangular meshes.One can use Half-Edge-Based representation [11], inwhich the triangulation is perceived as a set of half-edges. Each half-edge is represented at least with oneof its incident vertices, the opposite half-edge, and theprevious (or the next) half-edge in the same incidentface. Moreover, each vertex stores a reference to anincident half-edge, which yields a global storage cost of19n references for a triangulation of n vertices.In the Face-Based representation [4], the objects ofthe triangulation are faces (triangles). Each face main-tains the three vertices references, and the three neigh-∗This work has been supported by the French ”ACIMasse de Donnes” program, via the GeoComp project,http://www.lix.polythechnique.fr/ shaeffer/GeoComp†INRIA & LIX, amturing@lix.polytechnique.fr‡INRIA, Olivier.Devillers@sophia.inria.fr§INRIA, Abdelkrim.Mebarki@sophia.inria.frbors references. In addition, each vertex maintains areference of an incident face, hence representing a tri-angulation with n vertices requires 13n references.These structures allow an efficient navigation over thetriangulation: standard adjacency queries, as visitingneighbors, and incident queries (testing incidence be-tween faces, edges and vertices), are all supported inO(1) time.From the encoding point of view, many solutions havebeen developed for compression purpose, mainly for tri-angular meshes. In this case, the triangulation is justimplicitly encoded (hence is not a data structure), andthere does not exist an efficient way to access to thestored data, without the decompression of the wholecode. From the information theory point of view, weknow that representing an arbitrary planar triangula-tion requires 3.24bpv (bits per vertex), and a linear timeoptimal encoding has been recently introduced [13]. Onthe practical side, several efficient compression schemeshave been proposed, achieving very interesting bit rates,especially in the case of regular meshes [2].Beyond the usual representation schemes (Half-Edgebased and Face based), some compact representationswere proposed to reduce the memory cost.In [10], a Vertex-Based representation is discussed.Each vertex handles a list of all of its adjacent vertices(the vertex stores the size of this list), resulting in 7nreferences to represent the whole triangulation. How-ever, the internal structure no longer have an explicitrepresentation of faces, and queries takes time propor-tional to the degree of an involved vertex.In [3], a compact data structure for representingsimplicial meshes is proposed, requiring in practice5 bytes/triangle. The representation does admit anEdge-based or Vertex-based representation, providingbasic update operations and standard local navigationbetween triangles (performing these operations takesO(1) time for the case of meshes with bounded ver-tex degree). To gain in memory, difference vertex labelsare used instead of real pointers, and a preprocessingstep consisting of relabelling vertices, for reducing thedifferences, is needed (this approach takes advantage ofproperties of graphs with small separators).Paper’s contributionRecently [6, 7], we have proposed an optimal way of rep-resenting a triangulation of n points using 3.24n bits,18th Canadian Conference on Computational Geometry, 2006with an additional storage which is asymptotically neg-ligible (in the case of a triangulation of a topologicalsphere ; for the triangulation bounded by a polygon ofarbitrary size the cost is 2.17 bits per triangle).The idea is to gather triangles in tiny patches of sizebetween logn12 andlogn4 , and to introduce a graph ofpatches to describe adjacency relations between them.Each patch is then represented by a reference to a cat-alog, consisting of all different tiny patches of size lessthan logn4 . The whole size of all references to the cata-log gives the dominant term of 3.24 bpv, while the rep-resentation of the graph of patches requires a negligibleamount of space.Unfortunately this negligible term is of the formO(n log lognlogn)with some non negligible constant whichmakes the approach essentially of theoretical interest.Nevertheless, the general idea is interesting and can beused in practice with some simplifications and this isthe object of the present paper, which presents somenon asymptotycal results based on the two following re-marks :1– Even if not negligible, the incidence graph ofpatches is still smaller than the incidence graph of orig-inal triangles, allowing to reduce the memory require-ments.2– logn12 is very small (log 70billions12 = 4) it would beinteresting to study in detail the composition of smallfixed catalogs suitable for our purpose.In the sequel we propose three catalogs and evaluatein details the amount of storage needed for representingtriangulations using this approach. Preliminary imple-mentation shows that the theoretical improvements areactually obtained in practice.2 DefinitionsA Catalog C is a collection {t1, . . . , tp} of planar triangu-lations with a simple boundary of arbitrary size, calledpatches (the ti are called tiny triangulations in [6, 7]).A stable catalog for a given operation is a catalog,where the result of each operation applied to any mem-bers of the catalog is either included in the catalog,or decomposable (within a restricted neighborhood) tosome elements of the catalog. The interest of such acatalog is that a triangulation T could be representedwith a disjoint union of patches in C: T = ⋃j∈J tji(1 ≤ ji ≤ p). Examples of stable catalogs are shown inFigure 1.A catalog C is minimal if no patch ti can be obtainas disjoint union of other patches in C.In this paper, we aim to support the inser-tion/deletion of degree 3 vertices and the edge flip asupdate operations.Now one may ask, given a parameter k, how to finda minimal catalog C whose patches have each at least kFigure 1: Three stable catalogs for the elementary opera-tions : insertion and deletion of degree 3 vertex and edgeflip. The first one is the Triangle-Quad catalog C1. C2 is anon-minimal catalog with at least 2 triangles/patch. C3 isminimal and contains at least 3 triangles/patch.triangles. We may proceed as follows:Firstly, the catalog should contains all of the patcheswith k triangles. Also, it includes all of the combinationswith 2k− 1 triangles, since there is no way to representa triangulation containing 2k − 1 triangles with onlypatches of k triangles. Then, we have to investigate allof the configurations produced when applying an updateoperation on the patches of the catalog. Whenever weobtain a configuration that is not decomposable intoelements of the catalog C, we add it to C.3 Simple CatalogsIn this section we present some simple catalogs, pro-vided with upper bounds on the memory requirementsof our structure, based on basic combinatorial assump-tions.3.1 The Triangle-quad catalogThe first catalog C1 is composed of two basic elements:triangles and quadrilaterals. It is clear that C1 is sta-ble for the update operations considered above, but notminimal.The triangulation is then represented by two sets, onestoring the list of triangles, and one for the quads. Thetriangle representation remains unchanged: 6 referencesare required (3 for vertices and 3 for neighbors). Foreach quadrilateral only 4 references to vertices and 4pointers to neighbors are needed, which makes save 2references over each triangle converted into a quadrilat-eral.The gain is then proportional to the number of con-structed quadrilaterals. The maximum we can obtainCCCG 2006, Kingston, Ontario, August 14–16, 2006is 9n references for a triangulation of n points (for aquadrangulation). However, it is not always feasible toconstruct a triangulation only with quadrilaterals.3.1.1 Static representationIn the static case we assume that the triangulation isentirely constructed: it only remains to get a partitioninto patches (as triangles and quads), as required by ourscheme.Several approaches has been proposed to convert tri-angulations to quadrangulations [9, 14], or create aquadrangulation from an input set of points [15]. Ingeneral, there is no obvious way to guarantee that anarbitrary triangulation could be converted to a quadran-gulation. This is even impossible in some cases (whenthe size of the boundary is odd, see [5]). Otherwise, it isalways possible to convert a triangulation of a topolog-ical sphere to a quadrangulation, since Petersen’s theo-rem [12] guarantees that each 3-regular bridgeless con-nected graph has a complete matching (and hence alltriangles can be gathered in quadrilaterals).3.1.2 Dynamic representationIn the dynamic case we allow an incremental con-struction of the triangulation, provided with the im-plementation of basic update operations (vertex inser-tion/deletion and edge flip). Using Euler’s formula,in the case of planar triangulations without boundary(triangulations of discrete surfaces homeomorphic to asphere), we can state the following:Lemma 1 Let be T a planar triangulation with n ver-tices. Then an explicit Face-based representation usingtriangles and quads requires 10.6n pointers.Proof. We may assume we are given a decompositionof T such that there are not pairs of adjacent trian-gles (since adjacent triangles can be gathered to form aquadrilateral). Let t and q denote respectively the num-ber of triangles and quadrilaterals in the decompositionof T . The total number of triangles is2n = t+ 2q (1)The number of edges is known to be 3n, and is equal to3n = eint + eext (2)where eint is the number of edges internal to a quadrilat-eral (diagonals), and eext is the number of edges sharedby quadrilaterals and triangles, and the other ones (be-tween quadrilaterals). The first number is equal to q,while the second term is at least 3t, since a triangleedge is always shared with a quad (triangles are not ad-jacent), and the number of quad should be positive, andhence :3n− 3t− q > 0 (3)Using 1 and 3, we get :q >35n (4)That means that we have at most 45n triangles. Thetriangulation can be represented with 535 n = 10.6n ref-erences, instead of 13n in the basic representation, in-ducing a gain of 19%. 3.2 Catalog with at least 2-triangles per packageWe consider the catalog C2 with no less then 2 trianglesin each patch (see Fig. 1): C2 is stable under our 3updates operations but non minimal.The quadrilaterals are represented as above. The pen-tagons are represented using 5 vertex references and 5neighbor references. The hexagons are represented with6 references for both vertices and neighbors. In thisway, we can save two references in each triangle con-verted into quadrilateral, 83 references in each triangleconverted into pentagon, and 3 references when dealingwith hexagons.Again we may assume that no two quadrilaterals areadjacent. This is feasible since any two adjacent quadri-laterals can be converted into a hexagon. In this case(assuming that there is no hexagons in the worst case)we cannot get more than 511n quadrilaterals. On theother hand, we have at least 411n pentagons, whichyields, in the worst case, a storing cost of 9111n = 8.27n,corresponding to a gain of 36% over the basic represen-tation (using the same counting argument as before).3.3 Minimal catalog with at least 3-triangles perpackageFig. 1 shows the minimal stable catalog C3 for the up-date operations with no less than three triangles perpatch. This catalog contains triangulations having be-tween 3 and 7 triangles, whose boundary has size be-tween 6 and 9. Heptagons, octagons and enneagons arerepresented respectively with 14, 20 and 24 references.This produces respectively gains of 165 ,103 and247 overeach triangle converted into one of these patches.The worst case cost for this catalog occurs when allthe patches are pentagons. In this case, the global stor-age cost of the triangulation is 233 n = 7.67n references,which is equivalent to 41% gain in memory space overthe basic representation.4 ImplementationEven thought we have implemented only the Triangle-Quad catalog C1, the following remarks hold for anyarbitrary catalog. Firstly, the triangulation consists ofmultiple containers. One for the vertices data, anda container for each patch type in the catalog, which18th Canadian Conference on Computational Geometry, 200620.000.000 pointsNumber Number Number ResidentPoints Triangles of Quads Memoryusing quads 8.110.920 15.944.539 1.352.146.944only triangles 39.999.998 - 1.691.283.456Table 1: Memory requirements (expressed in byteunits)using (1) a Quad/Triangle catalog, and (2) a basicrepresentation. The gain is about 20%.makes the references manipulated in the triangulationnon equivalent. To avoid this problem, an additionalinformation indicating the patch type has to be storedin the references. Another optional information to bestored, is a code concerning the pointed triangle in thepatch. This can make faster the localization of a giventriangle from a neighbor, avoiding checking all of thetriangles in the patch. This issue is used in our imple-mentation, since it takes insignificant size in our case(fortriangle and quads one bit suffices).5 ResultsWe have tested a 2D implementation of our package-based representation for triangulations using trianglesand quadrialterals. The experimental results shownin Table 1 are obtained by computing the Delaunaytriangulation, from uniform random distributions ofpoints, adopting the incremental algorithm provided byCGAL [4]. It should be mentioned that our Quad-Triangle data structure can directly replace the usualCGAL data structure, without any modification of theend-user code. In term of software engineering, the useof catalogs to code triangulations requires some inter-mediate levels between the storage level (containing thereal objects of the catalog that are elements of the cat-alog) and the user level (in which only logical faces ap-pear). In our case, the Triangulation is a triple con-tainer : the first one is for vertices ; the second one isfor triangles ; the third is for quadrilaterals. At higherlevel, the user manipulate faces, that are triangles, with-out worrying about the internal representation. Thesedetails affect the time processing of the triangulation inboth steps : construction and manipulation.6 ConclusionWe have presented a new approach for designing com-pact coding schemes for triangulations using stable cat-alogs. This scheme is inspired from previous theoreticalwork [6]. The idea induces some results concerning thebounds and the limits using some simple stable catalogsand promises a very efficient results if combined withother techniques. We have also implemented the small-est minimal catalog that includes triangles and quadri-laterals, and shown the practical bench results in termof memory requirements.References[1] P. K. Agarwal, J. Basch, L. J. Guibas, J. Hershberger,and L. Zhang. Deformable free space tilings for kineticcollision detection. In Proc. 5th Workshop AlgorithmicFound. Robotics), pages 83–96. A. K. Peters, 2001.[2] P. Alliez and C. Gotsman. Advances in Multiresolutionfor Geometric Modelling. Recent Advances in Compres-sion of 3D Meshes, 2005.[3] D. K. Blandford, G. E. Blelloch, D. E. Cardoze and C.Kadow. Compact Representations of Simplicial Meshesin Two and ThreeDimensions. In Proc. 12th Interna-tional Meshing Roundtable, 2003.[4] J.-D. Boissonat, O. Devillers, S. Pion, M. Teillaud, andM. Yvinec. Triangulations in CGAL Comput. Geom.Theory & Applications 22(13):519, May 2002[5] P. Bose and G. Toussaint. Characterizing and efficientlycomputing quadrangulations of planar point sets. Com-put. Aided Geom. Des., 14(8):763-785,1997.[6] L. Castelli Aleardi, O. Devillers and G. Schaeffer. Suc-cinct representation of triangulations with a boundary.In Proc. of WADS 2005, p. 134-145, 2005.[7] L. Castelli Aleardi, O. Devillers and G. Schaeffer. Op-timal succinct representations of planar maps. In Proc.SoCG, 2006, to appear.[8] The CGAL Manual” A. Fabri, E. Fogel, B. Ga¨rtner,M. Hoffmann, L. Kettner, S. Pion, M. Teillaud, R.Veltkamp and M. Yvinec Release 3.0 , 2003.[9] E. Heighway. A mesh generator for automatically sub-dividing irregular polygons into quadrilaterals. In IEEETransactions on Magnetics, , 19(6):2535–2538, 1983.[10] M. Kallmann and D. Thalmann Star-vertices: a com-pact representation for planar meshes with adjacencyinformation J. Graph. Tools, 6(1):7-18, 2001.[11] L. Kettner. Using generic programming for designing adata structure for polyhedral surfaces. ComputationalGeometry – Theory and Applications, 13:65–90, 1999.[12] J. Petersen. Die Theorie der regularen Graphs (Thetheory of regular graphs). Acta Mathematica, 15:193–220, 1891.[13] Dominique Poulalhon and Gilles Schaeffer. Optimalcoding and sampling of triangulations. In Proc. ofICALP, p. 1080-1094, 2003.[14] S. Ramaswami, P.A. Ramos and G.T. Toussaint. Con-verting triangulations to quadrangulations. Comput.Geom. Theory & Applications 9(4):257-276, 1998.[15] G. Toussaint. Quadrangulations of Planar Sets. In Proc.of WADS, 218–227, 1995.

2D Triangulation Representation Using Stable Catalogs

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HAL Id: inria-00090631https://hal.inria.fr/inria-00090631Submitted on 1 Sep 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.2D Triangulation Representation Using Stable CatalogsOlivier Devillers, Abdelkrim Mebarki, Luca Castelli AleardiTo cite this version:Olivier Devillers, Abdelkrim Mebarki, Luca Castelli Aleardi. 2D Triangulation Representation UsingStable Catalogs. Proc. 18th Canadian Conference on Computational Geometry, Aug 2006, Kingston,Canada, France. ￿inria-00090631￿CCCG 2006, Kingston, Ontario, August 14–16, 20062D Triangulation Representation Using Stable Catalogs.∗Luca Castelli Aleardi† Olivier Devillers‡ Abdelkrim Mebarki§AbstractThe problem of representing triangulations has beenwidely studied to obtain convenient encodings and spaceefficient data structures. In this paper we propose anew practical approach to reduce the amount of spaceneeded to represent in main memory an arbitrary trian-gulation, while maintaining constant time for some basicqueries. This work focuses on the connectivity informa-tion of the triangulation, rather than the geometry in-formation (vertex coordinates), since the combinatorialdata represents the main storage part of the structure.The main idea is to gather triangles into patches, to re-duce the number of pointers by eliminating the internalpointers in the patches and reducing the multiple refer-ences to vertices. To accomplish this, we define stablecatalogs of patches that are close under basic standardupdate operations such as insertion and deletion of ver-tices, and edge flips. We present some bounds and re-sults concerning special catalogs, and some experimen-tal results for the quadrilateral-triangle catalog.1 IntroductionThe triangulation is the basic data structure in a largespectrum of application domains, ranging from geo-metric applications, to finite element applications andinterpolation schemes. This data structure has beenwidely studied from different points of view, and sev-eral schemes have been recently proposed for represent-ing triangular meshes.One can use Half-Edge-Based representation [11], inwhich the triangulation is perceived as a set of half-edges. Each half-edge is represented at least with oneof its incident vertices, the opposite half-edge, and theprevious (or the next) half-edge in the same incidentface. Moreover, each vertex stores a reference to anincident half-edge, which yields a global storage cost of19n references for a triangulation of n vertices.In the Face-Based representation [4], the objects ofthe triangulation are faces (triangles). Each face main-tains the three vertices references, and the three neigh-∗This work has been supported by the French ”ACIMasse de Donnes” program, via the GeoComp project,http://www.lix.polythechnique.fr/ shaeffer/GeoComp†INRIA & LIX, amturing@lix.polytechnique.fr‡INRIA, Olivier.Devillers@sophia.inria.fr§INRIA, Abdelkrim.Mebarki@sophia.inria.frbors references. In addition, each vertex maintains areference of an incident face, hence representing a tri-angulation with n vertices requires 13n references.These structures allow an efficient navigation over thetriangulation: standard adjacency queries, as visitingneighbors, and incident queries (testing incidence be-tween faces, edges and vertices), are all supported inO(1) time.From the encoding point of view, many solutions havebeen developed for compression purpose, mainly for tri-angular meshes. In this case, the triangulation is justimplicitly encoded (hence is not a data structure), andthere does not exist an efficient way to access to thestored data, without the decompression of the wholecode. From the information theory point of view, weknow that representing an arbitrary planar triangula-tion requires 3.24bpv (bits per vertex), and a linear timeoptimal encoding has been recently introduced [13]. Onthe practical side, several efficient compression schemeshave been proposed, achieving very interesting bit rates,especially in the case of regular meshes [2].Beyond the usual representation schemes (Half-Edgebased and Face based), some compact representationswere proposed to reduce the memory cost.In [10], a Vertex-Based representation is discussed.Each vertex handles a list of all of its adjacent vertices(the vertex stores the size of this list), resulting in 7nreferences to represent the whole triangulation. How-ever, the internal structure no longer have an explicitrepresentation of faces, and queries takes time propor-tional to the degree of an involved vertex.In [3], a compact data structure for representingsimplicial meshes is proposed, requiring in practice5 bytes/triangle. The representation does admit anEdge-based or Vertex-based representation, providingbasic update operations and standard local navigationbetween triangles (performing these operations takesO(1) time for the case of meshes with bounded ver-tex degree). To gain in memory, difference vertex labelsare used instead of real pointers, and a preprocessingstep consisting of relabelling vertices, for reducing thedifferences, is needed (this approach takes advantage ofproperties of graphs with small separators).Paper’s contributionRecently [6, 7], we have proposed an optimal way of rep-resenting a triangulation of n points using 3.24n bits,18th Canadian Conference on Computational Geometry, 2006with an additional storage which is asymptotically neg-ligible (in the case of a triangulation of a topologicalsphere ; for the triangulation bounded by a polygon ofarbitrary size the cost is 2.17 bits per triangle).The idea is to gather triangles in tiny patches of sizebetween log n12 andlog n4 , and to introduce a graph ofpatches to describe adjacency relations between them.Each patch is then represented by a reference to a cat-alog, consisting of all different tiny patches of size lessthan log n4 . The whole size of all references to the cata-log gives the dominant term of 3.24 bpv, while the rep-resentation of the graph of patches requires a negligibleamount of space.Unfortunately this negligible term is of the formO(n log log nlog n)with some non negligible constant whichmakes the approach essentially of theoretical interest.Nevertheless, the general idea is interesting and can beused in practice with some simplifications and this isthe object of the present paper, which presents somenon asymptotycal results based on the two following re-marks :1– Even if not negligible, the incidence graph ofpatches is still smaller than the incidence graph of orig-inal triangles, allowing to reduce the memory require-ments.2– log n12 is very small (log 70billions12 = 4) it would beinteresting to study in detail the composition of smallfixed catalogs suitable for our purpose.In the sequel we propose three catalogs and evaluatein details the amount of storage needed for representingtriangulations using this approach. Preliminary imple-mentation shows that the theoretical improvements areactually obtained in practice.2 DefinitionsA Catalog C is a collection {t1, . . . , tp} of planar triangu-lations with a simple boundary of arbitrary size, calledpatches (the ti are called tiny triangulations in [6, 7]).A stable catalog for a given operation is a catalog,where the result of each operation applied to any mem-bers of the catalog is either included in the catalog,or decomposable (within a restricted neighborhood) tosome elements of the catalog. The interest of such acatalog is that a triangulation T could be representedwith a disjoint union of patches in C: T =⋃j∈J tji(1 ≤ ji ≤ p). Examples of stable catalogs are shown inFigure 1.A catalog C is minimal if no patch ti can be obtainas disjoint union of other patches in C.In this paper, we aim to support the inser-tion/deletion of degree 3 vertices and the edge flip asupdate operations.Now one may ask, given a parameter k, how to finda minimal catalog C whose patches have each at least kFigure 1: Three stable catalogs for the elementary opera-tions : insertion and deletion of degree 3 vertex and edgeflip. The first one is the Triangle-Quad catalog C1. C2 is anon-minimal catalog with at least 2 triangles/patch. C3 isminimal and contains at least 3 triangles/patch.triangles. We may proceed as follows:Firstly, the catalog should contains all of the patcheswith k triangles. Also, it includes all of the combinationswith 2k− 1 triangles, since there is no way to representa triangulation containing 2k − 1 triangles with onlypatches of k triangles. Then, we have to investigate allof the configurations produced when applying an updateoperation on the patches of the catalog. Whenever weobtain a configuration that is not decomposable intoelements of the catalog C, we add it to C.3 Simple CatalogsIn this section we present some simple catalogs, pro-vided with upper bounds on the memory requirementsof our structure, based on basic combinatorial assump-tions.3.1 The Triangle-quad catalogThe first catalog C1 is composed of two basic elements:triangles and quadrilaterals. It is clear that C1 is sta-ble for the update operations considered above, but notminimal.The triangulation is then represented by two sets, onestoring the list of triangles, and one for the quads. Thetriangle representation remains unchanged: 6 referencesare required (3 for vertices and 3 for neighbors). Foreach quadrilateral only 4 references to vertices and 4pointers to neighbors are needed, which makes save 2references over each triangle converted into a quadrilat-eral.The gain is then proportional to the number of con-structed quadrilaterals. The maximum we can obtainCCCG 2006, Kingston, Ontario, August 14–16, 2006is 9n references for a triangulation of n points (for aquadrangulation). However, it is not always feasible toconstruct a triangulation only with quadrilaterals.3.1.1 Static representationIn the static case we assume that the triangulation isentirely constructed: it only remains to get a partitioninto patches (as triangles and quads), as required by ourscheme.Several approaches has been proposed to convert tri-angulations to quadrangulations [9, 14], or create aquadrangulation from an input set of points [15]. Ingeneral, there is no obvious way to guarantee that anarbitrary triangulation could be converted to a quadran-gulation. This is even impossible in some cases (whenthe size of the boundary is odd, see [5]). Otherwise, it isalways possible to convert a triangulation of a topolog-ical sphere to a quadrangulation, since Petersen’s theo-rem [12] guarantees that each 3-regular bridgeless con-nected graph has a complete matching (and hence alltriangles can be gathered in quadrilaterals).3.1.2 Dynamic representationIn the dynamic case we allow an incremental con-struction of the triangulation, provided with the im-plementation of basic update operations (vertex inser-tion/deletion and edge flip). Using Euler’s formula,in the case of planar triangulations without boundary(triangulations of discrete surfaces homeomorphic to asphere), we can state the following:Lemma 1 Let be T a planar triangulation with n ver-tices. Then an explicit Face-based representation usingtriangles and quads requires 10.6n pointers.Proof. We may assume we are given a decompositionof T such that there are not pairs of adjacent trian-gles (since adjacent triangles can be gathered to form aquadrilateral). Let t and q denote respectively the num-ber of triangles and quadrilaterals in the decompositionof T . The total number of triangles is2n = t + 2q (1)The number of edges is known to be 3n, and is equal to3n = eint + eext (2)where eint is the number of edges internal to a quadrilat-eral (diagonals), and eext is the number of edges sharedby quadrilaterals and triangles, and the other ones (be-tween quadrilaterals). The first number is equal to q,while the second term is at least 3t, since a triangleedge is always shared with a quad (triangles are not ad-jacent), and the number of quad should be positive, andhence :3n− 3t− q > 0 (3)Using 1 and 3, we get :q >35n (4)That means that we have at most 45n triangles. Thetriangulation can be represented with 535 n = 10.6n ref-erences, instead of 13n in the basic representation, in-ducing a gain of 19%. 3.2 Catalog with at least 2-triangles per packageWe consider the catalog C2 with no less then 2 trianglesin each patch (see Fig. 1): C2 is stable under our 3updates operations but non minimal.The quadrilaterals are represented as above. The pen-tagons are represented using 5 vertex references and 5neighbor references. The hexagons are represented with6 references for both vertices and neighbors. In thisway, we can save two references in each triangle con-verted into quadrilateral, 83 references in each triangleconverted into pentagon, and 3 references when dealingwith hexagons.Again we may assume that no two quadrilaterals areadjacent. This is feasible since any two adjacent quadri-laterals can be converted into a hexagon. In this case(assuming that there is no hexagons in the worst case)we cannot get more than 511n quadrilaterals. On theother hand, we have at least 411n pentagons, whichyields, in the worst case, a storing cost of 9111n = 8.27n,corresponding to a gain of 36% over the basic represen-tation (using the same counting argument as before).3.3 Minimal catalog with at least 3-triangles perpackageFig. 1 shows the minimal stable catalog C3 for the up-date operations with no less than three triangles perpatch. This catalog contains triangulations having be-tween 3 and 7 triangles, whose boundary has size be-tween 6 and 9. Heptagons, octagons and enneagons arerepresented respectively with 14, 20 and 24 references.This produces respectively gains of 165 ,103 and247 overeach triangle converted into one of these patches.The worst case cost for this catalog occurs when allthe patches are pentagons. In this case, the global stor-age cost of the triangulation is 233 n = 7.67n references,which is equivalent to 41% gain in memory space overthe basic representation.4 ImplementationEven thought we have implemented only the Triangle-Quad catalog C1, the following remarks hold for anyarbitrary catalog. Firstly, the triangulation consists ofmultiple containers. One for the vertices data, anda container for each patch type in the catalog, which18th Canadian Conference on Computational Geometry, 200620.000.000 pointsNumber Number Number ResidentPoints Triangles of Quads Memoryusing quads 8.110.920 15.944.539 1.352.146.944only triangles 39.999.998 - 1.691.283.456Table 1: Memory requirements (expressed in byteunits)using (1) a Quad/Triangle catalog, and (2) a basicrepresentation. The gain is about 20%.makes the references manipulated in the triangulationnon equivalent. To avoid this problem, an additionalinformation indicating the patch type has to be storedin the references. Another optional information to bestored, is a code concerning the pointed triangle in thepatch. This can make faster the localization of a giventriangle from a neighbor, avoiding checking all of thetriangles in the patch. This issue is used in our imple-mentation, since it takes insignificant size in our case(fortriangle and quads one bit suffices).5 ResultsWe have tested a 2D implementation of our package-based representation for triangulations using trianglesand quadrialterals. The experimental results shownin Table 1 are obtained by computing the Delaunaytriangulation, from uniform random distributions ofpoints, adopting the incremental algorithm provided byCGAL [4]. It should be mentioned that our Quad-Triangle data structure can directly replace the usualCGAL data structure, without any modification of theend-user code. In term of software engineering, the useof catalogs to code triangulations requires some inter-mediate levels between the storage level (containing thereal objects of the catalog that are elements of the cat-alog) and the user level (in which only logical faces ap-pear). In our case, the Triangulation is a triple con-tainer : the first one is for vertices ; the second one isfor triangles ; the third is for quadrilaterals. At higherlevel, the user manipulate faces, that are triangles, with-out worrying about the internal representation. Thesedetails affect the time processing of the triangulation inboth steps : construction and manipulation.6 ConclusionWe have presented a new approach for designing com-pact coding schemes for triangulations using stable cat-alogs. This scheme is inspired from previous theoreticalwork [6]. The idea induces some results concerning thebounds and the limits using some simple stable catalogsand promises a very efficient results if combined withother techniques. We have also implemented the small-est minimal catalog that includes triangles and quadri-laterals, and shown the practical bench results in termof memory requirements.References[1] P. K. Agarwal, J. Basch, L. J. Guibas, J. Hershberger,and L. Zhang. Deformable free space tilings for kineticcollision detection. In Proc. 5th Workshop AlgorithmicFound. Robotics), pages 83–96. A. K. Peters, 2001.[2] P. Alliez and C. Gotsman. Advances in Multiresolutionfor Geometric Modelling. Recent Advances in Compres-sion of 3D Meshes, 2005.[3] D. K. Blandford, G. E. Blelloch, D. E. Cardoze and C.Kadow. Compact Representations of Simplicial Meshesin Two and ThreeDimensions. In Proc. 12th Interna-tional Meshing Roundtable, 2003.[4] J.-D. Boissonat, O. Devillers, S. Pion, M. Teillaud, andM. Yvinec. Triangulations in CGAL Comput. Geom.Theory & Applications 22(13):519, May 2002[5] P. Bose and G. Toussaint. Characterizing and efficientlycomputing quadrangulations of planar point sets. Com-put. Aided Geom. Des., 14(8):763-785,1997.[6] L. Castelli Aleardi, O. Devillers and G. Schaeffer. Suc-cinct representation of triangulations with a boundary.In Proc. of WADS 2005, p. 134-145, 2005.[7] L. Castelli Aleardi, O. Devillers and G. Schaeffer. Op-timal succinct representations of planar maps. In Proc.SoCG, 2006, to appear.[8] The CGAL Manual” A. Fabri, E. Fogel, B. Gärtner,M. Hoffmann, L. Kettner, S. Pion, M. Teillaud, R.Veltkamp and M. Yvinec Release 3.0 , 2003.[9] E. Heighway. A mesh generator for automatically sub-dividing irregular polygons into quadrilaterals. In IEEETransactions on Magnetics, , 19(6):2535–2538, 1983.[10] M. Kallmann and D. Thalmann Star-vertices: a com-pact representation for planar meshes with adjacencyinformation J. Graph. Tools, 6(1):7-18, 2001.[11] L. Kettner. Using generic programming for designing adata structure for polyhedral surfaces. ComputationalGeometry – Theory and Applications, 13:65–90, 1999.[12] J. Petersen. Die Theorie der regularen Graphs (Thetheory of regular graphs). Acta Mathematica, 15:193–220, 1891.[13] Dominique Poulalhon and Gilles Schaeffer. Optimalcoding and sampling of triangulations. In Proc. ofICALP, p. 1080-1094, 2003.[14] S. Ramaswami, P.A. Ramos and G.T. Toussaint. Con-verting triangulations to quadrangulations. Comput.Geom. Theory & Applications 9(4):257-276, 1998.[15] G. Toussaint. Quadrangulations of Planar Sets. In Proc.of WADS, 218–227, 1995.

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2D Triangulation Representation Using Stable Catalogs

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