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Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps

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Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps

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dc.contributor.author Soto Francés, Víctor Manuel es_ES
dc.contributor.author Sarabia Escrivà, Emilio José es_ES
dc.contributor.author Pinazo Ojer, José Manuel es_ES
dc.date.accessioned 2019-09-05T20:03:50Z
dc.date.available 2019-09-05T20:03:50Z
dc.date.issued 2019 es_ES
dc.identifier.issn 0188-7009 es_ES
dc.identifier.uri http://hdl.handle.net/10251/125086
dc.description.abstract [EN] The modelling of real world objects is not a straightforward subject. There are many different schemes; constructive solid geome-try (CSG), cell decomposition, boundary representation, etcetera. Obviously, somehow, any scheme will be related to any other since they have a common goal. The paper shows how to model general polyhedra as an unordered discrete and finite set of geometric numbers of a projective Clifford Algebra or Geometric Algebra (GA). Clearly, not any randomly generated finite set of geometric numbers will have the structure of an object, this set must have some well defined properties. The topological properties extracted from this set are mapped to a boundary representation scheme based on a type of combinatorial map called generalised map or n-gmap. The n-gmaps have different types of or-bits (in the mathematical sense) to which an attribute can be attached. When the attribute has a geometrical meaning, it is said that it is the geometrical embedding of the n-gmap. In this way the n-gmap holds explicitly the topology or structure already defined by the discrete geometry. In our proposal, each single element of a n-gmap is consistently embedded into a geometrical number also known as multi-vector. The scheme has been implemented by modifying an open source code [46] of n-gmaps. This representation has interesting properties. GA and n-gmaps complement and reinforce each other. For instance; it improves the robustness when computing the structure from the geometrical information. It is capable of computing lengths, areas and volumes of any polyhedral complex (with or without holes) using the orbits of the n-gmap (some examples are given). Finally the paper gives hints about other potentialities. es_ES
dc.language Inglés es_ES
dc.publisher Springer-Verlag es_ES
dc.relation.ispartof Advances in Applied Clifford Algebras es_ES
dc.rights Reserva de todos los derechos es_ES
dc.subject Geometric algebra es_ES
dc.subject Clifford algebra es_ES
dc.subject Multi-vectors es_ES
dc.subject N-gmaps es_ES
dc.subject Building energy simulation es_ES
dc.subject Solid modelling es_ES
dc.subject Combinatorial maps es_ES
dc.subject Flags es_ES
dc.subject Darts es_ES
dc.subject.classification MAQUINAS Y MOTORES TERMICOS es_ES
dc.title Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps es_ES
dc.type Artículo es_ES
dc.identifier.doi 10.1007/s00006-018-0927-y es_ES
dc.rights.accessRights Cerrado es_ES
dc.contributor.affiliation Universitat Politècnica de València. Departamento de Termodinámica Aplicada - Departament de Termodinàmica Aplicada es_ES
dc.description.bibliographicCitation Soto Francés, VM.; Sarabia Escrivà, EJ.; Pinazo Ojer, JM. (2019). Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps. Advances in Applied Clifford Algebras. 29(1). https://doi.org/10.1007/s00006-018-0927-y es_ES
dc.description.accrualMethod S es_ES
dc.relation.publisherversion https://doi.org/10.1007/s00006-018-0927-y es_ES
dc.type.version info:eu-repo/semantics/publishedVersion es_ES
dc.description.volume 29 es_ES
dc.description.issue 1 es_ES
dc.relation.pasarela S\375212 es_ES
dc.relation.references Alayrangues, S., Damiand, G., Lienhardt, P., Peltier, S.: A Boundary Operator for Computing the Homology of Cellular Structures, Technical report, 71 pages, (2011). HAL id: hal-00683031, https://hal.archives-ouvertes.fr/hal-00683031 . Accessed 07 May 2018 es_ES
dc.relation.references Alayrangues, S., Daragon, X., Lachaud, J., Lienhardt, P.: Equivalence between closed connected n-G-maps without multi-incidence and n-surfaces. J. Math. Imaging Vis. 32, 122 (2008). https://doi.org/10.1007/s10851-008-0084-3 es_ES
dc.relation.references Alayrangues, S., Fuchs, L., Lienhardt, P., Peltier, S.: Incremental Computation of the Homology of Generalized Maps: An Application of Effective Homology Results, (2015). HAL id: hal-01142760v2 https://hal.archives-ouvertes.fr/hal-01142760 . Accessed 07 May 2018 es_ES
dc.relation.references Baig, S.U., Alizadehashrafi, B.: 3D Generalization of Boundary Representation (B-Rep) of Buildings, FIG Congress 2014, Kuala Lumpur, Malaysia, 16-21 June (2014) es_ES
dc.relation.references Bellet, T., Arnould, A., Charneau, S., Fuchs, L.: Modélisation nD à base d’algèbres géométriques, (2008) HAL id: hal-00354183 https://hal.archives-ouvertes.fr/hal-00354183 . Accessed 07 May 2018 es_ES
dc.relation.references Bellet, T., Arnould, A., Fuchs, L.: Polyhedral embedding of a topological structure. Applied Geometric Algebras in Computer Science and Engineering (AGACSE 2010), Amsterdam, Netherlands (2010) HAL id: hal-00488183 https://hal.archives-ouvertes.fr/hal-00488183 . Accessed 07 May 2018 es_ES
dc.relation.references Braulio-Gonzalo, M., Bovea, M.D.: Environmental and cost performance of buildings envelope insulation materials to reduce energy demand: Thickness optimisation. Energy Build. 150, 527–545 (2017). https://doi.org/10.1016/j.enbuild.2017.06.005 es_ES
dc.relation.references Brisson, E.: Representing geometric structures in d dimensions: topology and order. Discrete Comput. Geometry 9, 387426 (1993) es_ES
dc.relation.references Chard, J.A., Shapiro, V.: A multivector data structure for differential forms and equations. Math. Comput. Simul. 54, 3364 (2000). https://doi.org/10.1016/S0378-4754(00),00198-1 es_ES
dc.relation.references Chaïm Zonnenberg, PhD. thesis: Conformal Geometric Algebra Package, Utrecht University Department of Information and Computing Sciences, (July 23, 2007) es_ES
dc.relation.references Chisolm, E.: Geometric Algebra, (2012) arXiv: 1205.5935v1 [math-ph] es_ES
dc.relation.references Conradt, O.: Mathematical physics in space and counterspace,(Arbeitshefte, KLEINE REIHE, Band 4), Mathematisch-Astronomische Sektion am Goetheanum und Verlag am Goetheanum, CH-4143 Dornarch, (2008), ISBN-13: 978-3723513330 es_ES
dc.relation.references Conradt, O.: Mechanics in space and counterspace, Journal of Mathematical Physics 41 (6995-7028) (Oct 2000). https://doi.org/10.1063/1.1288495 es_ES
dc.relation.references Crawley, D.B., Lawrie, L.K., Winkelmann, F.C., Buhl, W.F., Huang, Y.J., Pedersen, C.O., Strand, R.K., Liesen, R.J., Witte, D.E.J. Glazer, J.: EnergyPlus: creating a new-generation building energy simulation program. Energy Build. 33(4), 319-331 (2001) https://energyplus.net/ . Accessed 07 May 2018 es_ES
dc.relation.references Damiand, G., Lienhardt, P.: Combinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing. CRC Press, Boca Raton, (2015), ISBN: 978-1-4822-0653-1 es_ES
dc.relation.references Damiand, G., Teillaud, M.: A Generic Implementation of dD Combinatorial Maps in CGAL, International Meshing Roundtable, Oct 2014, Londres, United Kingdom. 82, pp.46 - 58, (2014), https://doi.org/10.1016/j.proeng.2014.10.372 HAL id: hal-01090011 http://doc.cgal.org/ (Linear Cell Complex). Accessed 07 May 2018 es_ES
dc.relation.references Damiand, G.: Contributions aux Cartes Combinatoires et Cartes Généralises : Simplification, Modèles, Invariants Topologiques et Applications, HAL Id: tel-00538456 (2010) es_ES
dc.relation.references Diakité, A.A., Damiand, G., Maercke, D.V.: Topological reconstruction of complex 3d buildings and automatic extraction of levels of detail, Eurographics Workshop on Urban Data Modelling and Visualisation.(Strasbourg, France.), hal-01011376, pp. 25–30, (2014). https://doi.org/10.2312/udmv.20141074 https://hal.archives-ouvertes.fr/hal-01011376 . Accessed 07 May 2018 es_ES
dc.relation.references Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry, A volume in The Morgan Kaufmann Series in Computer Graphics, (2007). ISBN: 978-0-12-369465-2 es_ES
dc.relation.references Dorst, L.: 3D Oriented Projective Geometry Through Versors of $${\mathbb{R}}^{3,3}$$ R 3 , 3 , Advances in Applied Clifford Algebras, https://doi.org/10.1007/s00006-015-0625-y es_ES
dc.relation.references Fradin, D., Meneveaux, D., Lienhardt, P.: Hierarchy of generalized maps for modeling and rendering complex indoor scenes, Tech. Rep., Rapport de recherche No 2005-04, Signal Image Communication laboratory, CNRS, University of Poitiers, France (November 2005) es_ES
dc.relation.references Francés, V.M.S.: Modified version of MOKA implementing the GA method presented in this paper, https://github.com/vsotofrances/MOKACLIFFORD/tree/multivect . Accessed 07 May 2018 es_ES
dc.relation.references Franklin, W.R.: Polygon properties calculated from the vertex neighborhoods, in: N. ACM New York (Ed.), Proceeding SCG ’87, Proceedings of the third annual symposium on Computational geometry, (1987) ISBN:0-89791-231-4 https://doi.org/10.1145/41958.41969 https://www.ecse.rpi.edu/~wrf/Research/Short_Notes/volume.html . Accessed 07 May 2018 es_ES
dc.relation.references Genera3D, automation of the 3D model creation of a building, http://vpclima2.ter.upv.es/ . Accessed 07 May 2018 es_ES
dc.relation.references Gunn, C.: Guide to Geometric Algebra in Practice. (Chapter: On the Homogeneous Model of Euclidean Geometry) Ed. Leo Dorst & Joan Lasenby,Springer-Verlag London Limited (2011), ISBN: 978-0-85729-810-2 https://doi.org/10.1007/978-0-85729-811-9 es_ES
dc.relation.references Gunn, C.: On the Homogeneous Model Of Euclidean Geometry. arXiv:1101.4542 [math.MG]. Accessed 07 May 2018 es_ES
dc.relation.references Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, D. Reidel Publishing Company, : ISBN-13: 978–9027725615. ISBN- 10, 9027725616 (1984) es_ES
dc.relation.references Hestenes, D., Ziegler, R.: Projective geometry with clifford algebra. Acta Appl. Math. 23, 25–63 (1991) es_ES
dc.relation.references Hitzer, E.: Introduction to Clifford’s geometric algebra. J. Soc. Instrum. Control Eng. 51(4), 338–350 (2012). arXiv:1306.1660 . Accessed 07 May 2018 es_ES
dc.relation.references HULC, Unified Tool LIDER-CALENER, Version 1.0.1493.1049. http://www.codigotecnico.org/index.php/menu-recursos/menu-aplicaciones/282-herramienta-unificada-lider-calener (2016). Accessed 01 July 2016 es_ES
dc.relation.references Kraemer, P., Untereiner, L., Jund, T., Thery, S., Cazier, D.: CGoGN:n-dimensional Meshes with Combinatorial Maps, J. Sarrate & M. Staten (eds.). In: Proceedings of the 22nd International Meshing Roundtable, Springer International Publishing Switzerland (2013) https://doi.org/10.1007/978-3-319-02335-9_27 https://github.com/cgogn/CGoGN . Accessed 07 May 2018 es_ES
dc.relation.references Li, H., Huang, L., Shao, C., Dong, L.: Three-Dimensional Projective Geometry with Geometric Algebra, (2015). arXiv:1507.06634v1 es_ES
dc.relation.references Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput. Aided Des. 23(1), 59–82 (1991). https://doi.org/10.1016/0010-4485(91)90082-8 es_ES
dc.relation.references Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. J. Comput. Geom. Appl. 4(3), 275324 (1994). https://doi.org/10.1142/S0218195994000173 es_ES
dc.relation.references OFF 3D graphics data format. http://paulbourke.net/dataformats/ . Accessed 07 May 2018 es_ES
dc.relation.references Pappas, R.: Chapter: Oriented projective geometry with Clifford Algebra, in book titled, Clifford algebras with numeric and symbolic computations, Editors Rafal Ablamowicz, Pertti Lounesto, Josep M. Parra, Birkhäuser, (1996) ISBN: 978-1-4615-8159-8, ISBN 978-1-4615-8157-4 (eBook) https://doi.org/10.1007/978-1-4615-8157-4 es_ES
dc.relation.references Sokolov, A.: A key to projective model of homogeneous metric spaces, (2014) arXiv:1412.8095v1 [math.MG] es_ES
dc.relation.references Sokolov, A.: Clifford algebra and the projective model of Elliptic spaces, (2013) arXiv:1310.2713v1 [math.MG] es_ES
dc.relation.references Sokolov, A.: Clifford algebra and the projective model of homogeneous metric spaces: Foundations, (2013) arxiv:1307.2917v1 [math.MG] es_ES
dc.relation.references Sokolov, A.: Clifford algebra and the projective model of Hyperbolic spaces (2016) arXiv:1602.08562v1 [math.MG] es_ES
dc.relation.references Sokolov, A.: Clifford algebra and the projective model of Minkowski (pseudo-Euclidean) spaces (2013) arXiv:1307.4179v2 [math.MG] es_ES
dc.relation.references Stein, P.: geoma v1.2.2007.08.20, C++ software, http://nklein.com/tags/geoma/ . Accessed 07 May 2018 es_ES
dc.relation.references Stolfi, J.: Oriented Projective Geometry. A Framework for Geometric Computations , 1st Edition ISBN 9781483265193 , Academic Press, Published 28th July 1991 es_ES
dc.relation.references Stolfi, J.: Primitives for computational geometry, Technical report SRC-RR-36, Systems research center (January 27 1989). http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-36.html . Accessed 07 May 2018 es_ES
dc.relation.references Tonti, E.: Why starting from differential equations for computational physics? J. Comput. Phys. 257, 1260–1290 (2014). https://doi.org/10.1016/j.jcp.2013.08.016 es_ES
dc.relation.references Vidil, F., Damiand, G., Dexet-Guiard, M., Guiard, N., Ledoux, F., Fousse, A., Fradin, D., Liang, Y., Meneveaux, D., Bertrand, Y.: MOKA. http://moka-modeller.sourceforge.net/ (2002). Accessed 07 May 2018 es_ES
dc.relation.references Zonnenberg, C.: Conformal Geometric Algebra Package, (2007), http://www.cs.uu.nl/groups/MG/gallery/CGAP/index.html . Accessed 07 May 2018 es_ES


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