{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:31:44Z","timestamp":1767339104799,"version":"3.41.0"},"reference-count":99,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T00:00:00Z","timestamp":1602633600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000161","name":"National Institute of Standards and Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100000161","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Sogei S.p.A."},{"name":"ICT company of the Italian Ministry of Economy and Finance","award":["2016-17"],"award-info":[{"award-number":["2016-17"]}]},{"name":"National Science Foundation"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Spatial Algorithms Syst."],"published-print":{"date-parts":[[2021,3,31]]},"abstract":"<jats:p>In many areas of applied geometric\/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences, adjacencies, and ordering of cells, generally using disparate and often incompatible data structures and algorithms. This article introduces computational topology algorithms to discover the two-dimensional (2D)\/3D space partition induced by a collection of geometric objects of dimension 1D\/2D, respectively. Methods and language are those of basic geometric and algebraic topology. Only sparse vectors and matrices are used to compute both spaces and maps, i.e., the chain complex, from dimension zero to three. The prototype software is written in Julia, the novel language for scientific computing. The applications may vary from 3D graphics to 3D printing, from images to scene understanding, and from games to building information modeling.<\/jats:p>","DOI":"10.1145\/3401988","type":"journal-article","created":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T11:32:54Z","timestamp":1602675174000},"page":"1-29","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["Topological Computing of Arrangements with (Co)Chains"],"prefix":"10.1145","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3958-8089","authenticated-orcid":false,"given":"Alberto","family":"Paoluzzi","sequence":"first","affiliation":[{"name":"Roma Tre University, Rome, Italy"}]},{"given":"Vadim","family":"Shapiro","sequence":"additional","affiliation":[{"name":"University of Wisconsin\u2014Madison 8 ICSI, United States"}]},{"given":"Antonio","family":"Dicarlo","sequence":"additional","affiliation":[{"name":"CECAM-IT-SIMUL Node, Rome, Italy"}]},{"given":"Francesco","family":"Furiani","sequence":"additional","affiliation":[{"name":"Roma Tre University, Rome, Italy"}]},{"given":"Giulio","family":"Martella","sequence":"additional","affiliation":[{"name":"Roma Tre University, Rome, Italy"}]},{"given":"Giorgio","family":"Scorzelli","sequence":"additional","affiliation":[{"name":"Scientific Computing and Imaging Institute (SCI), Salt Lake City, Utah, USA"}]}],"member":"320","published-online":{"date-parts":[[2020,10,14]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.124288"},{"key":"#cr-split#-e_1_2_2_2_1.1","doi-asserted-by":"crossref","unstructured":"S. Alayrangues G. Damiand P. Lienhardt and S. Peltier. 2015. Homology of cellular structures allowing multi-incidence. Discr. Comput. Geom. 54 1 (01 July 2015) 42--77. DOI:https:\/\/doi.org\/10.1007\/s00454-015-9662-5 10.1007\/s00454-015-9662-5","DOI":"10.1007\/s00454-015-9662-5"},{"key":"#cr-split#-e_1_2_2_2_1.2","doi-asserted-by":"crossref","unstructured":"S. Alayrangues G. Damiand P. Lienhardt and S. Peltier. 2015. Homology of cellular structures allowing multi-incidence. Discr. Comput. Geom. 54 1 (01 July 2015) 42--77. DOI:https:\/\/doi.org\/10.1007\/s00454-015-9662-5","DOI":"10.1007\/s00454-015-9662-5"},{"key":"e_1_2_2_3_1","volume-title":"Djinn: A Geometric Interface for Solid Modelling. Technical Report","author":"Armstrong C.","year":"1999","unstructured":"C. Armstrong , A. Bowyer , S. Cameron , J. Corney , G. Jared , R. Martin , A. Middleditch , M. Sabin , J. Salmon , and J. Woodwark . 1999 . Djinn: A Geometric Interface for Solid Modelling. Technical Report . Information Geometers Ltd ., Winchester, UK. C. Armstrong, A. Bowyer, S. Cameron, J. Corney, G. Jared, R. Martin, A. Middleditch, M. Sabin, J. Salmon, and J. Woodwark. 1999. Djinn: A Geometric Interface for Solid Modelling. Technical Report. Information Geometers Ltd., Winchester, UK."},{"volume-title":"Finite Element Exterior Calculus. CBMS-NSF Regional Conference Series in Applied Mathematics","author":"Arnold Douglas N.","key":"e_1_2_2_4_1","unstructured":"Douglas N. Arnold . 2018. Finite Element Exterior Calculus. CBMS-NSF Regional Conference Series in Applied Mathematics , Vol. 93 . Society for Industrial and Applied Mathematics (SIAM) , Philadelphia, PA . Douglas N. Arnold. 2018. Finite Element Exterior Calculus. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 93. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA."},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492906210018"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-10-01278-4"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.gmod.2014.09.002"},{"key":"e_1_2_2_8_1","first-page":"5","article-title":"Progressive conversion from B-rep to BSP for streaming geometric modeling","volume":"3","author":"Bajaj C.","year":"2006","unstructured":"C. Bajaj , A. Paoluzzi , and G. Scorzelli . 2006 . Progressive conversion from B-rep to BSP for streaming geometric modeling . Comput.-Aid. Des. Appl. 3 , 5 -- 6 (2006). C. Bajaj, A. Paoluzzi, and G. Scorzelli. 2006. Progressive conversion from B-rep to BSP for streaming geometric modeling. Comput.-Aid. Des. Appl. 3, 5--6 (2006).","journal-title":"Comput.-Aid. Des. Appl."},{"volume-title":"Encyclopedia of Mathematics","author":"Baladze D. O.","key":"e_1_2_2_9_1","unstructured":"D. O. Baladze . 2012. CW-complex . In Encyclopedia of Mathematics . Springer 8 European Mathematical Society. D. O. Baladze. 2012. CW-complex. In Encyclopedia of Mathematics. Springer 8 European Mathematical Society."},{"key":"e_1_2_2_10_1","volume-title":"Proceedings of the SIAM Conference on Applied Linear Algebra","author":"Ballard Grey","year":"2015","unstructured":"Grey Ballard and Alex Druinsky . 2015 . Sparse matrix-matrix multiplication: Applications, algorithms, and implementations . In Proceedings of the SIAM Conference on Applied Linear Algebra . Atlanta, GA. Grey Ballard and Alex Druinsky. 2015. Sparse matrix-matrix multiplication: Applications, algorithms, and implementations. In Proceedings of the SIAM Conference on Applied Linear Algebra. Atlanta, GA."},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.06.016"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPDS.2018.2872064"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/141000671"},{"volume-title":"Nef polyhedra: A brief introduction","author":"Bieri Hanspeter","key":"e_1_2_2_17_1","unstructured":"Hanspeter Bieri . 1995. Nef polyhedra: A brief introduction . In Geometric Modelling, H. Hagen, G. Farin, and H. Noltemeier (Eds.). Springer Vienna , Vienna , 43--60. Hanspeter Bieri. 1995. Nef polyhedra: A brief introduction. In Geometric Modelling, H. Hagen, G. Farin, and H. Noltemeier (Eds.). Springer Vienna, Vienna, 43--60."},{"volume-title":"Lattice Theory","author":"Birkhoff Garrett","key":"e_1_2_2_18_1","unstructured":"Garrett Birkhoff . 1948. Lattice Theory (Revised ed.). American Mathematical Society , New York, NY . Garrett Birkhoff. 1948. Lattice Theory (Revised ed.). American Mathematical Society, New York, NY."},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.2307\/2152753"},{"key":"e_1_2_2_20_1","unstructured":"A. Bowyer. 1995. SvLis Set-theoretic Kernel Modeller: Introduction and User Manual. Information Geometers.  A. Bowyer. 1995. SvLis Set-theoretic Kernel Modeller: Introduction and User Manual. Information Geometers."},{"key":"e_1_2_2_21_1","volume-title":"Introducing Djinn: A Geometric Interface for Solid Modelling. Information Geometers [for] the Geometric Modelling Society.","author":"A. Bowyer and Geometric Modelling Society","year":"1995","unstructured":"A. Bowyer and Geometric Modelling Society . 1995 . Introducing Djinn: A Geometric Interface for Solid Modelling. Information Geometers [for] the Geometric Modelling Society. A. Bowyer and Geometric Modelling Society. 1995. Introducing Djinn: A Geometric Interface for Solid Modelling. Information Geometers [for] the Geometric Modelling Society."},{"volume-title":"Deep Learning with TensorFlow","author":"Boxel Dan Van","key":"e_1_2_2_22_1","unstructured":"Dan Van Boxel . 2016. Deep Learning with TensorFlow . Packt Publishing . Dan Van Boxel. 2016. Deep Learning with TensorFlow. Packt Publishing."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/360715.360727"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/73833.73858"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1137\/110848244"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01609.x"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.14733\/cadconfP.2016.348-351"},{"key":"e_1_2_2_28_1","volume-title":"Introduction to Algorithms","author":"Cormen Thomas H.","unstructured":"Thomas H. Cormen , Charles E. Leiserson , Ronald L. Rivest , and Clifford Stein . 2009. Introduction to Algorithms ( 3 rd ed.). The MIT Press . Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. 2009. Introduction to Algorithms (3rd ed.). The MIT Press.","edition":"3"},{"key":"e_1_2_2_29_1","doi-asserted-by":"crossref","unstructured":"H. S. M. Coxeter and S. L. Greitzer. 1967. Geometry Revisited. Mathematical Association of America Washington D.C.  H. S. M. Coxeter and S. L. Greitzer. 1967. Geometry Revisited. Mathematical Association of America Washington D.C.","DOI":"10.5948\/UPO9780883859346"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1201\/b17403"},{"key":"e_1_2_2_31_1","series-title":"Fundamentals of Algorithms 2","volume-title":"Direct Methods for Sparse Linear Systems","author":"Davis Timothy A.","unstructured":"Timothy A. Davis . 2006. Direct Methods for Sparse Linear Systems ( Fundamentals of Algorithms 2 ) . Society for Industrial and Applied Mathematics, Philadelphia , PA. Timothy A. Davis. 2006. Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2). Society for Industrial and Applied Mathematics, Philadelphia, PA."},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2014.04.007"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/1185657.1185665"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1109\/TASE.2009.2021342"},{"key":"e_1_2_2_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/1629255.1629273"},{"key":"e_1_2_2_36_1","volume-title":"Linear algebraic representation for topological structures. Comput. Aid. Des. 46 (January","author":"Dicarlo Antonio","year":"2014","unstructured":"Antonio Dicarlo , Alberto Paoluzzi , and Vadim Shapiro . 2014. Linear algebraic representation for topological structures. Comput. Aid. Des. 46 (January 2014 ), 269--274. DOI:https:\/\/doi.org\/10.1016\/j.cad.2013.08.044 10.1016\/j.cad.2013.08.044 Antonio Dicarlo, Alberto Paoluzzi, and Vadim Shapiro. 2014. Linear algebraic representation for topological structures. Comput. Aid. Des. 46 (January 2014), 269--274. DOI:https:\/\/doi.org\/10.1016\/j.cad.2013.08.044"},{"key":"e_1_2_2_37_1","volume-title":"Proceedings of the 3rd Annual Symposium on Computational Geometry (SCG\u201987)","author":"Dobkin D. P.","year":"1958","unstructured":"D. P. Dobkin and M. J. Laszlo . 1987. Primitives for the manipulation of three-dimensional subdivisions . In Proceedings of the 3rd Annual Symposium on Computational Geometry (SCG\u201987) . ACM, New York, NY, 86--99. DOI:https:\/\/doi.org\/10.1145\/4 1958 .41967 10.1145\/41958.41967 D. P. Dobkin and M. J. Laszlo. 1987. Primitives for the manipulation of three-dimensional subdivisions. In Proceedings of the 3rd Annual Symposium on Computational Geometry (SCG\u201987). ACM, New York, NY, 86--99. DOI:https:\/\/doi.org\/10.1145\/41958.41967"},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(95)00016-Y"},{"volume-title":"Algorithms in Combinatorial Geometry","author":"Edelsbrunner H.","key":"e_1_2_2_39_1","unstructured":"H. Edelsbrunner . 1987. Algorithms in Combinatorial Geometry . Springer-Verlag , New York, NY . H. Edelsbrunner. 1987. Algorithms in Combinatorial Geometry. Springer-Verlag, New York, NY."},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/1185657.1185666"},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1002\/1097-024X(200009)30:11<1167::AID-SPE337>3.0.CO;2-B"},{"volume-title":"The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering","author":"Ferretti E.","key":"e_1_2_2_42_1","unstructured":"E. Ferretti . 2014. The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering . Momentum Press . E. Ferretti. 2014. The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering. Momentum Press."},{"volume-title":"Effective Computational Geometry for Curves and Surfaces, Jean-Daniel Boissonat and Monique Teillaud (Eds.)","author":"Fogel Efi","key":"e_1_2_2_43_1","unstructured":"Efi Fogel , Dan Halperin , Lutz Kettner , Monique Teillaud , Ron Wein , and Nicola Wolpert . 2007. Arrangements . In Effective Computational Geometry for Curves and Surfaces, Jean-Daniel Boissonat and Monique Teillaud (Eds.) . Springer , 1--66. Efi Fogel, Dan Halperin, Lutz Kettner, Monique Teillaud, Ron Wein, and Nicola Wolpert. 2007. Arrangements. In Effective Computational Geometry for Curves and Surfaces, Jean-Daniel Boissonat and Monique Teillaud (Eds.). Springer, 1--66."},{"key":"e_1_2_2_44_1","volume-title":"Geometric computing with chain complexes: Design and features of a Julia package. CoRR abs\/1710.07819v2","author":"Furiani Francesco","year":"2017","unstructured":"Francesco Furiani , Giulio Martella , and Alberto Paoluzzi . 2017. Geometric computing with chain complexes: Design and features of a Julia package. CoRR abs\/1710.07819v2 ( 2017 ). arxiv:1710.07819v2 Francesco Furiani, Giulio Martella, and Alberto Paoluzzi. 2017. Geometric computing with chain complexes: Design and features of a Julia package. CoRR abs\/1710.07819v2 (2017). arxiv:1710.07819v2"},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.1145\/304012.304039"},{"volume-title":"Deep Learning","author":"Goodfellow Ian","key":"e_1_2_2_46_1","unstructured":"Ian Goodfellow , Yoshua Bengio , and Aaron Courville . 2016. Deep Learning . The MIT Press . Ian Goodfellow, Yoshua Bengio, and Aaron Courville. 2016. Deep Learning. The MIT Press."},{"key":"e_1_2_2_47_1","unstructured":"Jacob E. Goodman Joseph O\u2019Rourke and Csaba D. T\u00f2th (Eds.). 2017. Handbook of Discrete and Computational Geometry (3rd ed.). CRC Press Boca Raton FL.  Jacob E. Goodman Joseph O\u2019Rourke and Csaba D. T\u00f2th (Eds.). 2017. Handbook of Discrete and Computational Geometry (3rd ed.). CRC Press Boca Raton FL."},{"key":"e_1_2_2_48_1","doi-asserted-by":"publisher","DOI":"10.1145\/282918.282923"},{"key":"e_1_2_2_49_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195998000096"},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2006.11.009"},{"key":"e_1_2_2_51_1","unstructured":"Allen Hatcher. 2002. Algebraic Topology. Cambridge University Press.  Allen Hatcher. 2002. Algebraic Topology. Cambridge University Press."},{"volume-title":"Geometric and Solid Modeling: An Introduction. Morgan Kaufmann","author":"Hoffmann Christoph M.","key":"e_1_2_2_54_1","unstructured":"Christoph M. Hoffmann . 1989. Geometric and Solid Modeling: An Introduction. Morgan Kaufmann , San Francisco, CA . Christoph M. Hoffmann. 1989. Geometric and Solid Modeling: An Introduction. Morgan Kaufmann, San Francisco, CA."},{"key":"e_1_2_2_55_1","volume-title":"Karasick","author":"Hoffmann Christoph M.","year":"1987","unstructured":"Christoph M. Hoffmann , John E. Hopcroft , and Michael S . Karasick . 1987 . Robust Set Operations on Polyhedral Solids. Technical Report. Ithaca, NY. Christoph M. Hoffmann, John E. Hopcroft, and Michael S. Karasick. 1987. Robust Set Operations on Polyhedral Solids. Technical Report. Ithaca, NY."},{"key":"e_1_2_2_56_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4485(00)00073-7"},{"key":"e_1_2_2_57_1","volume-title":"Hoffmann and George Van\u011b\u010dek","author":"Christoph","year":"1991","unstructured":"Christoph M. Hoffmann and George Van\u011b\u010dek . 1991 . Fundamental Techniques for Geometric and Solid Modeling. Technical Report, Report Number 91-044. Purdue University . Christoph M. Hoffmann and George Van\u011b\u010dek. 1991. Fundamental Techniques for Geometric and Solid Modeling. Technical Report, Report Number 91-044. Purdue University."},{"key":"e_1_2_2_58_1","doi-asserted-by":"publisher","DOI":"10.1145\/362248.362272"},{"key":"e_1_2_2_59_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(73)90020-3"},{"key":"e_1_2_2_60_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(89)90067-5"},{"key":"e_1_2_2_61_1","doi-asserted-by":"publisher","DOI":"10.1137\/0208040"},{"key":"e_1_2_2_62_1","doi-asserted-by":"publisher","DOI":"10.1145\/376957.376976"},{"key":"e_1_2_2_63_1","first-page":"1","article-title":"Topological models for boundary representation: A comparison with n-dimensional generalized maps","volume":"23","author":"Lienhardt Pascal","year":"1991","unstructured":"Pascal Lienhardt . 1991 . Topological models for boundary representation: A comparison with n-dimensional generalized maps . Comput. Aided Des. 23 , 1 (February 1991), 59--82. DOI:https:\/\/doi.org\/10.1016\/0010-4485(91)90082-8 10.1016\/0010-4485(91)90082-8 Pascal Lienhardt. 1991. Topological models for boundary representation: A comparison with n-dimensional generalized maps. Comput. Aided Des. 23, 1 (February 1991), 59--82. DOI:https:\/\/doi.org\/10.1016\/0010-4485(91)90082-8","journal-title":"Comput. Aided Des."},{"volume-title":"Introduction to Solid Modeling","author":"Mantyla M.","key":"e_1_2_2_64_1","unstructured":"M. Mantyla . 1988. Introduction to Solid Modeling . W. H. Freeman 8 Co., New York, NY. M. Mantyla. 1988. Introduction to Solid Modeling. W. H. Freeman 8 Co., New York, NY."},{"volume-title":"Proceedings of the 12th International Conference on Computer Graphics Theory and Applications.","author":"Marino E.","key":"e_1_2_2_65_1","unstructured":"E. Marino , F. Spini , D. Salvati , C. Vadal\u00e0 , M. Vicentino , A. Paoluzzi , and A. Bottaro . 2017. Modeling semantics for building deconstruction . In Proceedings of the 12th International Conference on Computer Graphics Theory and Applications. E. Marino, F. Spini, D. Salvati, C. Vadal\u00e0, M. Vicentino, A. Paoluzzi, and A. Bottaro. 2017. Modeling semantics for building deconstruction. In Proceedings of the 12th International Conference on Computer Graphics Theory and Applications."},{"key":"e_1_2_2_66_1","doi-asserted-by":"publisher","DOI":"10.1145\/27625.27627"},{"key":"e_1_2_2_68_1","doi-asserted-by":"publisher","DOI":"10.1109\/MC.2014.103"},{"key":"e_1_2_2_69_1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(78)90051-8"},{"volume-title":"Elements of Algebraic Topology","author":"Munkres J. R.","key":"e_1_2_2_70_1","unstructured":"J. R. Munkres . 1984. Elements of Algebraic Topology . Addison-Wesley, Reading , MA. J. R. Munkres. 1984. Elements of Algebraic Topology. Addison-Wesley, Reading, MA."},{"key":"e_1_2_2_71_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8396(95)00015-X"},{"key":"e_1_2_2_72_1","volume-title":"Palmer and Vadim Shapiro","author":"Richard","year":"1993","unstructured":"Richard S. Palmer and Vadim Shapiro . 1993 . Chain models of physical behavior for engineering analysis and design. Res. Eng. Des . 5, 3 (01 September 1993), 161--184. DOI:https:\/\/doi.org\/10.1007\/BF01608361 10.1007\/BF01608361 Richard S. Palmer and Vadim Shapiro. 1993. Chain models of physical behavior for engineering analysis and design. Res. Eng. Des. 5, 3 (01 September 1993), 161--184. DOI:https:\/\/doi.org\/10.1007\/BF01608361"},{"volume-title":"Geometric Programming for Computer Aided Design","author":"Paoluzzi A.","key":"e_1_2_2_73_1","unstructured":"A. Paoluzzi . 2003. Geometric Programming for Computer Aided Design . John Wiley 8 Sons, Chichester, UK. https:\/\/doi.org\/10.1002\/0470013885 10.1002\/0470013885 A. Paoluzzi. 2003. Geometric Programming for Computer Aided Design. John Wiley 8 Sons, Chichester, UK. https:\/\/doi.org\/10.1002\/0470013885"},{"key":"e_1_2_2_74_1","doi-asserted-by":"publisher","DOI":"10.1145\/169728.169719"},{"key":"e_1_2_2_75_1","doi-asserted-by":"publisher","DOI":"10.1080\/16864360.2016.1168216"},{"key":"e_1_2_2_76_1","volume-title":"Advances in Architectural Geometry","author":"Paoluzzi Alberto","year":"2014","unstructured":"Alberto Paoluzzi , Enrico Marino , and Federico Spini . 2015. LAR-ABC , a representation of architectural geometry from concept of spaces, to design of building fabric, to construction simulation . In Advances in Architectural Geometry 2014 , Philippe Block, Jan Knippers, Niloy J. Mitra, and Wenping Wang (Eds.). Springer International Publishing , 353--372. DOI:https:\/\/doi.org\/10.1007\/978-3-319-11418-7_23 10.1007\/978-3-319-11418-7_23 Alberto Paoluzzi, Enrico Marino, and Federico Spini. 2015. LAR-ABC, a representation of architectural geometry from concept of spaces, to design of building fabric, to construction simulation. In Advances in Architectural Geometry 2014, Philippe Block, Jan Knippers, Niloy J. Mitra, and Wenping Wang (Eds.). Springer International Publishing, 353--372. DOI:https:\/\/doi.org\/10.1007\/978-3-319-11418-7_23"},{"volume-title":"Proceedings of the 9th ACM Symposium on Solid Modeling and Applications (SM\u201904)","author":"Paoluzzi A.","key":"e_1_2_2_77_1","unstructured":"A. Paoluzzi , V. Pascucci , and G. Scorzelli . 2004. Progressive dimension-independent boolean operations . In Proceedings of the 9th ACM Symposium on Solid Modeling and Applications (SM\u201904) . Eurographics Association, Aire-La-Ville, Switzerland, 203--211. A. Paoluzzi, V. Pascucci, and G. Scorzelli. 2004. Progressive dimension-independent boolean operations. In Proceedings of the 9th ACM Symposium on Solid Modeling and Applications (SM\u201904). Eurographics Association, Aire-La-Ville, Switzerland, 203--211."},{"key":"e_1_2_2_78_1","doi-asserted-by":"publisher","DOI":"10.1145\/212332.212349"},{"key":"e_1_2_2_79_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(89)90055-9"},{"key":"e_1_2_2_80_1","doi-asserted-by":"publisher","DOI":"10.1145\/218013.218055"},{"volume-title":"Proceedings of the 14th International Conference on Atomic Physics. 1--7.","author":"Michael","key":"e_1_2_2_81_1","unstructured":"Michael J. Pratt and Bill D. Anderson. 1994. A shape modelling API for the STEP standard . In Proceedings of the 14th International Conference on Atomic Physics. 1--7. Michael J. Pratt and Bill D. Anderson. 1994. A shape modelling API for the STEP standard. In Proceedings of the 14th International Conference on Atomic Physics. 1--7."},{"key":"e_1_2_2_82_1","doi-asserted-by":"publisher","DOI":"10.1145\/304012.304019"},{"key":"e_1_2_2_83_1","doi-asserted-by":"publisher","DOI":"10.1145\/267734.267749"},{"key":"e_1_2_2_84_1","volume-title":"Technical Report TM-25. Production Automation Project","author":"Requicha A. A. G.","year":"1977","unstructured":"A. A. G. Requicha and H. B. Voelcker . 1977 . Constructive Solid Geometry . Technical Report TM-25. Production Automation Project , University of Rochester. A. A. G. Requicha and H. B. Voelcker. 1977. Constructive Solid Geometry. Technical Report TM-25. Production Automation Project, University of Rochester."},{"key":"e_1_2_2_85_1","doi-asserted-by":"publisher","DOI":"10.1145\/356827.356833"},{"key":"e_1_2_2_86_1","volume-title":"SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries. In Geometric Modeling for Product Engineering. North-Holland.","author":"Rossignac J. R.","year":"1990","unstructured":"J. R. Rossignac and M. A. O\u2019Connor . 1990 . SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries. In Geometric Modeling for Product Engineering. North-Holland. J. R. Rossignac and M. A. O\u2019Connor. 1990. SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries. In Geometric Modeling for Product Engineering. North-Holland."},{"key":"e_1_2_2_88_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(91)90078-B"},{"volume-title":"Introduction to Piecewise-Linear Topology","author":"Rourke Colin","key":"e_1_2_2_89_1","unstructured":"Colin Rourke and Brian Sanderson . 1982. Introduction to Piecewise-Linear Topology . Springer-Verlag , Berlin . http:\/\/doi.org\/10.1007\/978-3-642-81735-9 10.1007\/978-3-642-81735-9 Colin Rourke and Brian Sanderson. 1982. Introduction to Piecewise-Linear Topology. Springer-Verlag, Berlin. http:\/\/doi.org\/10.1007\/978-3-642-81735-9"},{"key":"e_1_2_2_90_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195908002714"},{"key":"e_1_2_2_92_1","volume-title":"Proceedings of the 3rd ACM Symposium on Solid Modeling and Applications (SMA\u201995)","author":"Shapiro Vadim","year":"1801","unstructured":"Vadim Shapiro and Donald L. Vossler . 1995. What is a parametric family of solids? In Proceedings of the 3rd ACM Symposium on Solid Modeling and Applications (SMA\u201995) . ACM, 43--54. DOI:https:\/\/doi.org\/10.1145\/2 1801 3.218029 10.1145\/218013.218029 Vadim Shapiro and Donald L. Vossler. 1995. What is a parametric family of solids? In Proceedings of the 3rd ACM Symposium on Solid Modeling and Applications (SMA\u201995). ACM, 43--54. DOI:https:\/\/doi.org\/10.1145\/218013.218029"},{"key":"e_1_2_2_93_1","doi-asserted-by":"publisher","DOI":"10.1145\/237218.237337"},{"key":"e_1_2_2_94_1","volume-title":"Triangle: Engineering a 2D quality mesh generator and delaunay triangulator. In Applied Computational Geometry: Towards Geometric Engineering, Ming C","author":"Shewchuk Jonathan Richard","year":"1996","unstructured":"Jonathan Richard Shewchuk . 1996 . Triangle: Engineering a 2D quality mesh generator and delaunay triangulator. In Applied Computational Geometry: Towards Geometric Engineering, Ming C . Lin and Dinesh Manocha (Eds.). Lecture Notes in Computer Science, Vol. 1148 . Springer-Verlag , 203--222. Jonathan Richard Shewchuk. 1996. Triangle: Engineering a 2D quality mesh generator and delaunay triangulator. In Applied Computational Geometry: Towards Geometric Engineering, Ming C. Lin and Dinesh Manocha (Eds.). Lecture Notes in Computer Science, Vol. 1148. Springer-Verlag, 203--222."},{"key":"e_1_2_2_95_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(01)00047-5"},{"key":"e_1_2_2_97_1","doi-asserted-by":"publisher","DOI":"10.1145\/2945292.2945309"},{"key":"e_1_2_2_98_1","volume-title":"HIJSON: Cartographic document for web modeling of interactive indoor mapping. In Proceedings of the Smart Tools and Apps for Graphics--Eurographics Italian","author":"Spini Federico","year":"2015","unstructured":"Federico Spini , Marco Sportillo , Marco Virgadamo , Enrico Marino , Antonio Bottaro , and Alberto Paoluzzi . 2015 . HIJSON: Cartographic document for web modeling of interactive indoor mapping. In Proceedings of the Smart Tools and Apps for Graphics--Eurographics Italian Chapter Conference, Andrea Giachetti, Silvia Biasotti , and Marco Tarini (Eds.). The Eurographics Association . DOI:https:\/\/doi.org\/10.2312\/stag.20151290 10.2312\/stag.20151290 Federico Spini, Marco Sportillo, Marco Virgadamo, Enrico Marino, Antonio Bottaro, and Alberto Paoluzzi. 2015. HIJSON: Cartographic document for web modeling of interactive indoor mapping. In Proceedings of the Smart Tools and Apps for Graphics--Eurographics Italian Chapter Conference, Andrea Giachetti, Silvia Biasotti, and Marco Tarini (Eds.). The Eurographics Association. DOI:https:\/\/doi.org\/10.2312\/stag.20151290"},{"volume-title":"The Mathematical Structure of Classical and Relativistic Physics","author":"Tonti Enzo","key":"e_1_2_2_100_1","unstructured":"Enzo Tonti . 2013. The Mathematical Structure of Classical and Relativistic Physics . Birkh\u00e4user . Enzo Tonti. 2013. The Mathematical Structure of Classical and Relativistic Physics. Birkh\u00e4user."},{"key":"e_1_2_2_101_1","unstructured":"T. Vialar. 2016. Handbook of Mathematics. HDBoM.  T. Vialar. 2016. Handbook of Mathematics. HDBoM."},{"key":"e_1_2_2_102_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCG.1985.276271"},{"volume-title":"Geometric Modelling for CAD Applications","author":"Weiler K. J.","key":"e_1_2_2_104_1","unstructured":"K. J. Weiler . 1988. The radial edge structure: A topological representation for non-manifold geometric modelling . In Geometric Modelling for CAD Applications , M. Wozny, H. McLaughlin, and J. Encarnacao (Eds.). Amsterdam , 3--12. K. J. Weiler. 1988. The radial edge structure: A topological representation for non-manifold geometric modelling. In Geometric Modelling for CAD Applications, M. Wozny, H. McLaughlin, and J. Encarnacao (Eds.). Amsterdam, 3--12."},{"key":"e_1_2_2_105_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCG.1985.276337"},{"volume-title":"Proceedings of the Geometric Modeling for Product Engineering: IFIP WG 5.2\/NSF Working Conference on Geometric Modeling. North-Holland.","author":"Wozny M. J.","key":"e_1_2_2_106_1","unstructured":"M. J. Wozny , J. U. Turner , and K. Preiss . 1990 . Proceedings of the Geometric Modeling for Product Engineering: IFIP WG 5.2\/NSF Working Conference on Geometric Modeling. North-Holland. M. J. Wozny, J. U. Turner, and K. Preiss. 1990. Proceedings of the Geometric Modeling for Product Engineering: IFIP WG 5.2\/NSF Working Conference on Geometric Modeling. North-Holland."},{"key":"e_1_2_2_107_1","doi-asserted-by":"crossref","unstructured":"F. Yamaguchi and T. Tokieda. 1985. Bridge edge and triangulation approach in solid modeling. In Frontiers in Computer Graphics T. L. Kunii (Ed.). Springer-Verlag Berlin.  F. Yamaguchi and T. Tokieda. 1985. Bridge edge and triangulation approach in solid modeling. In Frontiers in Computer Graphics T. L. Kunii (Ed.). Springer-Verlag Berlin.","DOI":"10.1007\/978-4-431-68025-3_4"},{"key":"e_1_2_2_108_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.364963"},{"key":"e_1_2_2_109_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925901"}],"container-title":["ACM Transactions on Spatial Algorithms and Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3401988","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3401988","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3401988","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T22:03:13Z","timestamp":1750197793000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3401988"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,14]]},"references-count":99,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,3,31]]}},"alternative-id":["10.1145\/3401988"],"URL":"https:\/\/doi.org\/10.1145\/3401988","relation":{},"ISSN":["2374-0353","2374-0361"],"issn-type":[{"type":"print","value":"2374-0353"},{"type":"electronic","value":"2374-0361"}],"subject":[],"published":{"date-parts":[[2020,10,14]]},"assertion":[{"value":"2017-08-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-05-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-10-14","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}