{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T09:25:50Z","timestamp":1780392350048,"version":"3.54.1"},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2011,7,1]],"date-time":"2011-07-01T00:00:00Z","timestamp":1309478400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2011,7]]},"abstract":"<jats:p>\n            While the theory and applications of discrete Laplacians on\n            <jats:italic>triangulated<\/jats:italic>\n            surfaces are well developed, far less is known about the general\n            <jats:italic>polygonal<\/jats:italic>\n            case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace--Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes.\n          <\/jats:p>","DOI":"10.1145\/2010324.1964997","type":"journal-article","created":{"date-parts":[[2011,7,26]],"date-time":"2011-07-26T14:17:46Z","timestamp":1311689866000},"page":"1-10","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":70,"title":["Discrete Laplacians on general polygonal meshes"],"prefix":"10.1145","volume":"30","author":[{"given":"Marc","family":"Alexa","sequence":"first","affiliation":[{"name":"TU Berlin"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Max","family":"Wardetzky","sequence":"additional","affiliation":[{"name":"Universit\u00e4t G\u00f6ttingen"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2011,7,25]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1377676.1377725"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-9006-1"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/040613950"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202505000832"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2010.02.002"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/311535.311576"},{"key":"e_1_2_2_7_1","doi-asserted-by":"crossref","unstructured":"Desbrun M. Meyer M. and Alliez P. 2002. Intrinsic parameterizations of triangle meshes. Computer Graphics Forum (Proc. of Eurographics) 209--218.","DOI":"10.1111\/1467-8659.00580"},{"key":"e_1_2_2_8_1","unstructured":"Desbrun M. Hirani A. Leok M. and Marsden J. E. 2005. Discrete exterior calculus. arXiv:math.DG\/0508341."},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.5555\/1873601.1873655"},{"key":"e_1_2_2_10_1","first-page":"793","article-title":"Distributed and Lumped Networks","volume":"8","author":"Duffin R.","year":"1959","unstructured":"Duffin, R. 1959. Distributed and Lumped Networks. Journal of Mathematics and Mechanics 8, 793--825.","journal-title":"Journal of Mathematics and Mechanics"},{"key":"e_1_2_2_11_1","volume-title":"Technical Report CMPT2009-6.","author":"Dyer R.","year":"2009","unstructured":"Dyer, R., and Schaefer, S., 2009. Circumcentric dual cells with negative area. 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Discrete Laplace--Beltrami operator determines discrete Riemannian metric. arXiv:1010.4070."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10711-006-9109-5"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566590"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2007.04.047"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/s002200000348"},{"key":"e_1_2_2_21_1","doi-asserted-by":"crossref","unstructured":"Meyer M. Desbrun M. Schr\u00f6der P. and Barr A. H. 2003. Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III H.-C. Hege and K. Polthier Eds. 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Alternative approach of numerical integration for rational functions related to linear convex quadrilateral finite elements. Journal of App. Sciences Research 2, 9, 533--540.","journal-title":"Journal of App. Sciences Research"},{"key":"e_1_2_2_28_1","doi-asserted-by":"crossref","unstructured":"Sullivan J. 2008. Curvatures of smooth and discrete surfaces. In Discrete Differential Geometry A. I. Bobenko J. M. Sullivan P. Schr\u00f6der and G. Ziegler Eds. Birkh\u00e4user Basel 175--188.","DOI":"10.1007\/978-3-7643-8621-4_9"},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2007.07.006"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","unstructured":"Wardetzky M. Mathur S. K\u00e4lberer F. and Grinspun E. 2007. Discrete Laplace operators: No free lunch. In Siggraph\/Eurographics Sympos. Geom. Processing 33--37.","DOI":"10.5555\/1281991.1281995"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","unstructured":"Xiong Y. Li G. and Han G. 2011. 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