{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:19:14Z","timestamp":1758824354095,"version":"3.41.0"},"reference-count":75,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2020,3,18]],"date-time":"2020-03-18T00:00:00Z","timestamp":1584489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["282535003"],"award-info":[{"award-number":["282535003"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-17-17268"],"award-info":[{"award-number":["CCF-17-17268"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fields Centre for Quantitative Analysis and Modelling"},{"DOI":"10.13039\/501100002790","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN2017-05235"],"award-info":[{"award-number":["RGPIN2017-05235"]}],"id":[{"id":"10.13039\/501100002790","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001804","name":"Canada Research Chairs","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001804","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2020,6,30]]},"abstract":"<jats:p>Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.<\/jats:p>","DOI":"10.1145\/3377406","type":"journal-article","created":{"date-parts":[[2020,3,18]],"date-time":"2020-03-18T12:20:39Z","timestamp":1584534039000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":15,"title":["A Smoothness Energy without Boundary Distortion for Curved Surfaces"],"prefix":"10.1145","volume":"39","author":[{"given":"Oded","family":"Stein","sequence":"first","affiliation":[{"name":"Columbia University, USA"}]},{"given":"Alec","family":"Jacobson","sequence":"additional","affiliation":[{"name":"University of Toronto, Toronto, Ontario, Canada"}]},{"given":"Max","family":"Wardetzky","sequence":"additional","affiliation":[{"name":"University of G\u00f6ttingen, G\u00f6ttingen, Germany"}]},{"given":"Eitan","family":"Grinspun","sequence":"additional","affiliation":[{"name":"University of Toronto, Canada and Columbia University, USA"}]}],"member":"320","published-online":{"date-parts":[[2020,3,18]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"E. D. Andersen and K. D. Andersen. 2000. The &lt;scp;&gt;MOSEK&lt;\/scp;&gt; interior point optimizer for linear programming: An implementation of the homogeneous algorithm. In High Performance Optimization. Kluwer Academic Publishers 197--232.  E. D. Andersen and K. D. Andersen. 2000. The &lt;scp;&gt;MOSEK&lt;\/scp;&gt; interior point optimizer for linear programming: An implementation of the homogeneous algorithm. In High Performance Optimization. Kluwer Academic Publishers 197--232.","DOI":"10.1007\/978-1-4757-3216-0_8"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/2723158"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276467"},{"key":"e_1_2_2_4_1","volume-title":"Proceedings of the 4th Eurographics Symposium on Geometry Processing. 227--230","author":"Bergou Miklos","year":"2006","unstructured":"Miklos Bergou , Max Wardetzky , David Harmon , Denis Zorin , and Eitan Grinspun . 2006 . A quadratic bending model for inextensible surfaces . In Proceedings of the 4th Eurographics Symposium on Geometry Processing. 227--230 . Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, and Eitan Grinspun. 2006. A quadratic bending model for inextensible surfaces. In Proceedings of the 4th Eurographics Symposium on Geometry Processing. 227--230."},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360662"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08647-4"},{"volume-title":"Mixed Finite Element Methods and Applications","author":"Boffi Daniele","key":"e_1_2_2_7_1","unstructured":"Daniele Boffi , Franco Brezzi , and Michel Fortin . 2013. Mixed Finite Element Methods and Applications . Springer-Verlag Berlin . Daniele Boffi, Franco Brezzi, and Michel Fortin. 2013. Mixed Finite Element Methods and Applications. Springer-Verlag Berlin."},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015772"},{"key":"e_1_2_2_9_1","volume-title":"Finite Elements","author":"Braess Dietrich","unstructured":"Dietrich Braess . 2007. Finite Elements ( 3 rd ed.). Cambridge University Press . Dietrich Braess. 2007. Finite Elements (3rd ed.). Cambridge University Press.","edition":"3"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/3177750"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1002\/num.21892"},{"key":"e_1_2_2_12_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3292480","article-title":"Functional characterization of deformation fields","volume":"38","author":"Corman Etienne","year":"2019","unstructured":"Etienne Corman and Maks Ovsjanikov . 2019 . Functional characterization of deformation fields . ACM Trans. Graph. 38 , 1 (Jan. 2019), 8:1--8:19. Etienne Corman and Maks Ovsjanikov. 2019. Functional characterization of deformation fields. ACM Trans. Graph. 38, 1 (Jan. 2019), 8:1--8:19.","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_2_13_1","doi-asserted-by":"crossref","unstructured":"R. Courant and D. Hilbert. 1924. Methoden der Mathematischen Physik\u2014Erster Band. Julius Springer Berlin.  R. Courant and D. Hilbert. 1924. Methoden der Mathematischen Physik\u2014Erster Band. Julius Springer Berlin.","DOI":"10.1007\/978-3-662-36445-1"},{"key":"e_1_2_2_14_1","unstructured":"Keenan Crane. 2018. Keenan\u2019s 3D Model Repository. Retrieved from https:\/\/www.cs.cmu.edu\/kmcrane\/Projects\/ModelRepository\/.  Keenan Crane. 2018. Keenan\u2019s 3D Model Repository. Retrieved from https:\/\/www.cs.cmu.edu\/kmcrane\/Projects\/ModelRepository\/."},{"key":"e_1_2_2_15_1","first-page":"1525","article-title":"Trivial connections on discrete surfaces","volume":"29","author":"Crane Keenan","year":"2010","unstructured":"Keenan Crane , Mathieu Desbrun , and Peter Schr\u00f6der . 2010 . Trivial connections on discrete surfaces . Comput. Graph. For. 29 , 5 (2010), 1525 -- 1533 . Keenan Crane, Mathieu Desbrun, and Peter Schr\u00f6der. 2010. Trivial connections on discrete surfaces. Comput. Graph. For. 29, 5 (2010), 1525--1533.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461986"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/2516971.2516977"},{"key":"e_1_2_2_18_1","volume-title":"R3","author":"Crouzeix Michel","year":"1973","unstructured":"Michel Crouzeix and P.-A. Raviart . 1973. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM: Math. Modell. Numer. Anal.\u2014Mod\u00e9l. Math. Anal. Num\u00e9r. 7 , R3 ( 1973 ), 33--75. Michel Crouzeix and P.-A. Raviart. 1973. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM: Math. Modell. Numer. Anal.\u2014Mod\u00e9l. Math. Anal. Num\u00e9r. 7, R3 (1973), 33--75."},{"volume-title":"Proceedings of the 26th Conference on Computer Graphics and Interactive Techniques (SIGGRAPH\u201999)","author":"Desbrun Mathieu","key":"e_1_2_2_19_1","unstructured":"Mathieu Desbrun , Mark Meyer , Peter Schr\u00f6der , and Alan H. Barr . 1999. Implicit fairing of irregular meshes using diffusion and curvature flow . In Proceedings of the 26th Conference on Computer Graphics and Interactive Techniques (SIGGRAPH\u201999) . ACM Press\/Addison-Wesley Publishing Co., New York, NY, 317--324. Mathieu Desbrun, Mark Meyer, Peter Schr\u00f6der, and Alan H. Barr. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. In Proceedings of the 26th Conference on Computer Graphics and Interactive Techniques (SIGGRAPH\u201999). ACM Press\/Addison-Wesley Publishing Co., New York, NY, 317--324."},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-009-0166-x"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1031596100"},{"key":"e_1_2_2_22_1","first-page":"793","article-title":"Distributed and lumped networks","volume":"8","author":"Duffin R. J.","year":"1959","unstructured":"R. J. Duffin . 1959 . Distributed and lumped networks . J. Math. Mech. 8 , 5 (1959), 793 -- 826 . R. J. Duffin. 1959. Distributed and lumped networks. J. Math. Mech. 8, 5 (1959), 793--826.","journal-title":"J. Math. Mech."},{"volume-title":"Finite Elements for the Beltrami Operator on Arbitrary Surfaces","author":"Dziuk Gerhard","key":"e_1_2_2_23_1","unstructured":"Gerhard Dziuk . 1988. Finite Elements for the Beltrami Operator on Arbitrary Surfaces . Springer Berlin , 142--155. Gerhard Dziuk. 1988. Finite Elements for the Beltrami Operator on Arbitrary Surfaces. Springer Berlin, 142--155."},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360665"},{"key":"e_1_2_2_25_1","volume-title":"Partial Differential Equations","author":"Evans Lawrence C.","unstructured":"Lawrence C. Evans . 2010. Partial Differential Equations : ( 2 nd ed.). American Mathematical Society . Lawrence C. Evans. 2010. Partial Differential Equations: (2nd ed.). American Mathematical Society.","edition":"2"},{"volume-title":"Descartes on Polyhedra","author":"Federico P. J.","key":"e_1_2_2_26_1","unstructured":"P. J. Federico . 1982. Descartes on Polyhedra . Springer-Verlag New York Inc . P. J. Federico. 1982. Descartes on Polyhedra. Springer-Verlag New York Inc."},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276447"},{"volume-title":"Polyharmonic Boundary Value Problems","author":"Gazzola Filippo","key":"e_1_2_2_28_1","unstructured":"Filippo Gazzola , Hans-Christoph Grunau , and Guido Sweers . 2010. Polyharmonic Boundary Value Problems . Springer-Verlag Berlin . Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers. 2010. Polyharmonic Boundary Value Problems. Springer-Verlag Berlin."},{"key":"e_1_2_2_29_1","volume-title":"Proceedings of the European Conference on Computer Vision.","author":"Georgiev Todor","year":"2004","unstructured":"Todor Georgiev . 2004 . Photoshop healing brush: A tool for seamless cloning . In Proceedings of the European Conference on Computer Vision. Todor Georgiev. 2004. Photoshop healing brush: A tool for seamless cloning. In Proceedings of the European Conference on Computer Vision."},{"key":"e_1_2_2_30_1","unstructured":"Stanford Graphics. 2014. The Stanford 3D Scanning Repository. Retrieved from http:\/\/graphics.stanford.edu\/data\/3Dscanrep\/.  Stanford Graphics. 2014. The Stanford 3D Scanning Repository. Retrieved from http:\/\/graphics.stanford.edu\/data\/3Dscanrep\/."},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/1185657.1185659"},{"key":"e_1_2_2_32_1","volume-title":"Beno\u00eet Jacob et al","author":"Guennebaud Ga\u00ebl","year":"2010","unstructured":"Ga\u00ebl Guennebaud , Beno\u00eet Jacob et al . 2010 . Eigen v3. Retrieved from http:\/\/eigen.tuxfamily.org. Ga\u00ebl Guennebaud, Beno\u00eet Jacob et al. 2010. Eigen v3. Retrieved from http:\/\/eigen.tuxfamily.org."},{"key":"e_1_2_2_33_1","first-page":"1","article-title":"On the convergence of metric and geometric properties of polyhedral surfaces","volume":"123","author":"Hildebrandt Klaus","year":"2006","unstructured":"Klaus Hildebrandt , Konrad Polthier , and Max Wardetzky . 2006 . On the convergence of metric and geometric properties of polyhedral surfaces . Geom. Dedic. 123 , 1 (Dec. 2006), 89--112. Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky. 2006. On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedic. 123, 1 (Dec. 2006), 89--112.","journal-title":"Geom. Dedic."},{"key":"e_1_2_2_35_1","unstructured":"Alec Jacobson. 2019. gptoolbox\u2014Geometry Processing Toolbox. Retrieved from https:\/\/github.com\/alecjacobson\/gptoolbox.  Alec Jacobson. 2019. gptoolbox\u2014Geometry Processing Toolbox. Retrieved from https:\/\/github.com\/alecjacobson\/gptoolbox."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964973"},{"key":"e_1_2_2_37_1","volume-title":"Daniele Panozzo et al","author":"Jacobson Alec","year":"2019","unstructured":"Alec Jacobson , Daniele Panozzo et al . 2019 a. libigl: A Simple C++ Geometry Processing Library . Retrieved from http:\/\/libigl.github.io\/libigl\/. Alec Jacobson, Daniele Panozzo et al. 2019a. libigl: A Simple C++ Geometry Processing Library. Retrieved from http:\/\/libigl.github.io\/libigl\/."},{"key":"e_1_2_2_38_1","volume-title":"Daniele Panozzo et al","author":"Jacobson Alec","year":"2019","unstructured":"Alec Jacobson , Daniele Panozzo et al . 2019 b. Libigl Tutorial Data. Retrieved from https:\/\/github.com\/libigl\/libigl-tutorial-data. Alec Jacobson, Daniele Panozzo et al. 2019b. Libigl Tutorial Data. Retrieved from https:\/\/github.com\/libigl\/libigl-tutorial-data."},{"key":"e_1_2_2_39_1","first-page":"1565","article-title":"Mixed finite elements for variational surface modeling","volume":"29","author":"Jacobson Alec","year":"2010","unstructured":"Alec Jacobson , Elif Tosun , Olga Sorkine , and Denis Zorin . 2010 . Mixed finite elements for variational surface modeling . Comput. Graph. For. 29 , 5 (2010), 1565 -- 1574 . Alec Jacobson, Elif Tosun, Olga Sorkine, and Denis Zorin. 2010. Mixed finite elements for variational surface modeling. Comput. Graph. For. 29, 5 (2010), 1565--1574.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_40_1","first-page":"5","article-title":"Smooth shape-aware functions with controlled extrema","volume":"31","author":"Jacobson Alec","year":"2012","unstructured":"Alec Jacobson , Tino Weinkauf , and Olga Sorkine . 2012 . Smooth shape-aware functions with controlled extrema . Comput. Graph. For. 31 , 5 (Aug. 2012), 1577--1586. Alec Jacobson, Tino Weinkauf, and Olga Sorkine. 2012. Smooth shape-aware functions with controlled extrema. Comput. Graph. For. 31, 5 (Aug. 2012), 1577--1586.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_41_1","unstructured":"jansentee3d. 2018. Dragon Tower. Retrieved from https:\/\/www.thingiverse.com\/thing:3155868.  jansentee3d. 2018. Dragon Tower. Retrieved from https:\/\/www.thingiverse.com\/thing:3155868."},{"key":"e_1_2_2_42_1","unstructured":"Javo. 2016. Big Gladiator Helmet. Retrieved from https:\/\/www.thingiverse.com\/thing:1345281.  Javo. 2016. Big Gladiator Helmet. Retrieved from https:\/\/www.thingiverse.com\/thing:1345281."},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276466"},{"volume-title":"Riemannian Geometry and Geometric Analysis","author":"Jost J\u00fcrgen","key":"e_1_2_2_44_1","unstructured":"J\u00fcrgen Jost . 2011. Riemannian Geometry and Geometric Analysis . Springer-Verlag Berlin . J\u00fcrgen Jost. 2011. Riemannian Geometry and Geometric Analysis. Springer-Verlag Berlin."},{"volume-title":"Advances in Neural Information Processing Systems 22","author":"Kim Kwang I.","key":"e_1_2_2_45_1","unstructured":"Kwang I. Kim , Florian Steinke , and Matthias Hein . 2009. Semi-supervised regression using Hessian energy with an application to semi-supervised dimensionality reduction . In Advances in Neural Information Processing Systems 22 , Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta (Eds.). Curran Associates, Inc. , 979--987. Kwang I. Kim, Florian Steinke, and Matthias Hein. 2009. Semi-supervised regression using Hessian energy with an application to semi-supervised dimensionality reduction. In Advances in Neural Information Processing Systems 22, Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta (Eds.). Curran Associates, Inc., 979--987."},{"key":"e_1_2_2_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462005"},{"key":"e_1_2_2_47_1","doi-asserted-by":"publisher","DOI":"10.1145\/2767000"},{"volume-title":"Springer-Verlag New York","author":"Lee John M.","key":"e_1_2_2_48_1","unstructured":"John M. Lee . 1997. Riemannian Manifolds . Springer-Verlag New York , Inc . John M. Lee. 1997. Riemannian Manifolds. Springer-Verlag New York, Inc."},{"key":"e_1_2_2_49_1","volume-title":"IEEE Trans. Image Proc. 21 (09","author":"Lefkimmiatis Stamatis","year":"2011","unstructured":"Stamatis Lefkimmiatis , Aur\u00e9lien Bourquard , and Michael Unser . 2011 . Hessian-based norm regularization for image restoration with biomedical applications . IEEE Trans. Image Proc. 21 (09 2011), 983--95. Stamatis Lefkimmiatis, Aur\u00e9lien Bourquard, and Michael Unser. 2011. Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Proc. 21 (09 2011), 983--95."},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1145\/1186562.1015780"},{"key":"e_1_2_2_51_1","doi-asserted-by":"publisher","DOI":"10.1145\/1805964.1805971"},{"key":"e_1_2_2_52_1","doi-asserted-by":"publisher","DOI":"10.1145\/2870629"},{"key":"e_1_2_2_53_1","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2014.904854"},{"volume-title":"Smooth Subdivision Surfaces Based on Triangles. Master\u2019s Thesis","author":"Loop Charles","key":"e_1_2_2_54_1","unstructured":"Charles Loop . 1987. Smooth Subdivision Surfaces Based on Triangles. Master\u2019s Thesis . University of Utah. Charles Loop. 1987. Smooth Subdivision Surfaces Based on Triangles. Master\u2019s Thesis. University of Utah."},{"key":"e_1_2_2_55_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2003.819229"},{"volume-title":"version 9.6 (R2019a)","author":"MATLAB.","key":"e_1_2_2_57_1","unstructured":"MATLAB. 2019. version 9.6 (R2019a) . The MathWorks Inc . MATLAB. 2019. version 9.6 (R2019a). The MathWorks Inc."},{"key":"e_1_2_2_58_1","volume-title":"Wright","author":"Nocedal Jorge","year":"2006","unstructured":"Jorge Nocedal and Stephen J . Wright . 2006 . Numerical Optimization (2nd ed.). Springer Science + Business Media, LLC. Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2nd ed.). Springer Science + Business Media, LLC."},{"key":"e_1_2_2_59_1","volume-title":"C (Aug.","author":"Peter Pascal","year":"2016","unstructured":"Pascal Peter , Sebastian Hoffmann , Frank Nedwed , Laurent Hoeltgen , and Joachim Weickert . 2016. Evaluating the true potential of diffusion-based inpainting in a compression context. Image Commun. 46 , C (Aug. 2016 ), 40--53. Pascal Peter, Sebastian Hoffmann, Frank Nedwed, Laurent Hoeltgen, and Joachim Weickert. 2016. Evaluating the true potential of diffusion-based inpainting in a compression context. Image Commun. 46, C (Aug. 2016), 40--53."},{"key":"e_1_2_2_60_1","volume-title":"Riemannian Geometry","author":"Petersen Peter","unstructured":"Peter Petersen . 2006. Riemannian Geometry ( 2 nd ed.). Springer Science + Business Media, LLC. Peter Petersen. 2006. Riemannian Geometry (2nd ed.). Springer Science + Business Media, LLC.","edition":"2"},{"volume-title":"Straightest Geodesics on Polyhedral Surfaces","author":"Polthier Konrad","key":"e_1_2_2_61_1","unstructured":"Konrad Polthier and Markus Schmies . 1998. Straightest Geodesics on Polyhedral Surfaces . Springer Berlin , 135--150. DOI:https:\/\/doi.org\/10.1007\/978-3-662-03567-2_11 10.1007\/978-3-662-03567-2_11 Konrad Polthier and Markus Schmies. 1998. Straightest Geodesics on Polyhedral Surfaces. Springer Berlin, 135--150. DOI:https:\/\/doi.org\/10.1007\/978-3-662-03567-2_11"},{"key":"e_1_2_2_63_1","doi-asserted-by":"publisher","DOI":"10.1145\/1640443.1640444"},{"volume-title":"The Theory of Sound","author":"Strutt Rayleigh John William","key":"e_1_2_2_64_1","unstructured":"John William Strutt Rayleigh . 1894. The Theory of Sound . Vol. 1 . Macmillan and Co. John William Strutt Rayleigh. 1894. The Theory of Sound. Vol. 1. Macmillan and Co."},{"key":"e_1_2_2_65_1","volume-title":"A mixed method for 4th order problems using linear finite elements. RAIRO. Anal. num\u00e9r. 12, 1","author":"Scholz Reinhard","year":"1978","unstructured":"Reinhard Scholz . 1978. A mixed method for 4th order problems using linear finite elements. RAIRO. Anal. num\u00e9r. 12, 1 ( 1978 ), 85--90. Reinhard Scholz. 1978. A mixed method for 4th order problems using linear finite elements. RAIRO. Anal. num\u00e9r. 12, 1 (1978), 85--90."},{"key":"e_1_2_2_66_1","volume-title":"The vector heat method. CoRR abs\/1805.09170","author":"Sharp Nicholas","year":"2018","unstructured":"Nicholas Sharp , Yousuf Soliman , and Keenan Crane . 2018. The vector heat method. CoRR abs\/1805.09170 ( 2018 ). Retrieved from http:\/\/arxiv.org\/abs\/1805.09170. Nicholas Sharp, Yousuf Soliman, and Keenan Crane. 2018. The vector heat method. CoRR abs\/1805.09170 (2018). Retrieved from http:\/\/arxiv.org\/abs\/1805.09170."},{"key":"e_1_2_2_67_1","doi-asserted-by":"publisher","DOI":"10.1145\/1057432.1057456"},{"key":"e_1_2_2_68_1","doi-asserted-by":"publisher","DOI":"10.1145\/3186564"},{"key":"e_1_2_2_69_1","doi-asserted-by":"publisher","DOI":"10.1145\/2591011"},{"key":"e_1_2_2_70_1","first-page":"545","article-title":"Directional field synthesis, design, and processing","volume":"35","author":"Vaxman Amir","year":"2016","unstructured":"Amir Vaxman , Marcel Campen , Olga Diamanti , Daniele Panozzo , David Bommes , Klaus Hildebrandt , and Mirela Ben-Chen . 2016 . Directional field synthesis, design, and processing . Comput. Graph. For. 35 , 2 (2016), 545 -- 572 . Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, and Mirela Ben-Chen. 2016. Directional field synthesis, design, and processing. Comput. Graph. For. 35, 2 (2016), 545--572.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_71_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766952"},{"key":"e_1_2_2_74_1","volume-title":"Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP\u201907)","author":"Wardetzky Max","year":"2007","unstructured":"Max Wardetzky , Saurabh Mathur , Felix K\u00e4lberer , and Eitan Grinspun . 2007 . Discrete laplace operators: No free lunch . In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP\u201907) . Eurographics Association, 33--37. Max Wardetzky, Saurabh Mathur, Felix K\u00e4lberer, and Eitan Grinspun. 2007. Discrete laplace operators: No free lunch. In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP\u201907). Eurographics Association, 33--37."},{"key":"e_1_2_2_75_1","first-page":"2409","article-title":"Biharmonic coordinates","volume":"31","author":"Weber Ofir","year":"2012","unstructured":"Ofir Weber , Roi Poranne , and Craig Gotsman . 2012 . Biharmonic coordinates . Comput. Graph. For. 31 , 8 (2012), 2409 -- 2422 . Ofir Weber, Roi Poranne, and Craig Gotsman. 2012. Biharmonic coordinates. Comput. Graph. For. 31, 8 (2012), 2409--2422.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_76_1","first-page":"265","article-title":"Context-aware skeletal shape deformationss","volume":"26","author":"Weber Ofir","year":"2007","unstructured":"Ofir Weber , Olga Sorkine , Yaron Lipman , and Craig Gotsman . 2007 . Context-aware skeletal shape deformationss . Comput. Graph. For. 26 , 3 (2007), 265 -- 274 . Ofir Weber, Olga Sorkine, Yaron Lipman, and Craig Gotsman. 2007. Context-aware skeletal shape deformationss. Comput. Graph. For. 26, 3 (2007), 265--274.","journal-title":"Comput. Graph. For."},{"key":"e_1_2_2_77_1","volume-title":"Topology-based smoothing of 2D scalar fields with C1-continuity. Comput. Graph. For. 29, 3","author":"Weinkauf Tino","year":"2010","unstructured":"Tino Weinkauf , Yotam Gingold , and Olga Sorkine . 2010. Topology-based smoothing of 2D scalar fields with C1-continuity. Comput. Graph. For. 29, 3 ( 2010 ). Tino Weinkauf, Yotam Gingold, and Olga Sorkine. 2010. Topology-based smoothing of 2D scalar fields with C1-continuity. Comput. Graph. For. 29, 3 (2010)."},{"key":"e_1_2_2_78_1","unstructured":"Eric W. Weisstein. 2019. Monge Patch. From MathWorld--A Wolfram Web Resource. Retrieved from http:\/\/mathworld.wolfram.com\/MongePatch.html.  Eric W. Weisstein. 2019. Monge Patch. From MathWorld--A Wolfram Web Resource. Retrieved from http:\/\/mathworld.wolfram.com\/MongePatch.html."},{"key":"e_1_2_2_79_1","unstructured":"Roland Weitzenb\u00f6ck. 1885. Invariantentheorie. P. Noordhoff.  Roland Weitzenb\u00f6ck. 1885. Invariantentheorie. P. Noordhoff."},{"key":"e_1_2_2_80_1","unstructured":"YahooJAPAN. 2013. Plane. Retrieved from https:\/\/www.thingiverse.com\/thing:182252.  YahooJAPAN. 2013. Plane. Retrieved from https:\/\/www.thingiverse.com\/thing:182252."}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3377406","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3377406","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3377406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T22:33:17Z","timestamp":1750199597000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3377406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,18]]},"references-count":75,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,6,30]]}},"alternative-id":["10.1145\/3377406"],"URL":"https:\/\/doi.org\/10.1145\/3377406","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"type":"print","value":"0730-0301"},{"type":"electronic","value":"1557-7368"}],"subject":[],"published":{"date-parts":[[2020,3,18]]},"assertion":[{"value":"2019-05-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-01-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-03-18","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}