{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:51:33Z","timestamp":1774630293660,"version":"3.50.1"},"reference-count":51,"publisher":"Association for Computing Machinery (ACM)","issue":"5","license":[{"start":{"date-parts":[[2011,10,1]],"date-time":"2011-10-01T00:00:00Z","timestamp":1317427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100003284","name":"Canadian Psychological Association","doi-asserted-by":"publisher","award":["811485"],"award-info":[{"award-number":["811485"]}],"id":[{"id":"10.13039\/501100003284","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["2008 BE1 00018"],"award-info":[{"award-number":["2008 BE1 00018"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["TIN2010-20590-C02-01"],"award-info":[{"award-number":["TIN2010-20590-C02-01"]}],"id":[{"id":"10.13039\/100000001","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":[[2011,10]]},"abstract":"<jats:p>\n            We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA). Steady affine motions, for which ARA=0, include translations, rotations, screws, and the golden spiral. To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation),\n            <jats:italic>B<\/jats:italic>\n            and\n            <jats:italic>C<\/jats:italic>\n            , we propose the Steady Affine Morph (SAM), defined as\n            <jats:italic>A<\/jats:italic>\n            <jats:sup>t<\/jats:sup>\n            \u2218\n            <jats:italic>B<\/jats:italic>\n            with\n            <jats:italic>A<\/jats:italic>\n            =\n            <jats:italic>C<\/jats:italic>\n            \u2218\n            <jats:italic>B<\/jats:italic>\n            <jats:sup>-1<\/jats:sup>\n            . A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count,\n            <jats:italic>n<\/jats:italic>\n            , the series of uniformly sampled poses,\n            <jats:italic>A<\/jats:italic>\n            <jats:sup>i\/n<\/jats:sup>\n            \u2218\n            <jats:italic>B<\/jats:italic>\n            , of a SAM form a regular pattern which may be easily controlled by changing\n            <jats:italic>B<\/jats:italic>\n            ,\n            <jats:italic>C<\/jats:italic>\n            , or\n            <jats:italic>n<\/jats:italic>\n            , and where consecutive poses are related by the same affinity\n            <jats:italic>A<\/jats:italic>\n            <jats:sup>1\/n<\/jats:sup>\n            . Although a real matrix\n            <jats:italic>A<\/jats:italic>\n            <jats:italic>\n              <jats:sup>t<\/jats:sup>\n            <\/jats:italic>\n            does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities\n            <jats:italic>A<\/jats:italic>\n            . Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes\n            <jats:italic>A<\/jats:italic>\n            <jats:italic>\n              <jats:sup>t<\/jats:sup>\n            <\/jats:italic>\n            , when it exists, using closed-form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation.\n          <\/jats:p>","DOI":"10.1145\/2019627.2019635","type":"journal-article","created":{"date-parts":[[2011,10,25]],"date-time":"2011-10-25T12:23:05Z","timestamp":1319545385000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":29,"title":["Steady affine motions and morphs"],"prefix":"10.1145","volume":"30","author":[{"given":"Jarek","family":"Rossignac","sequence":"first","affiliation":[{"name":"Georgia Institute of Technology"}]},{"given":"\u00c1lvar","family":"Vinacua","sequence":"additional","affiliation":[{"name":"Technical University of Catalonia"}]}],"member":"320","published-online":{"date-parts":[[2011,10,22]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218654306000858"},{"key":"e_1_2_2_2_1","volume-title":"Computer Graphics and Geometric Modeling: Mathematics","author":"Agoston M. K.","unstructured":"Agoston , M. K. 2005. Computer Graphics and Geometric Modeling: Mathematics . Springer . Agoston, M. K. 2005. Computer Graphics and Geometric Modeling: Mathematics. Springer."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/566570.566592"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.344859"},{"key":"e_1_2_2_5_1","unstructured":"Alexander C. Ishikawa S. and Silverstein M. 1977. A Pattern Language: Towns Buildings Construction. Oxford University Press Oxford UK.  Alexander C. Ishikawa S. and Silverstein M. 1977. A Pattern Language: Towns Buildings Construction. Oxford University Press Oxford UK."},{"key":"e_1_2_2_6_1","volume-title":"Ordinary Differential Equations","author":"Arnold V. I.","unstructured":"Arnold , V. I. 1981. Ordinary Differential Equations . The MIT Press . Arnold, V. I. 1981. Ordinary Differential Equations. The MIT Press."},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/133994.134086"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/566570.566595"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/502122.502124"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/502122.502124"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899364015"},{"key":"e_1_2_2_12_1","unstructured":"Choi J. and Szymczak A. 2003. On coherent rotation angles for as-rigid-as-possible shape interpolation. Tech. rep. Georgia Institute of Technology.  Choi J. and Szymczak A. 2003. On coherent rotation angles for as-rigid-as-possible shape interpolation. Tech. rep. Georgia Institute of Technology."},{"key":"e_1_2_2_13_1","volume-title":"Mathematical Papers","author":"Clifford W.","unstructured":"Clifford , W. 1882. Mathematical Papers . Macmillan , London . Clifford, W. 1882. Mathematical Papers. Macmillan, London."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1966-0202740-6"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479802410815"},{"key":"e_1_2_2_16_1","volume-title":"SIGGRAPH Course Notes on State of the Art Image Synthesis","author":"Duff T.","unstructured":"Duff , T. 1986. Splines in animation and modeling . In SIGGRAPH Course Notes on State of the Art Image Synthesis . ACM Press . Duff, T. 1986. Splines in animation and modeling. In SIGGRAPH Course Notes on State of the Art Image Synthesis. ACM Press."},{"key":"e_1_2_2_17_1","volume-title":"Octave version 2.0.13","author":"Eaton J. W. J. W.","unstructured":"Eaton , J. W. J. W. 2000. GNU Octave: A High-Level Interactive Language for Numerical Computations Ed. 3 , Octave version 2.0.13 . Network Theory Ltd . Eaton, J. W. J. W. 2000. GNU Octave: A High-Level Interactive Language for Numerical Computations Ed. 3, Octave version 2.0.13. Network Theory Ltd."},{"key":"e_1_2_2_18_1","volume-title":"Proceedings of the Pacific Graphics'05 Conference. 100--102","author":"Fu H.","unstructured":"Fu , H. , Tai , C. L. , and Au , O. K . 2005. Morphing with laplacian coordinates and spatial-temporal texture . In Proceedings of the Pacific Graphics'05 Conference. 100--102 . Fu, H., Tai, C. L., and Au, O. K. 2005. Morphing with laplacian coordinates and spatial-temporal texture. In Proceedings of the Pacific Graphics'05 Conference. 100--102."},{"key":"e_1_2_2_19_1","volume-title":"Matrix Computations","author":"Golub G. H.","unstructured":"Golub , G. H. and Van Loan , C. F. 1996. Matrix Computations 3 rd Ed. Johns Hopkins University Press , Baltimore, MD . Golub, G. H. and Van Loan, C. F. 1996. Matrix Computations 3rd Ed. Johns Hopkins University Press, Baltimore, MD.","edition":"3"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1080\/10867651.1998.10487493"},{"key":"e_1_2_2_21_1","doi-asserted-by":"crossref","unstructured":"Higham D. J. and Higham N. J. 2005. MATLAB Guide. SIAM.   Higham D. J. and Higham N. J. 2005. MATLAB Guide. SIAM.","DOI":"10.1137\/1.9780898717891"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.2307\/2007992"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1006\/gmod.2002.0569"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.gmod.2008.12.001"},{"key":"e_1_2_2_25_1","unstructured":"Johnston O. and Thomas F. 1995. The Illusion of Life: Disney Animation. Disney Press.  Johnston O. and Thomas F. 1995. The Illusion of Life: Disney Animation. Disney Press."},{"key":"e_1_2_2_26_1","volume-title":"Dual quaternions for rigid transformation blending. Tech. rep. TCD-CS-2006-46","author":"Kavan L.","unstructured":"Kavan , L. , Collins , S. , O'Sullivan , C. , and \u017d\u00e1ra , J. 2006. Dual quaternions for rigid transformation blending. Tech. rep. TCD-CS-2006-46 , Trinity College , Dublin . Kavan, L., Collins, S., O'Sullivan, C., and \u017d\u00e1ra, J. 2006. Dual quaternions for rigid transformation blending. Tech. rep. TCD-CS-2006-46, Trinity College, Dublin."},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/1409625.1409627"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/1053427.1053429"},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.1632526"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218486"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(95)96802-S"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1145\/1174429.1174501"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1109\/2945.998665"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1145\/1201775.882323"},{"key":"e_1_2_2_35_1","volume-title":"Proceedings of the Pacific Conference on Computer Graphics and Applications. IEEE Computer Society, 154--164","author":"Ma L.","unstructured":"Ma , L. , Chan , T. K. Y. , and Jiang , Z . 2000. Interpolating and approximating moving frames using B-splines . In Proceedings of the Pacific Conference on Computer Graphics and Applications. IEEE Computer Society, 154--164 . Ma, L., Chan, T. K. Y., and Jiang, Z. 2000. Interpolating and approximating moving frames using B-splines. In Proceedings of the Pacific Conference on Computer Graphics and Applications. IEEE Computer Society, 154--164."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803426656"},{"key":"e_1_2_2_37_1","volume-title":"Geometric Transformations for 3D Modeling","author":"Mortenson M.","unstructured":"Mortenson , M. 2007. Geometric Transformations for 3D Modeling . Industrial Press , New York . Mortenson, M. 2007. Geometric Transformations for 3D Modeling. Industrial Press, New York."},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360642"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCG.2008.14"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/300523.300548"},{"key":"e_1_2_2_41_1","volume-title":"Proceedings of the Computer Vision and Pattern Recognition (CVPR'88)","author":"Roberts K. S.","unstructured":"Roberts , K. S. , Bishop , G. , and Ganapathy , S. K . 1988. Smooth interpolation of rotational matrices . In Proceedings of the Computer Vision and Pattern Recognition (CVPR'88) . IEEE Computer Science Press, 724--729. Roberts, K. S., Bishop, G., and Ganapathy, S. K. 1988. Smooth interpolation of rotational matrices. In Proceedings of the Computer Vision and Pattern Recognition (CVPR'88). IEEE Computer Science Press, 724--729."},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4485(00)00086-5"},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cad.2007.02.016"},{"key":"e_1_2_2_44_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cad.2008.09.002"},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.1145\/325334.325242"},{"key":"e_1_2_2_46_1","volume-title":"SIGGRAPH Course Notes on State of the Art Image Synthesis","author":"Shoemake K.","unstructured":"Shoemake , K. 1987. Quaternion calculus and fast animation . In SIGGRAPH Course Notes on State of the Art Image Synthesis . ACM Press , 101--121. Shoemake, K. 1987. Quaternion calculus and fast animation. In SIGGRAPH Course Notes on State of the Art Image Synthesis. ACM Press, 101--121."},{"key":"e_1_2_2_47_1","volume-title":"Proceedings of Graphics Interface Conference. Morgan Kaufmann Publishers, 258--264","author":"Shoemake K.","unstructured":"Shoemake , K. and Duff , T . 1992. Matrix animation and polar decomposition . In Proceedings of Graphics Interface Conference. Morgan Kaufmann Publishers, 258--264 . Shoemake, K. and Duff, T. 1992. Matrix animation and polar decomposition. In Proceedings of Graphics Interface Conference. Morgan Kaufmann Publishers, 258--264."},{"key":"e_1_2_2_48_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01908447"},{"key":"e_1_2_2_49_1","volume-title":"Proceedings of the Graphics Interface Conference. 24--32","author":"Wang W.","unstructured":"Wang , W. and Joe , B . 1993. Orientation interpolation in quaternion space using spherical biarcs . In Proceedings of the Graphics Interface Conference. 24--32 . Wang, W. and Joe, B. 1993. Orientation interpolation in quaternion space using spherical biarcs. In Proceedings of the Graphics Interface Conference. 24--32."},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01630.x"},{"key":"e_1_2_2_51_1","doi-asserted-by":"publisher","DOI":"10.1145\/1201775.882324"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2019627.2019635","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2019627.2019635","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T19:07:42Z","timestamp":1750273662000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2019627.2019635"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10]]},"references-count":51,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["10.1145\/2019627.2019635"],"URL":"https:\/\/doi.org\/10.1145\/2019627.2019635","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10]]},"assertion":[{"value":"2009-11-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-05-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-10-22","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}