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The finite-dimensional parameters of this connection are optimally computed by minimizing a quadratic measure of the deviation to the (discontinuous) Levi-Civita connection induced by the embedding of the input triangle mesh, or to any metric connection with arbitrary cone singularities at vertices. From this discrete connection, a covariant derivative is constructed through exact differentiation, leading to explicit expressions for local integrals of first-order derivatives (such as divergence, curl, and the Cauchy-Riemann operator) and for\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>2<\/jats:sub>\n            -based energies (such as the Dirichlet energy). We finally demonstrate the utility, flexibility, and accuracy of our discrete formulations for the design and analysis of vector,\n            <jats:italic>n<\/jats:italic>\n            -vector, and\n            <jats:italic>n<\/jats:italic>\n            -direction fields.\n          <\/jats:p>","DOI":"10.1145\/2870629","type":"journal-article","created":{"date-parts":[[2016,3,18]],"date-time":"2016-03-18T13:50:44Z","timestamp":1458309044000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":28,"title":["Discrete Connection and Covariant Derivative for Vector Field Analysis and Design"],"prefix":"10.1145","volume":"35","author":[{"given":"Beibei","family":"Liu","sequence":"first","affiliation":[{"name":"Michigan State University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiying","family":"Tong","sequence":"additional","affiliation":[{"name":"Michigan State University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fernando De","family":"Goes","sequence":"additional","affiliation":[{"name":"California Institute of Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mathieu","family":"Desbrun","sequence":"additional","affiliation":[{"name":"California Institute of Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2016,3,15]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"R. 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Discrete differential forms for computational modeling. In Discrete Differential Geometry, A. I. Bobenko, P. Schr\u00f6der, J. M. Sullivan, and G. M. Ziegler (Eds.). Oberwolfach Seminars, Vol. 38. Birkh\u00e4user Basel, 287--324."},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276447"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218475"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.345074"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/1356682.1356685"},{"key":"e_1_2_1_10_1","volume-title":"Spaces of relative parallelism. Annals of Mathematics","author":"Knebelman M. S.","year":"1951","unstructured":"M. S. Knebelman . 1951. Spaces of relative parallelism. Annals of Mathematics ( 1951 ), 387--399. M. S. Knebelman. 1951. Spaces of relative parallelism. 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