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We achieve this by building a continuous normal field over the mesh via Phong interpolation and using minimal Rodrigues rotations to transport vertex\u2010based tangent vectors into triangle interiors. Unlike most existing discretizations, which typically sacrifice either continuity or the ability to evaluate derivatives pointwise, our approach supports both. Because it is pointwise evaluatable, and using the fact that the covariant derivative can be decomposed into its symmetric, antisymmetric, and scalar components, our discretization supports the construction of standard vector\u2010field processing operators including the connection and Hodge Laplacians, Killing energy, divergence, curl, and the Lie bracket. This framework provides a simple and practical finite\u2010element formulation for vector\u2010field processing on meshes, supporting both integration\u2010based operators and pointwise queries. To our knowledge, ours is the first discretization that jointly enables extrinsic continuous vector fields, bounded derivatives, and pointwise evaluation of this collection of operators.<\/jats:p>","DOI":"10.1111\/cgf.70515","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T11:30:51Z","timestamp":1786620651000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Phong\u2010Rodrigues Extrinsic Vector\u2010Field Processing"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5908-4822","authenticated-orcid":false,"given":"Hongyi","family":"Liu","sequence":"first","affiliation":[{"name":"Johns Hopkins University  USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9741-3175","authenticated-orcid":false,"given":"Oded","family":"Stein","sequence":"additional","affiliation":[{"name":"Technion \u2010 Israel Institute of Technology  Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6998-6689","authenticated-orcid":false,"given":"Amir","family":"Vaxman","sequence":"additional","affiliation":[{"name":"University of Edinburgh  UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1732-2327","authenticated-orcid":false,"given":"Mirela","family":"Ben\u2010Chen","sequence":"additional","affiliation":[{"name":"Technion \u2010 Israel Institute of Technology  Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6904-2167","authenticated-orcid":false,"given":"Misha","family":"Kazhdan","sequence":"additional","affiliation":[{"name":"Johns Hopkins University  USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"issue":"5","key":"e_1_2_10_2_2","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1111\/cgf.12174","article-title":"An operator approach to tangent vector field processing","volume":"32","author":"Azencot O.","year":"2013","journal-title":"Computer Graphics Forum"},{"issue":"3","key":"e_1_2_10_3_2","doi-asserted-by":"crossref","DOI":"10.1145\/2723158","article-title":"Discrete derivatives of vector fields on surfaces \u2013 an operator approach","volume":"34","author":"Azencot O.","year":"2015","journal-title":"ACM Trans. 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