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In this work, we derive closed-form expressions for biharmonic coordinates and their derivatives for 3D triangular cages. The core of our derivation lies in computing the closed-form expressions for the integral of the Euclidean distance over a triangle\n            <jats:italic>and<\/jats:italic>\n            its derivatives. The derived 3D biharmonic coordinates not only fill a missing component in methods of generalized barycentric coordinates but also pave the way for various interesting applications in practice, including producing a family of biharmonic deformations, solving variational shape deformations, and even unlocking the closed-form expressions for recently-introduced Somigliana coordinates for both fast and accurate evaluations.\n          <\/jats:p>","DOI":"10.1145\/3658208","type":"journal-article","created":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T14:47:57Z","timestamp":1721400477000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Biharmonic Coordinates and their Derivatives for Triangular 3D Cages"],"prefix":"10.1145","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6568-2642","authenticated-orcid":false,"given":"Jean-Marc","family":"Thiery","sequence":"first","affiliation":[{"name":"Adobe Research, Paris, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2147-3427","authenticated-orcid":false,"given":"\u00c9lie","family":"Michel","sequence":"additional","affiliation":[{"name":"Adobe Research, Paris, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9411-1689","authenticated-orcid":false,"given":"Jiong","family":"Chen","sequence":"additional","affiliation":[{"name":"Inria, Saclay, France"}]}],"member":"320","published-online":{"date-parts":[[2024,7,19]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531340"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015772"},{"key":"e_1_2_2_3_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3197517.3201387","article-title":"Hyper-reduced projective dynamics","volume":"37","author":"Brandt Christopher","year":"2018","unstructured":"Christopher Brandt, Elmar Eisemann, and Klaus Hildebrandt. 2018. 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