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Harmonic function\u2013based approaches, grounded in a strong mathematical framework, are widely employed. In particular, harmonic maps are valuable for guaranteeing injectivity under suitable boundary conditions. They are also computationally efficient, as they operate within a linear subspace [FW22]. However, this restriction to a limited subspace often introduces substantial isometric distortion, especially on highly curved surfaces. In contrast, methods that explore the full space of piecewise linear maps [SPSH*17] achieve much lower isometric distortion, but at the expense of increased computational cost. We propose a hybrid method that combines the speed and robustness of harmonic maps with the generality of full\u2010space methods to produce injective maps with low isometric distortion up to 50 times faster than state\u2010of\u2010the\u2010art methods. The core concept is simple but powerful. Instead of searching for the optimal parameterization over the original mesh via a full\u2010space method, we simplify the input fine mesh and parameterize the coarse mesh. We then prolong the Beltrami coefficients from the coarse mesh parameterization to the fine mesh, resulting in a customized Laplacian matrix that gives rise to a harmonic map in a modified metric [WGS23, FW25]. Our method ensures injectivity, offers great computational efficiency, and produces significantly lower isometric distortion compared to harmonic maps.<\/jats:p>","DOI":"10.1111\/cgf.70341","type":"journal-article","created":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T13:14:26Z","timestamp":1775826866000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fast Injective Mesh Parameterization via Beltrami Coefficient Prolongation"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7124-6783","authenticated-orcid":false,"given":"G.","family":"Fargion","sequence":"first","affiliation":[{"name":"Bar\u2010Ilan University  Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3724-5031","authenticated-orcid":false,"given":"O.","family":"Weber","sequence":"additional","affiliation":[{"name":"Bar\u2010Ilan University  Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,4,10]]},"reference":[{"key":"e_1_2_7_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530141"},{"key":"e_1_2_7_3_2","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461931"},{"key":"e_1_2_7_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/2816795.2818099"},{"key":"e_1_2_7_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601158"},{"key":"e_1_2_7_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462014"},{"key":"e_1_2_7_7_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073646"},{"key":"e_1_2_7_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-9006-1"},{"key":"e_1_2_7_9_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13243"},{"key":"e_1_2_7_10_2","doi-asserted-by":"publisher","DOI":"10.1145\/3478513.3480557"},{"key":"e_1_2_7_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925926"},{"key":"e_1_2_7_12_2","unstructured":"CGAL Project:CGAL user and reference manual 2024. 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