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Using a two-stage process, we compute several points radially emanating from the map origin, and interpolate between them with a spline surface. The narrow interface of our method allows it to support several kinds of geometry such as signed distance functions, general analytic implicit functions, triangle meshes, neural implicits, and point clouds. We demonstrate the high quality of our generated parameterizations on a variety of examples, and show applications in local texturing and surface curve drawing.<\/jats:p>","DOI":"10.1145\/3767323","type":"journal-article","created":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T10:35:01Z","timestamp":1758105301000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Local Surface Parameterizations via Smoothed Geodesic Splines"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6119-7500","authenticated-orcid":false,"given":"Abhishek","family":"Madan","sequence":"first","affiliation":[{"name":"Computer Science, University of Toronto","place":["Toronto, Canada"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7079-1934","authenticated-orcid":false,"given":"David","family":"Levin","sequence":"additional","affiliation":[{"name":"Computer Science, University of Toronto","place":["Toronto, Canada"]},{"name":"NVIDIA Corp","place":["Toronto, Canada"]}]}],"member":"320","published-online":{"date-parts":[[2025,10,13]]},"reference":[{"key":"e_1_3_3_2_1","article-title":"Controlling neural level sets","author":"Atzmon Matan","year":"2019","unstructured":"Matan Atzmon, Niv Haim, Lior Yariv, Ofer Israelov, Haggai Maron, and Yaron Lipman. 2019. 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