{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T16:54:32Z","timestamp":1770742472367,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020-DyCA","award":["UIDP\/00048\/2020"],"award-info":[{"award-number":["UIDP\/00048\/2020"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia (FCT)","award":["UIDP\/00048\/2020"],"award-info":[{"award-number":["UIDP\/00048\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Simple closed formulas for endpoint geodesics on Gra\u00dfmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated with the geometric de Casteljau construction offering an excellent alternative to the variational approach which gives rise to Riemannian polynomials and splines.<\/jats:p>","DOI":"10.3390\/math11163545","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T10:08:09Z","timestamp":1692180489000},"page":"3545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Endpoint Geodesic Formulas on Gra\u00dfmannians Applied to Interpolation Problems"],"prefix":"10.3390","volume":"11","author":[{"given":"Knut","family":"H\u00fcper","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Julius-Maximilians-Universit\u00e4t W\u00fcrzburg, 97074 W\u00fcrzburg, Germany"}]},{"given":"F\u00e1tima","family":"Silva Leite","sequence":"additional","affiliation":[{"name":"Institute of Systems and Robotics-Coimbra, 3030-290 Coimbra, Portugal"},{"name":"Department of Mathematics, University of Coimbra, 3001-143 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1093\/imamci\/6.4.465","article-title":"Cubic splines on curved spaces","volume":"6","author":"Noakes","year":"1989","journal-title":"IMA J. 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