{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:14:51Z","timestamp":1760242491727,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,9,20]],"date-time":"2017-09-20T00:00:00Z","timestamp":1505865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well.<\/jats:p>","DOI":"10.3390\/axioms6030026","type":"journal-article","created":{"date-parts":[[2017,9,20]],"date-time":"2017-09-20T10:57:41Z","timestamp":1505905061000},"page":"26","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["From Normal Surfaces to Normal Curves to Geodesics on Surfaces"],"prefix":"10.3390","volume":"6","author":[{"given":"Eli","family":"Appleboim","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering, Technion, Haifa 3200, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF02559591","article-title":"Theorie der normalflachen","volume":"105","author":"Haken","year":"1961","journal-title":"Acta Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"493","DOI":"10.4310\/jdg\/1214442006","article-title":"PL-Minimal surfaces in 3-manifolds","volume":"27","author":"Jaco","year":"1988","journal-title":"J. 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Elementary differential topology, Princeton University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s00454-004-1096-4","article-title":"Approximation of the normal vector field and the area of a smooth surface","volume":"32","author":"Morvan","year":"2004","journal-title":"Disc. Comp. Geom."},{"key":"ref_16","unstructured":"Cheeger, J., and Ebin, D. (1975). Comparison Theorems in Riemannian Geometry, American Mathematical Society."},{"key":"ref_17","first-page":"137","article-title":"Curves of finite total curvature","volume":"38","author":"Sullivan","year":"2008","journal-title":"Disc. Diff. Geom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01386390","article-title":"A note on two problems in connexion with graphs","volume":"1","author":"Dijekstra","year":"1959","journal-title":"Numer. Math."},{"key":"ref_19","first-page":"248","article-title":"Geschlossene flachen in drei-dimesnionalen Mannigfaltigkeiten","volume":"38","author":"Kneser","year":"1929","journal-title":"Jahresbericht der Deutschen Mathematiker-Vereinigung"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"613","DOI":"10.4310\/MRL.1994.v1.n5.a9","article-title":"Thin position and the recognition of \ud835\udd4a3","volume":"1","author":"Thompson","year":"1994","journal-title":"Math. Res. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0040-9383(84)90039-9","article-title":"An Algorithm to decide if a 3-Manifold is a Haken Manifold","volume":"23","author":"Jaco","year":"1984","journal-title":"Topology"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1090\/S0002-9947-99-02296-5","article-title":"Almost normal surfaces in 3-manifolds","volume":"352","author":"Stocking","year":"2003","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"441","DOI":"10.2307\/1971088","article-title":"Topology of three dimensional manifolds and the embedding problem in minimal surface theory","volume":"112","author":"Meeks","year":"1980","journal-title":"Ann. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0040-9383(82)90021-0","article-title":"The Classical Plateau problem and the topology of three dimensional manifolds","volume":"21","author":"Meeks","year":"1982","journal-title":"Topology"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/BF01214308","article-title":"The Existence of embedded minimal surfaces and the problem of uniqueness","volume":"179","author":"Meeks","year":"1982","journal-title":"Math. 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