2012/04/30 by Jeanne N. Clelland, Clelland, Jeanne, Edward Y. Estrada +9
Engineering · Mathematics · #3D Shape Modeling and Analysis #53 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1205.0065
openalex publication_date 2012/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Euclidean geometry, all metric notions (arc length for curves, the first\nfundamental form for surfaces, etc.) are derived from the Euclidean inner\nproduct on tangent vectors, and this inner product is preserved by the full\nsymmetry group of Euclidean space (translations, rotations, and reflections).\nIn equiaffine geometry, there is no invariant notion of inner product on\ntangent vectors that is preserved by the full equiaffine symmetry group.\nNevertheless, it is possible to define an invariant notion of arc length for\nnondegenerate curves, and an invariant first fundamental form for nondegenerate\nsurfaces in equiaffine space. This leads to two possible notions of arc length\nfor a curve contained in a surface, and these two arc length functions do not\nnecessarily agree. In this paper we will derive necessary and sufficient\nconditions under which the two arc length functions do agree, and illustrate\nwith examples.\n