2011/02/22 by Shohei Shiba, Shiba, Shohei, Masaaki Umehara +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #math.DG
paper · pdf · doi:10.48550/arxiv.1102.4478
18pages, 3 figures
arxiv created 2011/02/22 · openalex publication_date 2011/02/22 · arxiv updated 2011/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
At a 3/2-cusp of a given plane curve γ(t), both of the Euclidean curvature κg and the affine curvature κA diverge. In this paper, we show that each of √(|sg|)κg and (sA)2 κA (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable t, where sg (resp. sA) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp sg=0 (resp. sA=0). Moreover, we give a characterization of the behaviour of the curvature functions κg and κA at 3/2-cusps. On the other hand, inflection points are also singular points of curves in affine geometry. We give a similar characterization of affine curvature functions near generic inflection points. As an application, new affine invariants of 3/2-cusps and generic inflection points are given.