Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal for Geometry and Graphics 18 (2014), No. 1, 045--059Copyright Heldermann Verlag 2014 Asymptotic Behaviour of the Maximum Curvature of Lamé Curves Masaya Matsuura Dept. of Mathematics, Graduate School of Science and Engineering, Ehime Univesity, 2-5 Bunkyo-cho, Matsuyama 790-8577, Japan masaya@ehime-u.ac.jp [Abstract-pdf] The curve $|x/a|^p + |y/b|^p = 1$ for $a,b,p>0$ in the $xy$-plane is called a Lam\'e curve. It is also known as a superellipse and is one of the symbols of Scandinavian design. For fixed $a$ and $b$, the above curve expands as $p$ increases and shrinks as $p$ decreases. The curve converges to a rectangle as $p\to\infty$, while it converges to a cross shape as $p\to 0^+$. In general, if $p>2$, Lam\'e curves have shapes which lie between ellipses and rectangles. From the viewpoint of application, one of the fundamental problems is to detect the optimal'' value of the exponent $p$ which creates the most refined'' shape. With this in mind, we closely examine how the curvature of Lam\'e curves depends on $p$. In particular, we derive an explicit expression of the asymptote of the maximum curvature, which is the main result of this paper. Keywords: Lame curve, superellipse, curvature, maximum curvature. MSC: 53A04 [ Fulltext-pdf  (1010  KB)] for subscribers only.