
Journal for Geometry and Graphics 18 (2014), No. 1, 045059 Copyright 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, 25 Bunkyocho, Matsuyama 7908577, Japan masaya@ehimeu.ac.jp [Abstractpdf] 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 [ Fulltextpdf (1010 KB)] for subscribers only. 