Journal for Geometry and Graphics 18 (2014), No. 1, 045--059
Copyright Heldermann Verlag 2014
Asymptotic Behaviour of the Maximum Curvature of Lamé Curves
Dept. of Mathematics, Graduate School of Science and Engineering, Ehime Univesity, 2-5 Bunkyo-cho, Matsuyama 790-8577, Japan
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.
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