Asymptotic Behaviour of the Maximum Curvature of Lam� Curves

M. Matsuura

Journal for geometry and graphics, Tome 18 (2014) no. 1, pp. 45-59.

Résumé

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.

Détail

Classification : 53A04
Mots-clés : Lame curve, superellipse, curvature, maximum curvature