For any smooth, regular (planar) curve \(C:I\rightarrow\mathbb{R}^2\) parameterized by the arc length \(s\), its anisotropic curvature is defined by $$\lambda(s):=\frac{\kappa(s)}{\mu\left(\theta(s)\right)}\,,$$ where \(\kappa(s)\) is the (Frenet) curvature of \(C\) and \(\mu\left(\theta(s)\right)\) denotes the curvature of a convex curve, referred to as the Wulff shape.

The anisotropic bending energy of \(C\) is then given by the functional $$\mathcal{E}_\beta[C]:=\int_C\left(\lambda^2+\beta\right)ds\,,$$ where \(\beta\in\mathbb{R}\).

In (Palmer & ---, 2020-1) critical curves for this energy in \(\mathbb{R}^3\) were studied as well as equilibria for more general functionals including twisting. Here, we describe the classification in \(\mathbb{R}^2\) proved in (Palmer & ---, 2020-2), which for functionals possesing an essential symmetry coincides with Euler's classification, although the order in which the nine types of curves occur may be more complicated than in the isotropic case.

Theorem. (Palmer & ---, 2020-2) Let \(C\) be a critical curve for \(\mathcal{E}_\beta\) for a symmetric Wulff shape. Then, up to quasi-rotations and rescalings, \(C\) is one of the followings:

1. A suitable rotation of the Wulff shape \((\lambda^2=\beta)\).


2. An orbit-like anisotropic elasticae \((\beta>1)\).


3. A borderline anisotropic elasticae \((\beta=1)\).


4. A wave-like anisotropic elasticae \((-1<\beta<1)\).


• A multiloop \((0<\beta<1)\).


• A lemniscate \((0<\beta<1)\).


• A deep wave \((0<\beta<1)\).


• A rectangular \((\beta=0)\).


• A shallow wave \((-1<\beta<0)\).


5. A straight line \((\lambda=0)\).

We now show some illustrations of above classification for the Wulff shape described by $$\frac{1}{\mu}:=1-\frac{24}{25}\cos\left(5\theta\right).$$