Stability problem of equilibrium discrete planar curves

Discrete differential geometry is an active research field in connection with the smooth manifold theory and the visualization on the computer. In this field, from the theoretical point of view, one approach is based on the variational problems and the other is the integrability of equations. We take the former approach and focus on discrete planar curves. Surprisingly, such a discrete curve theory is not well-understood, therefore we try to develop them inspired by the work [10]. Moreover, we consider the stability problem for equilibrium discrete curves. Stability problem for discrete objects have not developed well after the work by Polthier and Rossman [11].

First we derive the first variation formula and extract the vector from the formula which should be called the “discrete curvature vector” in §2. In §3, by using this curvature vector, we show an unifed interpretation which derives various kinds of discrete curvature notions introduced in [7], [8]. The important viewpoint here is that there is no natural notion of the line element on the vertices. In §4, we characterize equilibrium curves for the length functional under the area-constraint condition as regular polygons. In §5, we derive the Steiner-type formula for parallel curves by using the “vertex normal” constructed from the “discrete curvature vector” derived in §2. In §6, we will consider the stability problem of the regular polygons. We can derive the second variation formula similar to the smooth case by decomposing the variation vector field to the “normal” and “tangential” directions as in [1]. Moreover, we show the instability for the non-convex regular polygons by using the second variation formula for “normal” variations in the section 7.

In this section we will consider the variation of discrete curves and extract the “curvature vector” from the first variation formula. We can expect that this vector gives a notion of the curvature and normal at vertices. Let us recall the basic definition of discrete planar curves.

For each oriented edge $e_{k}:=[p_{k},p_{k+1}]$ of $\Gamma_{h}$ we can assign an unit normal vector

$$\nu_{k}:=R\left(\frac{p_{k+1}-p_{k}}{l_{k}}\right):=R\left(\frac{p_{k+1}-p_{k}}{|p_{k+1}-p_{k}|}\right),\quad k=0,\ldots,n-1,$$

where $R$ is the $\pi/2$-rotation or $-\pi/2$-rotation in $\mathbb{R}^{2}$. It does not matter whichever we choose but we choose the same $R$ for all $k$.

For a discrete curve $\Gamma_{h}=\{p_{k}\}_{k}$ with an orientation, the length of $\Gamma_{h}$ and the $2$-dimensional volume (area) bounded by $\Gamma_{h}$ are defined by

$$L(\Gamma_{h}):=\sum_{k}l_{k}=\sum_{k}|p_{k+1}-p_{k}|,\quad\operatorname{Vol}(\Gamma_{h}):=\frac{\,1\,}{2}\sum_{k}\langle p_{k},\nu_{k}\rangle|p_{k+1}-p_{k}|$$

In these settings, we consider the following question:

What is the unit normal, curvature and line element at the vertices ?

A realization of the dual graph may give an answer for this question, but we do not consider such a realization. In order to approach the problem, we will extract the discrete curvature vector from the first variation of the length. Therefore we first derive the first variation formula of the length functional. In the discrete setting we consider the variation of vertices, that is, piecewise linear variations. We consider a variation

$$p_{k}(t)=p_{k}+tv_{k}+O(t^{2}),\quad k=0,\ldots,n-1$$

where ${}^{t}\vec{v}=({}^{t}v_{0},\ldots,{}^{t}v_{n-1})\in\mathbb{R}^{2n}$ is the “variation vector field”. If $p_{k}$ is a boundary point of $\Gamma_{h}$, then we assume $v_{k}=0$.

We want to find the vector ${}^{t}\nabla L\in\mathbb{R}^{2n}$ satisfies

$$\delta L:=\frac{d}{dt}_{|t=0}L=\langle\vec{v},\nabla L\rangle_{\mathbb{R}^{2n}}=\sum_{k}\langle v_{k},\nabla_{p_{k}}L\rangle_{\mathbb{R}^{2}},$$

where we write ${}^{t}\nabla L=({}^{t}\nabla_{p_{1}}L,\ldots,{}^{t}\nabla_{p_{n}}L)\in\mathbb{R}^{2n}$. Since this is just a direction derivative in $\mathbb{R}^{2n}$, the following proposition is immediately:

By using this formula we have

$$\delta L=\sum_{k}\langle\nabla_{p_{k}}L,v_{k}\rangle=-\sum_{k}\left\langle-\frac{\,1\,}{L_{k}}\nabla_{p_{k}}L,v_{k}\right\rangle L_{k},$$

where we inserted some auxilary function $L_{k}$ defined on the vertices. This kind of observation is essentially remarked in the paper [3].

In this section, non-uniqueness of the line element at the vertices gives various notions of the discrete curvature defined in [7], [8].

To describe the curvature notions, we have to define the angles at vertices. We define (the absolute value of) the angle between $\nu_{k-1}$ and $\nu_{k}$ as $\theta_{k}$, i.e.,

$$\cos\theta_{k}:=\langle\nu_{k},\nu_{k-1}\rangle=\frac{\langle p_{k+1}-p_{k},p_{k}-p_{k-1}\rangle}{|p_{k+1}-p_{k}|\cdot|p_{k}-p_{k-1}|}.$$

We have to care about the signature of $\theta_{k}$. Let $R_{\theta}$ be the $\theta$-rotation in $\mathbb{R}^{2}$. We assign the signature depends on the choice of the rotation $R$ appeared in the definition of the edge normal:

$$\sigma:=\begin{cases}+1\quad&{\rm if}\quad R=R_{\pi/2}\\ -1\quad&{\rm if}\quad R=R_{-\pi/2}\end{cases}$$

In this situation the signature of $\theta_{k}$ is determined by the equation $R_{\sigma\theta_{k}}(\nu_{k-1})=\nu_{k}$. We easily see that $\sum_{k}\theta_{k}=2m\pi$ for some integer $m\in\mathbb{Z}$ for any closed curve. We define the discrete curvature with respect to the choice of $L_{k}$ by the length of the discrete curvature vector with respect to $L_{k}$:

In the lecture note by Tim Hoffmann [8], three kinds of notions of the curvature for discrete curves are introduced:

1. $(1)$

   The curvature at vertices by using the vertex osculating circle method,

2. $(2)$

   The curvature at edges by using the edge osculating circle method,

3. $(3)$

   The curvature at vertices by using edge osculating circle for “arclength parametrized” curve.

Moreover, Hatakeyama [7] also defined the curvature for discrete curves another way. We will show that these curvature notions can be derived from our viewpoint.

In the paper [7], the discrete curvature at the vertex is defined as

$$\kappa(p_{k}):=\frac{1}{|p_{k}-p_{k-1}|}\left|\frac{p_{k+1}-p_{k}}{|p_{k+1}-p_{k}|}-\frac{p_{k}-p_{k-1}}{|p_{k}-p_{k-1}|}\right|=-\frac{|\nabla_{p_{k}}L|}{|p_{k}-p_{k-1}|}.$$

Then we immediately have the following result:

Before considering the edge osculating circle method, we shall modify the first variation formula from the vertex-based expression to the edge-based expression. If we put $v_{k}=(w_{k}+w_{k-1})/2$, then we have

	$\displaystyle\delta L$	$\displaystyle=\frac{1}{2}\sum_{k}\langle\nabla_{p_{k}}L,w_{k}+w_{k-1}\rangle$
		$\displaystyle=\frac{1}{2}\sum_{k}\langle\nabla_{p_{k}}L+\nabla_{p_{k+1}}L,w_{k}\rangle=-\sum_{k}\left\langle\frac{R(\nu_{k-1}-\nu_{k+1})}{2L_{k}^{\prime}},w_{k}\right\rangle L_{k}^{\prime}$

where $L_{k}^{\prime}$ is some auxiliary function. As in the vertex case, we call the value

$$\kappa(e_{k}):=\frac{1}{L_{k}^{\prime}}\cdot\sin\frac{\theta_{k}+\theta_{k+1}}{2}$$

the discrete curvature at the edge $e_{k}=[p_{k},p_{k+1}]$ with respect to $L_{k}^{\prime}$.

In the previous section, we showed that non-uniqueness of the line element at vertices gives various discrete curvature notions. However, the equilibrium curves for the length functional under the area-constraint condition should be characterized as some “constant curvature” objects by virture of the smooth case. In this section, we show that such equilibrium curves can be characterized as regular polygons and that they are certainly regarded as “constant curvature” objects.

By a direct calculation we have the following result.

We will show that these regular polygons are the only equilibrium curves for the functional $L+\kappa\operatorname{Vol}$.

We found that equilibrium closed curves of the functional $L+\kappa\operatorname{Vol}$ must satisfy $l_{k}\equiv l_{0}$, i.e., they must have “good coordinates (arclength parameter)”. If we note that we can define the curvature at vertices for an arclength parametrized curve, the previous result can be restated as follows:

In this section we will derive the discrete version of the Steiner-type formula in order to show an effectiveness of the vertex normal which we will define. The following type of Steiner formula is essentially appeared in some papers, e.g. [2], [4]. Although they try to find the curvature notion from the Steiner-type formula, we will derive the Steiner-type formula by using our vertex normal and connect with the well-known curvature notion.

Let $\Gamma_{h}=\{p_{k}\}_{k}$ be a discrete curve and take an interior vertex $p_{k}$. We can expect that if we normalize the discrete curvature vector, then we have the “vertex normal”. Recall that the length of the length gradient $\nabla_{p_{k}}L$ can be computed as $2\sin(\theta_{k}/2)$ up to the signature. In the discrete case, we should consider another factor $2\sin(\theta_{k}/2)\cos(\theta_{k}/2)=\sin\theta_{k}$ and put

$$N_{k}:=-\frac{\nabla_{p_{k}}\operatorname{Length}}{\sin\theta_{k}}=\frac{R(\nu_{k-1}-\nu_{k})}{\sin\theta_{k}}=\frac{1}{1+\cos\theta_{k}}(\nu_{k}+\nu_{k-1}),$$

(5.1)

then we shall call the vector $N_{k}$ as the vertex normal at the vertex $p_{k}$. The second expression in (5.1) allows us to define the vertex normal even if $\theta_{k}=0$. Then we consider the following deformation of the curve:

$$p_{k}(t)=p_{k}+tN_{k},\quad k=1,\ldots,n.$$

Before giving the proof, we give an intuitive explanation for a special case.

In the Figure 2, the similarity ratio of the triangles gives

$\displaystyle|p_{k+1}(t)-p_{k}(t)|:|p_{k+1}-p_{k}|=(-r_{k}+t):-r_{k}\iff|p_{k+1}(t)-p_{k}(t)|=|p_{k+1}-p_{k}|(1-t\cdot\kappa(e_{k})).$

Note that the signature of the curvature radius $r_{k}$ is negative in the figure.

In this section we consider the second variation formula of the length functional. We will follow the argument developed in [11].

Let $\Gamma_{h}=\{p_{k}\}_{k}$ be an equilibrium closed curve for the functional $L+\kappa\operatorname{Vol}$. We say a variation is admissible (or permissible) if the variation is volume-preserving and fixes the boundary. We recall the first variation formula of the length and the $2$-dimensional volume:

	$\displaystyle\frac{d}{dt}L$	$\displaystyle=\sum_{k}\langle\nabla_{p_{k}}L,p_{k}^{\prime}\rangle,\quad\nabla_{p_{k}}L=R(\nu_{k}-\nu_{k-1}),$
	$\displaystyle\frac{d}{dt}\operatorname{Vol}$	$\displaystyle=\sum_{k}\langle\nabla_{p_{k}}\operatorname{Vol},p_{k}^{\prime}\rangle,\quad\nabla_{p_{k}}\operatorname{Vol}=\frac{1}{2}R(p_{k+1}-p_{k-1}).$

Note that if the variation is admissible, then we have

$$0=\delta^{2}\operatorname{Vol}=\sum_{k}\langle\delta(\nabla_{p_{k}}\operatorname{Vol}),\delta p_{k}\rangle+\langle\nabla_{p_{k}}\operatorname{Vol},\delta^{2}p_{k}\rangle.$$