1 Introduction

Discrete differential geometry is an active research field with connections to the smooth manifold theory and visualization on the computer. In this field, from the theoretical point of view, one approach is based on the variational problems and the other on 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 further develop it inspired by the work [10]. Moreover, we consider the stability problem for equilibrium discrete curves. The stability problem for discrete objects has not developed well after the work by Polthier and Rossman [11].

First we derive the first variation formula and extract a vector from the formula, called the discrete curvature vector in Sect. 2. In Sect. 3, by using this curvature vector, we show a unified 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 Sect. 4, we characterize equilibrium curves for the length functional under the area-constraint condition as regular polygons. In Sect. 5, we derive a Steiner-type formula for parallel curves by using the “vertex normal” constructed from the discrete curvature vector derived in Sect. 2. In Sect. 6, we 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 into 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 Sect. 7.

2 The First Variation Formula

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.

Definition 2.1

Let n be a non-negative integer. A standard (abstract) n-path is a simplicial complex \(G = (V, E)\) formed by

  • \(n+1\) abstract points \(V = \{ v_0, \ldots , v_n \}\),

  • the set of n edges \(e_k = [v_k, v_{k+1}]\), \(k = 0, \ldots , n-1\).

A standard abstract n-circle is the union of a standard n-path and the “final” edge \(e_n = [v_n, v_0]\). A discrete (planar) curve is a geometric realization of a standard \((n-1)\)-path or \((n-1)\)-circle, which is a map \(X:V \rightarrow {\mathbb {R}}^2\) satisfying \(l_k := |p_{k+1} - p_k| \ne 0\) for all \(k = 0,1,\ldots ,n-1\), where we denote \(p_k := X(v_k)\). We denote such a discrete curve as \(\Gamma _h = \{ p_k \}_k\).

To each oriented edge \(e_k := [p_k, p_{k+1}]\) of \(\Gamma _h\) we can assign a unit normal vector

$$\begin{aligned} \nu _k:= R \biggl ( \dfrac{p_{k+1} - p_k}{l_k} \biggr )= R \biggl ( \dfrac{p_{k+1} - p_k}{|p_{k+1} - p_k|} \biggr ), \qquad k = 0, \ldots , n-1, \end{aligned}$$

where R is the \((\pi /2)\)-rotation or \((-\pi /2)\)-rotation in \({\mathbb {R}}^2\). It does not matter which one 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

$$\begin{aligned} L (\Gamma _h) := \sum _k l_k = \sum _k |p_{k+1} - p_k|, \quad {\text {Area}}\Gamma _h:= \dfrac{1}{2} \sum _k \langle p_k, \nu _k \rangle |p_{k+1} - p_k|. \end{aligned}$$

In these settings, we consider the following question:

$$\begin{aligned} \textit{What are the unit normal, curvature, and line element at the vertices?} \end{aligned}$$

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

$$\begin{aligned} p_k(t) = p_k + t v_k +O(t^2), \qquad k = 0, \ldots , n-1, \end{aligned}$$

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 \({}^t \nabla L \in {\mathbb {R}}^{2n}\) satisfying

$$\begin{aligned} \delta L:= \dfrac{dL}{dt}\Big |_{t=0}\!= \langle \vec {v}, \nabla L \rangle _{{\mathbb {R}}^{2n}}= \sum _k \langle v_k, \nabla _{\!p_k} L \rangle _{{\mathbb {R}}^2}, \end{aligned}$$

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 directional derivative in \({\mathbb {R}}^{2n}\), the following proposition is immediate:

Proposition 2.2

Let \(\Gamma _h = \{ p_k \}_k\) be a discrete closed curve and \(p_k\) an interior vertex. Then the gradient of the length can be expressed as in the following formula:

$$\begin{aligned} \nabla _{\!p_k} L= R (\nu _k - \nu _{k-1})= - \dfrac{p_{k+1}-p_k}{l_k} + \dfrac{p_k - p_{k-1}}{l_{k-1}}. \end{aligned}$$
(2.1)

By using this formula we have

$$\begin{aligned} \delta L= \sum _k \langle \nabla _{\!p_k} L, v_k \rangle = - \sum _k \biggl \langle \!{-}\dfrac{1}{L_k} \nabla _{\!p_k} L, v_k \!\biggr \rangle L_k, \end{aligned}$$

where we inserted some auxiliary function \(L_k\) defined on the vertices. This kind of observation is essentially remarked on in the paper [3].

Definition 2.3

(discrete curvature vector)   For a positive function \(L_k\) defined on the vertices, we call the vector

$$\begin{aligned} \widetilde{N}_k= - \dfrac{1}{L_k} \nabla _{\!p_k} L = \dfrac{1}{L_k} \biggl ( \dfrac{p_{k+1} - p_k}{l_k} - \dfrac{p_k - p_{k-1}}{l_{k-1}} \biggr ) \end{aligned}$$
(2.2)

the discrete curvature vector with respect to \(L_k\).

Remark 2.4

There are some reasons why we can regard the vector \(\widetilde{N}_k\) as the curvature vector. First, the vector \(\widetilde{N}_k\) is independent of the choice of the unit normal, i.e., an intrinsic quantity. Moreover, the second expression (2.2) can be regarded as a discretization of a part of the Frenet–Serret formula. As we will remark below, suitable choices of the function \(L_k\) derive various notions of discrete curvature defined in [7, 8]. Non-uniqueness of the function \(L_k\) comes from the fact that there is no natural line element, i.e., the metric, at vertices.

3 Relation with Other Notions of the Discrete Curvature

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.,

$$\begin{aligned} \cos \theta _k:= \langle \nu _k, \nu _{k-1} \rangle = \dfrac{ \langle p_{k+1} - p_k, p_k - p_{k-1} \rangle }{|p_{k+1} - p_k| \cdot |p_k - p_{k-1}|}. \end{aligned}$$

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 depending on the choice of the rotation R appearing in the definition of the edge normal:

$$\begin{aligned} \sigma :={\left\{ \begin{array}{ll} +1 \quad &{}\text {if }R = R_{\pi /2}, \\ -1 \quad &{}\text {if }R = R_{-\pi /2}. \end{array}\right. } \end{aligned}$$

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 \(L_k\):

Definition 3.1

For a positive function \(L_k\) defined on the vertices, we call the value

$$\begin{aligned} \kappa (p_k) := \dfrac{2 \sin (\theta _k/2)}{L_k} \end{aligned}$$

the discrete curvature with respect to \(L_k\).

Remark 3.2

If we use the expression \(\nabla _{\!p_k} L = R(\nu _k - \nu _{k-1})\), the discrete curvature with respect to \(L_k\) can be written as \(\kappa (p_k) = |\nu _k - \nu _{k-1}|/L_k\) up to the signature. This can be regarded as a discretization of the curvature for a regular planar curve.

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

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

  • the curvature at edges by using the edge osculating circle method, and

  • 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.

Proposition 3.3

(vertex osculating circle method [8])   If we choose

$$\begin{aligned} L_k= \dfrac{|p_{k+1} - p_{k-1}|}{2 \cos (\theta _k / 2)}= \dfrac{|p_{k+1} - p_k + p_k - p_{k-1}|}{2 \cos (\theta _k / 2)}, \end{aligned}$$

then the discrete curvature with respect to \(L_k\) becomes

$$\begin{aligned} \kappa (p_k)= \dfrac{2 \sin \theta _k}{|p_{k+1} - p_{k-1}|}= \dfrac{2 \sin \theta _k}{|p_{k+1} - p_k + p_k - p_{k-1}|} \end{aligned}$$

and this value coincides with the curvature based on the vertex osculating circle method.

Proposition 3.4

(for the arclength parametrized curves [8])   Assume \(l_k = l_{k-1} = l_0\). If we choose

$$\begin{aligned} L_k = l_0 \cos \dfrac{\theta _k}{2} = \dfrac{l_k + l_{k-1}}{2} \cdot \cos \dfrac{\theta _k}{2}, \end{aligned}$$

then the discrete curvature with respect to \(L_k\) becomes

$$\begin{aligned} \kappa (p_k) = \dfrac{2}{l_0} \tan \dfrac{\theta _k}{2} \end{aligned}$$

and this value coincides with the curvature of arclength parametrized curve.

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

$$\begin{aligned} \kappa (p_k):= \dfrac{1}{|p_k - p_{k-1}|} \cdot \biggl | \dfrac{p_{k+1} - p_k}{|p_{k+1} - p_k|} - \dfrac{p_k - p_{k-1}}{|p_k - p_{k-1}|}\biggr |= - \dfrac{|\nabla _{\!p_k} L|}{|p_k - p_{k-1}|}. \end{aligned}$$

Then we immediately have the following result:

Proposition 3.5

If we choose \(L_k = l_{k-1} = |p_k - p_{k-1}|\), then the discrete curvature with respect to \(L_k\) coincides with the discrete curvature defined in [7].

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

$$\begin{aligned} \delta L&= \dfrac{1}{2} \sum _k \langle \nabla _{\!p_k} L, w_k + w_{k-1} \rangle \\&=\dfrac{1}{2}\sum _k\langle \nabla _{\!p_k}L+\nabla _{\!p_{k+1}} L,w_k\rangle =-\sum _k\biggl \langle \dfrac{R(\nu _{k-1}-\nu _{k+1})}{2L_k'},w_k \biggr \rangle L_k' , \end{aligned}$$

where \(L_k'\) is some auxiliary function. As in the vertex case, we call the value

$$\begin{aligned} \kappa (e_k) := \dfrac{1}{L_k'} \cdot \sin \dfrac{\theta _k + \theta _{k+1}}{2} \end{aligned}$$

the discrete curvature at the edge \(e_k = [p_k, p_{k+1}]\) with respect to \(L_k'\).

Proposition 3.6

(edge osculating circle method [8])   If we choose

$$\begin{aligned} L_k'= l_k \cos \dfrac{\theta _k}{2} \cos \dfrac{\theta _{k+1}}{2}= |p_{k+1} - p_k| \cdot \cos \dfrac{\theta _k}{2} \cos \dfrac{\theta _{k+1}}{2}, \end{aligned}$$

then the discrete curvature with respect to \(L_k'\) becomes

$$\begin{aligned} \kappa (e_k) = \dfrac{\tan (\theta _k/2) + \tan (\theta _{k+1}/2)}{|p_{k+1} - p_k|} \end{aligned}$$

and this value coincides with the curvature based on the edge osculating circle method.

Remark 3.7

To define the discrete curvature, we have to choose \(L_k\) (respectively \(L_k'\)) properly. That means if \(l_k,l_{k-1} \rightarrow ds\) and \(\theta _k \rightarrow 0\), then \(L_k\) (respectively \(L_k'\)) must converge to ds, i.e., \(L_k\) must be a “good” candidate for a discrete line element. We can check that \(L_k\) and \(L_k'\) satisfy this condition in the above examples.

Remark 3.8

(relations with the discrete Laplacian)   On a discrete curve \(\Gamma _h\) with the vertex set V, we consider a function \(\psi :V \rightarrow {\mathbb {R}}\). Then the gradient and the Laplacian of \(\psi \) can be defined as

$$\begin{aligned} \nabla \psi _k:= & {} \dfrac{\psi _{k+1} - \psi _k}{l_k},\\ \Delta \psi _k:= & {} \dfrac{1}{L_k} (\nabla \psi _k - \nabla \psi _{k-1}) =\dfrac{1}{L_k} \biggl ( \dfrac{\psi _{k+1} - \psi _k}{l_k} - \dfrac{\psi _k - \psi _{k-1}}{l_{k-1}} \biggr ), \end{aligned}$$

where we denote \(\psi _k := \psi (k)\). Since the Laplacian should be a linear operator and determined by a local property, this definition can be seen as a special case of the discrete Laplacian from [12]. Note that the gradient is an “edge-based operator” but the Laplacian is a “vertex-based” operator. In addition, the discrete curvature vector \(\widetilde{N}_k\) with respect to \(L_k\) can be written as \(\Delta p_k\).

From another point of view, if we define the Dirichlet energy of \(\psi \) as

$$\begin{aligned} E_h(\psi ):= \dfrac{1}{2} \sum _k |\nabla \psi _k|^2l_k= \dfrac{1}{2} \sum _k \dfrac{|\psi _{k+1} - \psi _k|^2}{l_k}, \end{aligned}$$

then the first variation of the energy becomes

$$\begin{aligned} \delta E_h (\psi )= & {} \sum _k \dfrac{\langle \psi _{k+1} - \psi _k, \varphi _{k+1} - \varphi _k \rangle }{l_k}= \sum _k \langle \nabla \psi _{k-1} - \nabla \psi _k, \varphi _k \rangle \\= & {} - \sum _k \langle \Delta \psi _k, \varphi _k \rangle L_k, \end{aligned}$$

where we take the variation of \(\psi \) as \(\psi _k (t) = \psi _k + t \varphi _k + O(t^2)\). Therefore \(\delta E_h (\psi ) = 0\) if and only if \(\Delta \psi _k=0\). Note that the condition \(\Delta \psi _k = 0\) is independent of the choice of \(L_k\).

As in the curvature case, the Laplacian can be changed since there is no natural “line element divisor \(L_k\)”. However, with another function \(\varphi :V \rightarrow {\mathbb {R}}\), we still have the following properties, since the quantities \(\Delta \psi _k L_k\) are independent of \(L_k\):

  • If \(\psi \) is constant, then \(\Delta \psi = 0\).

  • The condition \(\Delta \psi = 0\) is independent of the choice of \(L_k\) and in this case we have the mean value property:

    $$\begin{aligned} \psi _k = \dfrac{l_{k-1}}{l_k + l_{k-1}} \psi _{k+1} + \dfrac{l_k}{l_k + l_{k-1}}\psi _{k-1}. \end{aligned}$$

  • \(L^2\) symmetric property:

    $$\begin{aligned} \sum _k \psi _k \cdot \Delta \varphi _k \cdot L_k = \sum _k \Delta \psi _k \cdot \varphi _k \cdot L_k. \end{aligned}$$

    Note that the summation is vertex-based.

  • Integration by parts:

    $$\begin{aligned} - \sum _k \psi _k \cdot \Delta \varphi _k \cdot L_k = \sum _k \nabla \psi _k \cdot \nabla \varphi _k \cdot l_k. \end{aligned}$$

    Note that the right hand side is an edge-based summation but the left hand side is a vertex-based summation. As a corollary, the operator \(-\Delta \) is positive semi-definite.

4 Equilibrium Curves of the Length Functional

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 virtue of the smooth case. In this section, we show that such equilibrium curves can be characterized as regular polygons and that they certainly can be regarded as “constant curvature” objects. By a direct calculation we have the following result.

Lemma 4.1

For any vertex \(p_k\) of \(\Gamma _h\) the gradient of the area, \(\nabla _{\!p_k}{{{\,\mathrm{Area}\,}}}\), is given by

$$\begin{aligned} \nabla _{\!p_k}{{{\,\mathrm{Area}\,}}} = \dfrac{1}{2} R (p_{k+1} - p_{k-1}). \end{aligned}$$

Remark 4.2

We can modify the first variation formula of the volume as follows:

$$\begin{aligned} \delta {{\,\mathrm{Area}\,}}= \dfrac{1}{2} \sum _k \langle R(p_{k+1} - p_k + p_k - p_{k-1}), v_k \rangle = \sum _k \biggl \langle \dfrac{l_k \nu _k + l_{k-1} \nu _{k-1}}{2 L_k}, v_k\!\biggr \rangle L_k. \end{aligned}$$

It is natural to choose \(2L_k = l_k + l_{k-1} = {{\,\mathrm{Length}\,}}(\star (p))\) and this is also frequently used as a “vertex normal” (a weighted sum of the edge normals):

$$\begin{aligned} N_k^A := \dfrac{l_k \nu _k + l_{k-1} \nu _{k-1}}{l_k + l_{k-1}}. \end{aligned}$$

In addition, we have

$$\begin{aligned} -\nabla _{\!p_k}L=R(\nu _{k-1}-\nu _k)=\dfrac{\sin \theta _k}{1+\cos \theta _k}(\nu _k +\nu _{k-1})= 2 \tan \dfrac{\theta _k}{2}\cdot \dfrac{\nu _k+ \nu _{k-1}}{2}, \end{aligned}$$

by a simple calculation. Therefore, unless the curve is arclength parameterized, there are (at least) two choices of the “vertex normal” from the variational viewpoint: using the length gradient (length descent direction) or using the volume gradient (volume descent direction). This suggests that, in contrast to the smooth case, we have to choose the “preferred” vertex normal according to the energy in question.

Example 4.3

(regular polygons)   Let us take a discrete curve \(\Gamma _h^{m,n} = \{ p_k \}_k\) as follows (inducing non-convex regular n-gon with radius a):

$$\begin{aligned} p_k = a \exp \dfrac{2mk \pi \sqrt{-1}}{n},\qquad k=0, \ldots , n-1, \quad m=1,\ldots ,n-1, \end{aligned}$$

where we assume that \(m/n \ne 1/2\). In particular, we sometimes call the curve \(\Gamma _h^{1,n}\) a convex regular n-gon. Note that \(\Gamma _h^{n-1,n}\) is also convex but it has an opposite unit normal with \(\Gamma _h^{1,n}\) (usually we assume that \(\Gamma _h^{1,n}\) has the outward-pointing unit normal). Then the curve \(\Gamma _h^{m,n}\) is a critical point of the functional \({{{\,\mathrm{Length}\,}}} + \kappa \cdot {{{\,\mathrm{Area}\,}}}\) with \(\kappa = - 1/ (a \cos (m \pi /n))\). This value is the reciprocal of the radius of the inscribed circle of the polygon (up to the signature). We sometimes say that a convex regular n-gon with radius a (and outward-pointing unit normal) has constant curvature \(\kappa _n = -1/(a \cos (\pi /n))\). Note that \(\cos (\pi m/n)=-\cos (\pi (n-m)/n)\) and \(\kappa _n \rightarrow -1/a\) when \(n \rightarrow \infty \).

Fig. 1
figure 1

Convex and non-convex regular 5-gons with radii \(a=1\)

Full size image

We will show that these regular polygons are the only equilibrium curves for the functional \(L + \kappa \cdot {{{\,\mathrm{Area}\,}}}\).

Theorem 4.4

Let \(\Gamma _h = \{ p_k \}_{k=1}^n\) be a closed discrete curve and take \(\kappa \in {\mathbb {R}}\setminus \{ 0 \}\). Then the following two conditions are equivalent:

  • \(\Gamma _h\) is an equilibrium curve of the functional \(L + \kappa \cdot { {{\,\mathrm{Area}\,}}}\).

  • There exist numbers \(l_0\) and \(\theta _0\) such that \(l_k \equiv l_0\), \(\theta _k \equiv \theta _0\), and \(\kappa l_0 = 2 \tan (\theta _0/2)\), i.e., \(\Gamma _h\) must be a regular polygon.

Proof

We put

$$\begin{aligned} A_k := (\nu _k - \nu _{k-1}) + \dfrac{\kappa }{2} (p_{k+1} - p_{k-1}), \qquad k=1, \ldots , n. \end{aligned}$$

Then the discrete curve \(\Gamma _h\) is a critical point of the functional \({{{\,\mathrm{Length}\,}}} + \kappa \cdot {{{\,\mathrm{Area}\,}}}\) if and only if \(A_k = 0\) for all k. By a simple calculation we have

$$\begin{aligned} \langle A_k, \nu _{k-1} \rangle&= \biggl ( \dfrac{\kappa l_k}{2} - \tan \dfrac{\theta _k}{2} \biggr )\sin \theta _k , \end{aligned}$$
(4.1)
$$\begin{aligned} \langle A_k, \nu _k \rangle&= \biggl ( \tan \dfrac{\theta _k}{2} - \dfrac{\kappa l_{k-1}}{2} \biggr )\sin \theta _k , \end{aligned}$$
(4.2)
$$\begin{aligned} \langle A_{k+1}, \nu _{k+1} \rangle&= \biggl ( \tan \dfrac{\theta _{k+1}}{2} - \dfrac{\kappa l_k}{2} \biggr ) \sin \theta _{k+1}. \end{aligned}$$
(4.3)

For the necessity, that is, if we assume \(A_k = 0\) for all k, then it follows from (4.1) and (4.2) that \(\kappa l_k=2 \tan (\theta _k /2) = \kappa l_{k-1}\). And it also follows from (4.2), (4.3), and using \(l_k = l_{k-1}\) that

$$\begin{aligned} \tan \dfrac{\theta _k}{2} = \dfrac{\kappa l_k}{2} = \dfrac{\kappa l_{k-1}}{2} = \tan \dfrac{\theta _{k-1}}{2}. \end{aligned}$$

To prove the sufficiency, since \(\nu _k\) and \(p_k - p_{k-1}\) form a basis of \({\mathbb {R}}^2\) and \(\langle A_k, \nu _k \rangle = 0\) by (4.1) and the assumption, all we have to prove is \(\langle A_k, p_k - p_{k-1} \rangle = 0\) for all k. By using the assumption \(l_k = l_{k-1} = l_0\) and \(\theta _k = \theta _0\), we have

$$\begin{aligned}&\biggl \langle A_k, \dfrac{p_k - p_{k-1}}{l_k} \biggr \rangle = - \sin \theta _k +\dfrac{\kappa }{2} (l_k \cos \theta _k + l_{k-1}) \\&\quad = (1 + \cos \theta _0) \biggl ( \dfrac{\kappa l_0}{2} - \dfrac{\sin \theta _0}{1+\cos \theta _0} \biggr )= (1 + \cos \theta _0) \biggl ( \dfrac{\kappa l_0}{2} - \tan \dfrac{\theta _0}{2} \biggr )= 0. \end{aligned}$$

This shows \(A_k=0\) and proves the statement. \(\square \)

Remark 4.5

The equilibrium condition \(A_k = \nu _k - \nu _{k-1} + (\kappa /2)(p_{k+1}-p_{k-1})=0\) is equivalent to the condition \(\nu _k + (\kappa /2) (p_{k+1} + p_k) \equiv c\) for some constant vector \(c\in {\mathbb {R}}^2\). The latter condition can be considered as a conservation law for the Euler–Lagrange equation \(A_k=0\). Since the vector \(c\in {\mathbb {R}}^2\) is just a translation of the curve, we can put \(c=0\) and in this case we have \((p_{k+1}+p_k)/2=-\nu _k/\kappa \). Therefore, the edge midpoints must be tangent to the unit circle.

We found that equilibrium closed curves of the functional \(L + \kappa \cdot {{{\,\mathrm{Area}\,}}}\) 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:

Corollary 4.6

Let \(\Gamma _h\) be an arclength parametrized discrete closed curve, i.e., \(l_k \equiv l_0\), and \(\kappa \in {\mathbb {R}}\setminus \{ 0 \}\). Then the following two conditions are equivalent:

  • \(\Gamma _h\) is an equilibrium curve of the functional \({{{\,\mathrm{Length}\,}}} + \kappa \cdot {{{\,\mathrm{Area}\,}}}\).

  • The discrete curvature \((2/l_0) \tan (\theta _k /2)\) is constant and \(=\kappa \).

5 Parallel Curves and Steiner-Type Formula

In this section we will derive the discrete version of the Steiner-type formula in order to show effectiveness of the vertex normal we will define. The following type of Steiner formula has in fact appeared in some papers, e.g., [2, 4]. There, however, the notion of curvature is tried to be found with the Steiner-type formula; here we will derive the Steiner-type formula by using our vertex normal and connect it with the well-known notion of curvature.

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

$$\begin{aligned} N_k :=-\dfrac{\nabla _{\!p_k}{L}}{\sin \theta _k}= \dfrac{R (\nu _{k-1} - \nu _k)}{\sin \theta _k}= \dfrac{1}{1+\cos \theta _k}(\nu _k + \nu _{k-1}). \end{aligned}$$
(5.1)

Then we shall call the vector \(N_k\) the vertex normal at \(p_k\). 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:

$$\begin{aligned} p_k(t) = p_k + t N_k, \qquad k=1,\ldots , n. \end{aligned}$$

Lemma 5.1

We have \(\langle p_{k+1}(t) - p_k(t), \nu _k \rangle = 0\); therefore we call this deformation parallel curves.

Proof

This is a direct calculation:

$$\begin{aligned} \langle p_{k+1}(t) - p_k(t), \nu _k \rangle&=\bigl \langle (p_{k+1} - p_k) + t (N_{k+1} - N_k), \nu _k\bigr \rangle \\&= t \biggl ( \dfrac{\langle \nu _{k+1} + \nu _k, \nu _k \rangle }{1 + \cos \theta _{k+1}}- \dfrac{\langle \nu _k + \nu _{k-1}, \nu _k \rangle }{1 + \cos \theta _k} \biggr ) \\&= t \biggl ( \dfrac{1 + \cos \theta _{k+1}}{1 + \cos \theta _{k+1}}-\dfrac{1+\cos \theta _k}{1+\cos \theta _k}\biggr )=0. \end{aligned}$$

\(\square \)

Theorem 5.2

(discrete Steiner-type formula)   For parallel curves \(\{ p_k(t) \}_k\), we have

$$\begin{aligned} |p_{k+1} (t) - p_k(t)| = |p_{k+1} - p_k| \cdot (1 - t \kappa (e_k)), \end{aligned}$$

where \(\kappa (e_k)\) is the discrete curvature based on the edge osculating circle method [8]:

$$\begin{aligned} \kappa (e_k) = \dfrac{\tan (\theta _k/2) + \tan (\theta _{k+1}/2)}{|p_{k+1} - p_k|}. \end{aligned}$$

Before giving the proof, we give an intuitive explanation for a special case. In Fig. 2, the similarity ratio of the triangles gives

Fig. 2
figure 2

Parallel curves

Full size image
$$\begin{aligned}&|p_{k+1}(t) - p_k(t)| : |p_{k+1} - p_k| = (-r_k + t) : -r_k\\&\quad \iff \quad |p_{k+1}(t) - p_k(t)| = |p_{k+1} - p_k|\cdot (1 - t\kappa (e_k)). \end{aligned}$$

Note that the signature of the curvature radius \(r_k\) is negative in the figure.

Proof

By direct calculation we have

$$\begin{aligned} \biggl \langle \dfrac{p_{k+1} -p_k}{|p_{k+1} -p_k|}, N_{k+1} - N_k \biggr \rangle&= \biggl \langle \!{ -} R \nu _k, \dfrac{\nu _k + \nu _{k+1}}{1+\cos \theta _{k+1}} - \dfrac{\nu _{k-1} + \nu _k}{1 + \cos \theta _k} \biggr \rangle \\&= - \biggl ( \dfrac{\sin \theta _{k+1}}{1+\cos \theta _{k+1}} + \dfrac{\sin \theta _k}{1+\cos \theta _k} \biggr )= - \biggl ( \tan \dfrac{\theta _{k+1}}{2} + \tan \dfrac{\theta _k}{2} \biggr ), \end{aligned}$$
$$\begin{aligned}&|N_{k+1} - N_k|^2= \dfrac{1}{\cos ^2 (\theta _{k+1}/2)} + \dfrac{1}{\cos ^2 (\theta _k/2)} -2 \langle N_k, N_{k+1} \rangle \\&\qquad \quad = 2 \!+\! \tan ^2 \dfrac{\theta _{k+1}}{2} \!+ \!\tan ^2 \dfrac{\theta _k}{2}\!-\!2 \dfrac{1 \!+\! \cos \theta _k\!+\!\cos \theta _{k+1}\!+\!\cos (\theta _k \!+\! \theta _{k+1})}{(1 \!+\! \cos \theta _k) (1 \!+ \!\cos \theta _{k+1})}\\&\qquad \quad = \biggl ( \tan \dfrac{\theta _k}{2} + \tan \dfrac{\theta _{k+1}}{2} \biggr )^{\!2}. \end{aligned}$$

Therefore we conclude

$$\begin{aligned}&|p_{k+1}(t)-p_k(t)|^2 \\ \quad&= |p_{k+1} - p_k|^2 + 2t\langle p_{k+1} -p_k, N_{k+1} -N_k \rangle \\&\quad + t^2 |N_{k+1} - N_k|^2 \\ \\ \quad&= |p_{k+1} - p_k|^2 - 2t |p_{k+1} - p_k| \biggl ( \tan \dfrac{\theta _k}{2} + \tan \dfrac{\theta _{k+1}}{2} \biggr ) \\&\quad + t^2 \biggl (\tan \dfrac{\theta _k}{2} + \tan \dfrac{\theta _{k+1}}{2} \biggr )^{\!2} \\ \quad&= |p_{k+1} - p_k|^2 \biggl ( 1 - t \cdot \dfrac{\tan (\theta _k/2) + \tan (\theta _{k+1}/2)}{|p_{k+1} - p_k|} \biggr )^{\!2}. \end{aligned}$$

\(\square \)

Remark 5.3

This formula itself has already appeared in some papers aiming to find the discrete curvature at the vertices by using the offsets, see [2, 4]. If we take the offset of the edges, then we have the following (at least) three possibilities. By computing the length of dotted curves in the figure, we can write the total length of each offset as follows:

$$\begin{aligned} {{\,\mathrm{Length}\,}}\Gamma _{h, t}^{(1)}= & {} {{\,\mathrm{Length}\,}}\Gamma _h- t \sum _k2\sin \dfrac{\theta _k}{2},\qquad {{\,\mathrm{Length}\,}}\Gamma _{h,t}^{(2)}={{\,\mathrm{Length}\,}}\Gamma _h-t\sum _k\theta _k,\\ {{\,\mathrm{Length}\,}}\Gamma _{h, t}^{(3)}= & {} {{\,\mathrm{Length}\,}}\Gamma _h- t \sum _k 2 \tan \dfrac{\theta _k}{2}. \end{aligned}$$

Note the signature of the angles. The second curve \(\Gamma _h^{(2)}\) is nothing but the normal cone method (or the boundary of the Minkowski sum with the disk) known in convex geometry. However, the only possible way to keep the number of the vertices unchanged during the offset procedure is the third one, and modifying the third formula gives our Steiner-type formula:

$$\begin{aligned} {{\,\mathrm{Length}\,}}\Gamma _{h, t}^{(3)}&= \sum _k\biggl (l_k - 2t \tan \dfrac{\theta _k}{2}\biggr )\\&= \sum _k\biggl (l_k - t \tan \dfrac{\theta _k}{2} + t\tan \dfrac{\theta _{k+1}}{2}\biggr )= \sum _k l_k (1 - t \kappa (e_k)), \end{aligned}$$

where we put \(\kappa (e_k) = (\tan (\theta _k / 2) + \tan (\theta _{k+1} /2)) / l_k\).

Remark 5.4

(discrete Frenet–Serret formula)   In Sect. 2, we remarked that the relation

$$\begin{aligned} \widetilde{N}_k = \dfrac{1}{L_k} (t_k - t_{k-1}), \quad t_k = \dfrac{p_{k+1} - p_k}{|p_{k+1} - p_k|} \end{aligned}$$

can be considered as a part of the Frenet–Serret formula. On the other hand, by some calculation we have

$$\begin{aligned} \dfrac{1}{|p_{k+1} - p_k|} (N_{k+1} - N_k)= - \dfrac{\tan (\theta _k/2) + \tan (\theta _{k+1}/2)}{|p_{k+1} - p_k|}t_k= - \kappa (e_k)t_k. \end{aligned}$$

This formula can be seen as the other part of the Frenet–Serret formula. Note that the former formula is a formula on the vertex \(p_k\) while the latter formula is a formula on the edge \([p_k, p_{k+1}]\).

Fig. 3
figure 3

Three possibilities for constructing a new curve

Full size image

6 The Second Variation Formula

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 \(L + \kappa \cdot { {{\,\mathrm{Area}\,}}}\). 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:

$$\begin{aligned} \dfrac{d}{dt} L&= \sum _k \langle \nabla _{\!p_k} L, p_k' \rangle ,&\nabla _{\!p_k} L&= R ( \nu _k - \nu _{k-1} ), \\ \dfrac{d}{dt}{ {{\,\mathrm{Area}\,}}}&= \sum _k \langle \nabla _{\!p_k} {{{\,\mathrm{Area}\,}}}, p_k' \rangle ,&\nabla _{\!p_k}{ {{\,\mathrm{Area}\,}}}&= \dfrac{1}{2}R(p_{k+1} - p_{k-1}). \end{aligned}$$

Note that if the variation is admissible, then we have

$$\begin{aligned} 0 = \delta ^2{{{\,\mathrm{Area}\,}}} = \sum _k \,\bigl (\langle \delta \nabla _{\!p_k}{ {{\,\mathrm{Area}\,}}}, \delta p_k \rangle + \langle \nabla _{\!p_k}{ {{\,\mathrm{Area}\,}}}, \delta ^2 p_k \rangle \bigr ). \end{aligned}$$

Lemma 6.1

Let \(\Gamma _h = \{ p_k \}_k\) be an equilibrium closed curve for \(L + \kappa \cdot {{{\,\mathrm{Area}\,}}}\) and \(p_k(t) = p_k + t v_k + O(t^2)\) be an admissible variation. Then we have

$$\begin{aligned} \delta ^2 L:= \dfrac{d^2L}{d t^2}\Big |_{t=0}\!= \sum _k \bigl \langle \delta (\nabla _{\!p_k} L + \kappa \nabla _{\!p_k}{ {{\,\mathrm{Area}\,}}}), v_k\bigr \rangle . \end{aligned}$$

Definition 6.2

(stability of discrete curves)   Let \(\Gamma _h = \{ p_k \}_k\) be a closed equilibrium curve for the functional \(L + \kappa \cdot {{{\,\mathrm{Area}\,}}}\). Then \(\Gamma _h\) is said to be stable if \(\delta ^2 L \ge 0\) for any admissible variation.

We introduce the matrices \(Q^L\) and \(Q^A\) as follows:

$$\begin{aligned} {}^t \vec {v}Q^L \vec {v} = \sum _k \langle \delta \nabla _{\!p_k} L, v_k \rangle , \qquad {}^t \vec {v}Q^A \vec {v} = \sum _k \langle \delta \nabla _{\!p_k}{ {{\,\mathrm{Area}\,}}}, v_k \rangle . \end{aligned}$$

Then we can write \(\delta ^2 L = {}^t \vec {v}(Q^L + \kappa Q^A)\vec {v}\).

Lemma 6.3

\({}^t \vec {v}Q^A \vec {v} = \sum _k \langle v_k, R v_{k+1} \rangle \).

Proof

The proof follows from direct computation:

$$\begin{aligned} {}^t \vec {v}Q^A \vec {v}\,&= \sum _k\langle \delta \nabla _{\!p_k} {{{\,\mathrm{Area}\,}}} , v_k \rangle = \dfrac{1}{2} \sum _k \langle \delta R(p_{k+1} - p_{k-1}), v_k \rangle \\&= \dfrac{1}{2} \sum _k \langle R(v_{k+1} - v_{k-1}), v_k \rangle = \sum _k \langle v_k, R v_{k+1} \rangle . \end{aligned}$$

\(\square \)

Proposition 6.4

(second variation formula for the length functional)

$$\begin{aligned} Q ^L= \sum _k\dfrac{1}{l_k}\bigl (|v_{k+1} - v_k|^2 - \langle v_{k+1} - v_k, R \nu _k \rangle ^2\bigr )= \sum _k( |\nabla v_k|^2 - \langle \nabla v_k, R \nu _k \rangle ^2)l_k, \end{aligned}$$

and therefore we have the following second variation formula for the length functional:

$$\begin{aligned} \delta ^2 L= \sum _k\bigl (( |\nabla v_k|^2 - \langle \nabla v_k, R \nu _k \rangle ^2)l_k + \kappa \langle v_k, R v_{k+1} \rangle \bigr ). \end{aligned}$$
(6.1)

Proof

By using the fact that \(l_k \delta \nu _k = R(v_{k+1} - v_k) - \langle R(v_{k+1} - v_k), \nu _k \rangle \nu _k\), we have

$$\begin{aligned} Q^L&= \sum _k \langle \delta \nu _k, R(v_{k+1} - v_k) \rangle = \sum _k\bigl \langle R(\nabla v_k) - \langle R(\nabla v_k), \nu _k \rangle \nu _k, R(\nabla v_k)\bigr \rangle l_k \\&= \sum _k( |\nabla v_k|^2 - \langle R(\nabla v_k), \nu _k \rangle ^2)l_k= \sum _k( |\nabla v_k|^2 - \langle \nabla v_k, R \nu _k \rangle ^2)l_k. \end{aligned}$$

\(\square \)

In the section on the Steiner-type formula, we used the vector

$$\begin{aligned} N_k := \dfrac{R(\nu _{k-1} - \nu _k)}{\sin \theta _k}= \dfrac{1}{1 + \cos \theta _k}(\nu _k + \nu _{k-1}) \end{aligned}$$

as the “normal vector” at the vertex \(p_k\). If we define the “tangent vector” \(T_k\) as \(T_k = -RN_k\), then we can decompose the variation vector \(v_k\) as

$$\begin{aligned} v_k = \psi _k N_k + \eta _k T_k = \psi _k N_k - \eta _k RN_k, \end{aligned}$$

where \(\psi , \eta :V \rightarrow {\mathbb {R}}\) are some functions on the vertices. If \(\eta _k = 0\) for all k, we call the variation the normal variation. In the following we will use this notation.

Lemma 6.5

The first variation formula of the volume can be written as

$$\begin{aligned} \delta { {{\,\mathrm{Area}\,}}}= \dfrac{1}{2} \sum _k\biggl ( \psi _k (l_k + l_{k-1}) + \eta _k (l_k - l_{k-1}) \tan \dfrac{\theta _k}{2} \biggr ). \end{aligned}$$

In particular, if the curve \(\Gamma _h = \{ p_k \}_k\) satisfies \(l_k \equiv l_0\), then a variation \(p_k(t) = p_k + t (\psi _k N_k + \eta _k T_k) + O(t^2)\) is volume-preserving if \(\sum _k \psi _k = 0\).

Proof

Using the first variation formula, we have

$$\begin{aligned} \delta {{\,\mathrm{Area}\,}}&= \dfrac{1}{2} \sum _k \langle R(p_{k+1} - p_{k-1}), v_k \rangle = \dfrac{1}{2} \sum _k \langle R(p_{k+1} - p_{k-1}), \psi _k N_k + \eta _kT_k\rangle \\&=\dfrac{1}{2} \sum _k\biggl [ \psi _k l_k \dfrac{\langle \nu _k, R \nu _{k-1} \rangle }{\sin \theta _k}+ \psi _k l_{k-1} \dfrac{\langle \nu _{k-1}, -R\nu _k \rangle }{\sin \theta _k}\\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \,\, + \dfrac{\eta _k}{\sin \theta _k}(l_k - l_{k-1}) (1- \cos \theta _k)\biggr ] \\&=\dfrac{1}{2} \sum _k\biggl ( \psi _k (l_k + l_{k-1})+ \eta _k (l_k - l_{k-1}) \tan \dfrac{\theta _k}{2}\biggr ). \end{aligned}$$

\(\square \)

Remark 6.6

For functions \(\psi _k\) satisfying \(\sum _k \psi _k = 0\), we can find variations whose variation vector fields are \(\psi _kN_k\). The proof is completely the same as in [1].

Recall that if \(\{p_k\}_k\) is an equilibrium curve of the functional \(L+\kappa \cdot {{{\,\mathrm{Area}\,}}}\), then we have \(l_k\equiv l_0\), \(\theta _k\equiv \theta _0\), and \(\kappa l_0=2 \tan (\theta _0/2)\).

Lemma 6.7

$$\begin{aligned} |v_{k+1} - v_k|^2 - \langle v_{k+1} - v_k, R \nu _k \rangle ^2=\biggl [(\psi _{k+1}-\psi _k)+(\eta _{k+1}+\eta _k)\tan \dfrac{\theta _0}{2} \biggr ]^2. \end{aligned}$$

Therefore we have

$$\begin{aligned} |\nabla v_k|^2 - \langle \nabla v_k, R \nu _k \rangle ^2=\biggl ( \nabla \psi _k + \dfrac{\kappa }{2} (\eta _k + \eta _{k+1}) \biggr )^{\!2} \end{aligned}$$

Proof

Recall that

$$\begin{aligned} \langle N_k, -RN_{k+1}\rangle&= \tan \dfrac{\theta _k}{2} + \tan \dfrac{\theta _{k+1}}{2}= 2 \tan \dfrac{\theta _0}{2}, \\ \langle N_k, N_{k+1} \rangle&= 1 - \tan \dfrac{\theta _k}{2} \tan \dfrac{\theta _{k+1}}{2}= 1- \tan ^2 \dfrac{\theta _0}{2}. \end{aligned}$$

If we note that \(|N_k| = 1/{\cos (\theta _0/2)}\), then

$$\begin{aligned}&|v_{k+1} - v_k|^2 \\&\quad =|\psi _{k+1}N_{k+1}-\psi _kN_k|^2+2\langle \psi _{k+1}N_{k+1}- \psi _kN_k,\eta _{k+1}T_{k+1}-\eta _kT_k\rangle \\&\qquad +|\eta _{k+1}T_{k+1}-\eta _kT_k|^2\\&\quad = \dfrac{\psi _{k+1}^2}{\cos ^2(\theta _0/2)} -2 \psi _k \psi _{k+1}\biggl (1 - \tan ^2\dfrac{\theta _0}{2}\biggr )+ \dfrac{\psi _k^2}{\cos ^2(\theta _0/2)}\\&\qquad -2\langle N_k, RN_{k+1} \rangle (\psi _{k+1} \eta _k - \psi _k \eta _{k+1})\\&\qquad + \dfrac{\eta _{k+1}^2}{\cos ^2(\theta _0/2)} -2 \eta _k \eta _{k+1} \biggl (1 - \tan ^2\dfrac{\theta _0}{2}\biggr )+ \dfrac{\eta _k^2}{\cos ^2 (\theta _0/2)} \\&\quad = (\psi _{k+1}^2 + \psi _k^2 + \eta _{k+1}^2 + \eta _k^2)\biggl (1+ \tan ^2\dfrac{\theta _0}{2}\biggr ) \\&\qquad -2(\psi _k \psi _{k+1} + \eta _k \eta _{k+1})\biggl (1 -\tan ^2\dfrac{\theta _0}{2}\biggr )+ 4(\eta _k\psi _{k+1}-\eta _{k+1}\psi _k)\tan \dfrac{\theta _0}{2}. \end{aligned}$$

Similarly, since

$$\begin{aligned}&\langle v_{k+1} - v_k, R \nu _k \rangle \\&\quad = \langle \psi _{k+1} N_{k+1} - \psi _k N_k + \eta _{k+1} T_{k+1} -\eta _k T_k, R \nu _k \rangle \\&\quad = \psi _{k+1} \dfrac{\langle \nu _k - \nu _{k+1}, \nu _k \rangle }{\sin \theta _{k+1}}-\psi _k \dfrac{\langle \nu _{k-1} - \nu _k, \nu _k \rangle }{\sin \theta _k}\\&\qquad - \eta _{k+1} \dfrac{\langle \nu _{k+1} -\nu _k, R\nu _k \rangle }{\sin \theta _{k+1}} + \eta _k \dfrac{\langle \nu _k - \nu _{k-1}, R \nu _k\rangle }{\sin \theta _k} \\&\quad = (\psi _{k+1} + \psi _k) \tan \dfrac{\theta _0 }{2}- (\eta _{k+1} - \eta _k), \end{aligned}$$

we have

$$\begin{aligned} \langle v_{k+1} - v_k, R \nu _k \rangle ^2&= (\psi _{k+1} + \psi _k)^2 \tan ^2\dfrac{\theta _0}{2}\\&\quad - 2 (\psi _{k+1} + \psi _k) (\eta _{k+1} - \eta _k)\tan \dfrac{\theta _0}{2} + (\eta _{k+1} - \eta _k)^2. \end{aligned}$$

Subtracting these factors we have

$$\begin{aligned}&|v_{k+1} - v_k|^2 - \langle v_{k+1} - v_k, R \nu _k \rangle ^2 \\&\quad =\psi _{k+1}^2 + \psi _k^2 + (\eta _{k+1}^2 + \eta _k^2) \tan ^2\dfrac{\theta _0}{2}-2 \psi _k \psi _{k+1} \tan ^2\dfrac{\theta _0}{2} + 2 \eta _k \eta _{k+1}\\&\qquad + 2(2 \eta _k \psi _{k+1} - 2 \eta _{k+1} \psi _k+ \psi _{k+1} \eta _{k+1} - \psi _{k+1} \eta _k + \psi _k \eta _{k+1} - \psi _k \eta _k)\tan \dfrac{\theta _0}{2} \\&\qquad -2(\psi _k \psi _{k+1} + \eta _k \eta _{k+1})+ 2 (\psi _k \psi _{k+1} + \eta _k \eta _{k+1})\tan ^2 \dfrac{\theta _0}{2}\\&\quad = (\psi _{k+1} - \psi _k)^2 + (\eta _{k+1} + \eta _k)^2\tan ^2 \dfrac{\theta _0}{2}+ 2 (\psi _{k+1} - \psi _k) (\eta _{k+1} + \eta _k) \tan \dfrac{\theta _0}{2}\\&\qquad = \biggl [(\psi _{k+1} - \psi _k) + (\eta _{k+1} + \eta _k)\tan \dfrac{\theta _0}{2} \biggr ]^2. \end{aligned}$$

\(\square \)

Lemma 6.8

$$\begin{aligned} \langle v_k, R v_{k+1} \rangle = - 2(\psi _k \psi _{k+1} + \eta _k \eta _{k+1})\tan \dfrac{\theta _0}{2}- (\eta _k \psi _{k+1} - \eta _{k+1} \psi _k)\biggl (1-\tan ^2\dfrac{\theta _0}{2}\biggr ). \end{aligned}$$

Proof

This is also a simple calculation:

$$\begin{aligned} \langle v_k, R v_{k+1} \rangle&= \langle \psi _k N_k - \eta _k R N_k, \psi _{k+1} R N_{k+1} + \eta _{k+1} N_{k+1} \rangle \\&= (\psi _k \psi _{k+1} + \eta _k \eta _{k+1}) \langle N_k, R N_{k+1} \rangle - (\eta _k \psi _{k+1} - \eta _{k+1} \psi _k) \langle N_k, N_{k+1} \rangle \\&= - 2 (\psi _k \psi _{k+1} + \eta _k \eta _{k+1})\tan \dfrac{\theta _0}{2}-(\eta _k \psi _{k+1} - \eta _{k+1} \psi _k)\biggl (1- \tan ^2\dfrac{\theta _0}{2}\biggr ). \end{aligned}$$

\(\square \)

Theorem 6.9

(second variation formula for the length functional)

$$\begin{aligned} \delta ^2 L= \sum _k\biggl [ |\nabla \psi _k|^2 - \kappa ^2 \psi _k \psi _{k+1}+ (\kappa \nabla \psi _k (\eta _{k+1} + \eta _k) + |\nabla \eta _k|^2)\tan ^2\dfrac{\theta _0}{2}\biggr ]l_0 . \end{aligned}$$

In particular, for the normal variation we have

$$\begin{aligned} \delta ^2 L= \sum _k(|\nabla \psi _k|^2 - \kappa ^2 \psi _k \psi _{k+1})l_0=-\sum _k\psi _k(\Delta \psi _k+\kappa ^2\psi _{k+1})l_0, \end{aligned}$$

where we use the integration by parts and take the line element at the vertex as \(L_k = (l_k + l_{k-1})/2 = l_0\).

Proof

By using the previous lemmas, we have

$$\begin{aligned}&(|\nabla v_k|^2 - \langle \nabla v_k, R \nu _k \rangle ^2)l_k + \kappa \langle v_k, R v_{k+1} \rangle \\&\quad = \biggl ( |\nabla \psi _k|^2 + \kappa \nabla \psi _k (\eta _{k+1} + \eta _k)+ \dfrac{\kappa ^2}{4} (\eta _{k+1} + \eta _k)^2 \biggr )l_0 \\&\qquad - \kappa ^2 l_0 (\psi _k \psi _{k+1} + \eta _k \eta _{k+1})- \kappa (\eta _k \psi _{k+1} - \eta _{k+1} \psi _k)\biggl (1-\tan ^2\dfrac{\theta _0}{2}\biggr )\\&\quad = |\nabla \psi _k|^2 l_0 + \kappa \nabla \psi _k (\eta _{k+1} + \eta _k) l_0+ \dfrac{\kappa ^2}{4} (\eta _{k+1} - \eta _k)^2 l_0 - \kappa ^2 \psi _k \psi _{k+1} l_0 \\&\qquad - \kappa \bigl ( (\psi _{k+1} - \psi _k)(\eta _{k+1} + \eta _k) - (\psi _{k+1} \eta _{k+1} - \psi _k \eta _k)\bigr )\biggl (1-\tan ^2\dfrac{\theta _2}{2}\biggr ) \\&\quad = |\nabla \psi _k|^2 l_0 + \kappa \tan ^2\dfrac{\theta _0}{2}\cdot \nabla \psi _k (\eta _{k+1} + \eta _k)l_0+ \dfrac{\kappa ^2}{4}(\eta _{k+1} - \eta _k)^2 l_0 - \kappa ^2 \psi _k \psi _{k+1} l_0 \\&\qquad + \kappa (\psi _{k+1} \eta _{k+1}-\psi _k \eta _k)\biggl (1 - \tan ^2\dfrac{\theta _0}{2}\biggr ). \end{aligned}$$

Summing up, we get the desired result. \(\square \)

7 Instability of Non-Convex Regular Polygons

In this section we will prove that non-convex regular polygons and convex regular polygons with multiplicity are unstable. To prove this, we find a special variation with a help of the following discrete version of Wirtinger’s inequality:

Theorem 7.1

(discrete Wirtinger’s inequality [6])   Let \(\psi _0,\ldots ,\psi _n\) be \(n+1\) real numbers such that

$$\begin{aligned} \psi _0 = \psi _n, \qquad \sum _{k=0}^{n-1} \psi _k = 0. \end{aligned}$$

Then we have

$$\begin{aligned} \sum _{k=0}^{n-1} (\psi _{k+1} - \psi _k)^2 \ge 4 \sin ^2 \dfrac{\pi }{n} \sum _{k=0}^{n-1} \psi _k^2 \end{aligned}$$
(7.1)

and the equality holds if and only if there exist \(A, B \in {\mathbb {R}}\) such that

$$\begin{aligned} \psi _k = A \cos \dfrac{2 \pi k}{n} + B \sin \dfrac{2 \pi k}{n}. \end{aligned}$$
(7.2)

In the following, we will consider the normal variation, i.e., the variation of the form

$$\begin{aligned} p_k(t) = p_k + t \psi _k N_k + O(t^2), \qquad N_k =\dfrac{\nu _k + \nu _{k-1}}{1 + \cos \theta _0}, \end{aligned}$$

where \(\psi _k\) satisfies \(\sum _k \psi _k = 0\). By the second variation formula (Theorem 6.9) we have

$$\begin{aligned} \delta ^2 L= \sum _k(|\nabla \psi _k|^2 - \kappa ^2 \psi _k \psi _{k+1})l_0= \sum _k\dfrac{1}{l_0}\biggl [(\psi _{k+1} - \psi _k)^2 - 4 \psi _k \psi _{k+1} \tan ^2 \dfrac{m \pi }{n} \biggr ] \end{aligned}$$

for any admissible variations, where we use the relation \(\kappa l_0 = 2 \tan (\theta _0/2)\), put \(\theta _0 = 2 m \pi / n\) for some \(m \in {\mathbb {Z}}\), and assume that \(m/n \ne 1/2\).

Theorem 7.2

(instability of non-convex regular polygons)   Let \(n \ge 5\). By taking \(\psi _k\) as in (7.2), \((A, B) \ne (0, 0)\), we have

$$\begin{aligned} \delta ^2 {L}= \dfrac{4}{l_0} \biggl [ \sin ^2 \dfrac{\pi }{n} - \cos \dfrac{2 \pi }{n} \tan ^2 \dfrac{m \pi }{n} \biggr ] \sum _k \psi _k^2. \end{aligned}$$

In particular, \(\delta ^2 {L} < 0\) for \(2 \le m \le n-2\), i.e., non-convex regular polygons are unstable.

Proof

By the discrete Wirtinger’s inequality we have

$$\begin{aligned} \delta ^2 {L} \ge \sum _k\dfrac{4}{l_0} \biggl (\psi _k^2 \sin ^2 \dfrac{\pi }{n} - \psi _k \psi _{k+1} \tan ^2 \dfrac{m \pi }{ n}\biggr ). \end{aligned}$$

In the following we use (7.2) and put \(\varphi _k = -A \sin (2 \pi k/n) + B \cos (2\pi k/n)\). Then we have

$$\begin{aligned} \psi _{k+1}=\psi _k\cos \dfrac{2\pi }{n} +\varphi _k \sin \dfrac{2 \pi }{n},\qquad \psi _k \varphi _k = \dfrac{B^2 - A^2}{2}\sin \dfrac{4 k \pi }{n} + AB \cos \dfrac{4 k \pi }{n}. \end{aligned}$$

If we note that \(\sum _k \psi _k \varphi _k = 0\), then

$$\begin{aligned} \delta ^2 {L}&= \sum _k \dfrac{4}{l_0} \biggl [ \psi _k^2 \sin ^2 \dfrac{\pi }{n} - \psi _k \biggl ( \psi _k \cos \dfrac{2 \pi }{n} +\varphi _k \sin \dfrac{2 \pi }{n} \biggr ) \tan ^2 \dfrac{m \pi }{n} \biggr ] \\&= \dfrac{ 4}{l_0} \biggl [ \sin ^2 \dfrac{\pi }{n} - \cos \dfrac{2 \pi }{n} \tan ^2 \dfrac{m \pi }{n} \biggr ] \sum _k \psi _k^2. \end{aligned}$$

If \(m=1\) or \(m=n-1\), then

$$\begin{aligned} \delta ^2 {L} \ge \dfrac{4}{l_0} \sin ^2 \dfrac{\pi }{n} \tan ^2 \dfrac{\pi }{n} \sum _k \psi _k^2 \ge 0. \end{aligned}$$

On the other hand, for \(2 \le m \le n-2\) and \((A, B) \ne (0, 0)\) we have

$$\begin{aligned} \delta ^2 {L}\le - \dfrac{4 \sin ^2(\pi /n) (1 + 2 \cos ^2 (\pi /n))}{l_0 \cos (2 \pi /n)} \sum _k \psi _k^2 < 0, \end{aligned}$$

where we use the fact \(\tan ^2 (m \pi /n) \ge \tan ^2(2\pi /n)\) if \(2\le m\le n-2\), and \(\cos (2\pi /n)>0\) if \(n\ge 5\). This proves the statement. \(\square \)

8 Concluding Remarks

We observe the second variation formula from the analysis of the Jacobi operator. We can modify the equation (7.1) as follows:

$$\begin{aligned} \delta ^2 L= \dfrac{1}{l_0} \sum _k(- \alpha \psi _{k-1} + 2 \psi _k - \alpha \psi _{k+1})\psi _k= \dfrac{\langle H \Psi , \Psi \rangle }{l_0} , \end{aligned}$$

where we put \(\alpha = 1 + 2 \tan ^2 (m\pi /n)\) and

$$\begin{aligned} H =\begin{pmatrix} 2 &{}\quad -\alpha &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -\alpha \\ -\alpha &{}\quad 2 &{}\quad -\alpha &{}\quad \cdots &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -\alpha &{}\quad 2 &{}\quad \cdots &{}\quad 0 &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 2 &{}\quad -\alpha \\ -\alpha &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad -\alpha &{}\quad 2 \\ \end{pmatrix},\quad \Psi =\begin{pmatrix} \psi _1 \\ \psi _2 \\ \psi _3 \\ \vdots \\ \psi _n \end{pmatrix}. \end{aligned}$$

Since H is a circulant matrix, the eigenvalues \(\lambda _j\) can be calculated explicitly, see e.g. [5]:

$$\begin{aligned} \lambda _j= 2 - 2 \alpha \cos \dfrac{2\pi j}{n}= \dfrac{4 \cos ^2 (j \pi /n)}{\cos ^2 (m\pi /n)}\biggl ( \tan ^2 \dfrac{j \pi }{n} - \sin ^2 \dfrac{m \pi }{n} \biggr ),\qquad j = 1, \ldots , n. \end{aligned}$$

The corresponding eigenvectors are

$$\begin{aligned} e_j = {}^t (1, \omega ^j, \ldots , \omega ^{j(n-1)}) \in {\mathbb {C}}^n, \qquad \omega = \exp \dfrac{2 \pi \sqrt{-1}}{n}. \end{aligned}$$
(8.1)

The condition \(\sum _k \psi _k = 0\) is equivalent to the condition that \(\Psi = {}^t (\psi _1, \ldots , \psi _n)\) is perpendicular to \(e_n = {}^t (1, \ldots , 1)\). Therefore we only consider the eigenvalues \(\lambda _1, \ldots , \lambda _{n-1}\).

Lemma 8.1

For \(m \le j \le n-m\), \(\lambda _j > 0\). In particular, \(\lambda _j > 0\) for \(1 \le j \le n-1\) if \(m=1\), i.e., the convex regular polygon case.

Proof

This is a direct calculation:

$$\begin{aligned} \qquad \qquad \quad \tan ^2\dfrac{j\pi }{n}{-}\sin ^2\dfrac{m\pi }{n}\ge \tan ^2\dfrac{m \pi }{n}{-} \sin ^2 \dfrac{m \pi }{n}{=} \sin ^2 \dfrac{m \pi }{n} \tan ^2 \dfrac{m \pi }{n}{>}0. \qquad \end{aligned}$$

\(\square \)

Theorem 8.2

Assume \(m \ge 2\). For \(1 \le j \le m/2\) or \(n - m/2 \le j \le n-1\), we have \(\lambda _j < 0\). Therefore, the index of a non-convex regular polygon or a convex polygon with multiplicity is at least \(\lfloor m/2 \rfloor \).

Proof

Under the above conditions, we have

$$\begin{aligned} \qquad \qquad \qquad \quad \tan ^2 \dfrac{j \pi }{n} - \sin ^2 \dfrac{m \pi }{n}\le \tan ^2 \dfrac{j \pi }{n} - \sin ^2 \dfrac{2 j \pi }{n} < 0.\qquad \qquad \qquad \quad \square \qquad \end{aligned}$$

\(\square \)

Remark 8.3

More precisely, \(\lambda _j < 0\) for \(j < (n /\pi ) \arctan (\sin (m\pi /n))\).

Remark 8.4

From another point of view, a non-convex regular polygon is the high-frequency component of the discrete Fourier expansion of the polygon. More precisely, any polygon in \({\mathbb {R}}^2\) with n vertices can be regarded as a point in \({\mathbb {R}}^{2n}\simeq {\mathbb {C}}^n\). Each eigenvector \(e_k\) in (8.1) corresponds to the regular n-gon and \(\{e_1,\ldots ,e_n\}\) forms a basis of \({\mathbb {C}}^n\). Therefore, any polygon in \({\mathbb {R}}^2\) can be written as a linear combination of regular n-gons and this fact corresponds to the discrete Fourier expansion.