On the Stability Problem of Equilibrium Discrete Planar Curves

We study planar polygonal curves with the variational methods. We show a unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.


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, Editor in Charge: János Pach Yoshiki Jikumaru y-jikumaru@imi.kyushu-u.ac.jp 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.

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. 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 → R 2 satisfying l k := | p k+1 − p k | = 0 for all k = 0, 1, . . . , n − 1, where we denote p k := X (v k ). We denote such a discrete curve as h = {p k } k .
To each oriented edge e k := [ p k , p k+1 ] of h we can assign a unit normal vector where R is the (π/2)-rotation or (−π/2)-rotation in R 2 . It does not matter which one we choose, but we choose the same R for all k. For a discrete curve h = {p k } k with an orientation, the length of h and the 2-dimensional volume (area) bounded by h are defined by In these settings, we consider the following question: What are 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 is the "variation vector field". If p k is a boundary point of h , then we assume v k = 0. We want to find the vector t ∇ L ∈ R 2n satisfying Since this is just a directional derivative in R 2n , the following proposition is immediate:

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: By using this formula we have where we inserted some auxiliary function L k defined on the vertices. This kind of observation is essentially remarked on in the paper [3].
the discrete curvature vector with respect to L k .

Remark 2.4
There are some reasons why we can regard the vector N k as the curvature vector. First, the vector 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.

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 ν k−1 and ν k as θ k , i.e., We have to care about the signature of θ k . Let R θ be the θ -rotation in R 2 . We assign the signature depending on the choice of the rotation R appearing in the definition of the edge normal: In this situation the signature of θ k is determined by the equation R σ θ k (ν k−1 ) = ν k . We easily see that k θ k = 2mπ for some integer m ∈ 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 : the discrete curvature with respect to L k .

Remark 3.2
If we use the expression ∇ p k L = R(ν k − ν k−1 ), the discrete curvature with respect to L k can be written as κ( p k ) = |ν k − ν 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.
then the discrete curvature with respect to L k becomes 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 then the discrete curvature with respect to L k becomes and this value coincides with the curvature of arclength parametrized curve.
In the paper [7], the discrete curvature at a vertex is defined as Then we immediately have the following result: 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 where L k is some auxiliary function. As in the vertex case, we call the value 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 then the discrete curvature with respect to L k becomes 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 → ds and θ k → 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 h with the vertex set V , we consider a function ψ : V → R. Then the gradient and the Laplacian of ψ can be defined as where we denote ψ k := ψ(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 N k with respect to L k can be written as p k . From another point of view, if we define the Dirichlet energy of ψ as then the first variation of the energy becomes where we take the variation of ψ as ψ k (t) = ψ k +tϕ k + O(t 2 ). Therefore δ E h (ψ) = 0 if and only if ψ k = 0. Note that the condition ψ 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 ϕ : V → R, we still have the following properties, since the quantities ψ k L k are independent of L k : • If ψ is constant, then ψ = 0.
• The condition ψ = 0 is independent of the choice of L k and in this case we have the mean value property: • L 2 symmetric property: Note that the summation is vertex-based. • Integration by parts: 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 − is positive semi-definite.

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.

Remark 4.2
We can modify the first variation formula of the volume as follows: It is natural to choose 2L k = l k + l k−1 = Length( ( p)) and this is also frequently used as a "vertex normal" (a weighted sum of the edge normals): In addition, we have 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. where we assume that m/n = 1/2. In particular, we sometimes call the curve 1,n h a convex regular n-gon. Note that n−1,n h is also convex but it has an opposite unit normal with 1,n h (usually we assume that 1,n h has the outward-pointing unit normal). Then the curve m,n h is a critical point of the functional Length + κ · Area with κ = −1/(a cos(mπ/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 κ n = −1/(a cos(π/n)). Note that cos(π m/n) = − cos(π(n − m)/n) and κ n → −1/a when n → ∞.
We will show that these regular polygons are the only equilibrium curves for the functional L + κ · Area. • h is an equilibrium curve of the functional L + κ · Area.

Proof We put
Then the discrete curve h is a critical point of the functional Length + κ · Area if and only if A k = 0 for all k. By a simple calculation we have For the necessity, that is, if we assume A k = 0 for all k, then it follows from (4.1) and (4.2) that κl k = 2 tan(θ k /2) = κl k−1 . And it also follows from (4.2), (4.3), and using l k = l k−1 that To prove the sufficiency, since ν k and p k − p k−1 form a basis of R 2 and A k , ν k = 0 by (4.1) and the assumption, all we have to prove is A k , p k − p k−1 = 0 for all k. By using the assumption l k = l k−1 = l 0 and θ k = θ 0 , we have This shows A k = 0 and proves the statement.

Remark 4.5 The equilibrium condition
The latter condition can be considered as a conservation law for the Euler-Lagrange equation A k = 0. Since the vector c ∈ 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 = −ν k /κ. Therefore, the edge midpoints must be tangent to the unit circle.
We found that equilibrium closed curves of the functional L + κ · Area must satisfy l k ≡ 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 h be an arclength parametrized discrete closed curve, i.e., l k ≡ l 0 , and κ ∈ R \ {0}. Then the following two conditions are equivalent: • h is an equilibrium curve of the functional Length + κ · Area.

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 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 ∇ p k L can be computed as 2 sin(θ k /2) up to the signature. In the discrete case, we should consider another factor 2 sin(θ k /2) cos(θ k /2) = sin θ k and put 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 θ k = 0. Then we consider the following deformation of the curve: Lemma 5. 1 We have p k+1 (t) − p k (t), ν k = 0; therefore we call this deformation parallel curves.
Proof. This is a direct calculation:

Fig. 2 Parallel curves
where κ(e k ) is the discrete curvature based on the edge osculating circle method [8]: Before giving the proof, we give an intuitive explanation for a special case. In Fig. 2, the similarity ratio of the triangles gives Note that the signature of the curvature radius r k is negative in the figure.

Proof. By direct calculation we have
Therefore we conclude

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: Note the signature of the angles. The second curve (2) h 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: where we put κ(e k ) = (tan(θ k /2) + tan(θ k+1 /2))/l k .

Remark 5.4 (discrete Frenet-Serret formula) In Sect. 2, we remarked that the relation
can be considered as a part of the Frenet-Serret formula. On the other hand, by some calculation we have 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 ]. h,t , connect by a segment (b) Γ (2) h,t , connect by an arc (c) Γ h,t , connect by a wedge Fig. 3 Three possibilities for constructing a new curve

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 h = {p k } k be an equilibrium closed curve for the functional L + κ · 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: Note that if the variation is admissible, then we have 0 = δ 2 Area = k δ∇ p k Area, δ p k + ∇ p k Area, δ 2 p k . Lemma 6.1 Let h = {p k } k be an equilibrium closed curve for L + κ · Area and p k (t) = p k + tv k + O(t 2 ) be an admissible variation. Then we have Definition 6.2 (stability of discrete curves) Let h = {p k } k be a closed equilibrium curve for the functional L + κ · Area. Then h is said to be stable if δ 2 L ≥ 0 for any admissible variation.
We introduce the matrices Q L and Q A as follows: Then we can write Proof. The proof follows from direct computation: Proposition 6.4 (second variation formula for the length functional) and therefore we have the following second variation formula for the length functional: Proof. By using the fact that In the section on the Steiner-type formula, we used the vector as the "normal vector" at the vertex p k . If we define the "tangent vector" T k as T k = −R N k , then we can decompose the variation vector v k as where ψ, η : V → R are some functions on the vertices. If η 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 In particular, if the curve h = {p k } k satisfies l k ≡ l 0 , then a variation p k (t) = Proof. Using the first variation formula, we have Remark 6.6 For functions ψ k satisfying k ψ k = 0, we can find variations whose variation vector fields are ψ k N k . The proof is completely the same as in [1].
Therefore we have If we note that |N k | = 1/cos(θ 0 /2), then Subtracting these factors we have Proof. This is also a simple calculation: Theorem 6.9 (second variation formula for the length functional) In particular, for the normal variation we have 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 Summing up, we get the desired result.

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: In the following, we will consider the normal variation, i.e., the variation of the form where ψ k satisfies k ψ k = 0. By the second variation formula (Theorem 6.9) we have δ 2 L = k (|∇ψ k | 2 − κ 2 ψ k ψ k+1 )l 0 = k 1 l 0 (ψ k+1 − ψ k ) 2 − 4ψ k ψ k+1 tan 2 mπ n for any admissible variations, where we use the relation κl 0 = 2 tan(θ 0 /2), put θ 0 = 2mπ/n for some m ∈ Z, and assume that m/n = 1/2. In particular, δ 2 L < 0 for 2 ≤ m ≤ n − 2, i.e., non-convex regular polygons are unstable.
Theorem 8.2 Assume m ≥ 2. For 1 ≤ j ≤ m/2 or n − m/2 ≤ j ≤ n − 1, we have λ j < 0. Therefore, the index of a non-convex regular polygon or a convex polygon with multiplicity is at least m/2 .

Remark 8.4
From another point of view, a non-convex regular polygon is the highfrequency component of the discrete Fourier expansion of the polygon. More precisely, any polygon in R 2 with n vertices can be regarded as a point in R 2n C n . Each eigenvector e k in (8.1) corresponds to the regular n-gon and {e 1 , . . . , e n } forms a basis of C n . Therefore, any polygon in R 2 can be written as a linear combination of regular n-gons and this fact corresponds to the discrete Fourier expansion.