Isogonal Deformation of Discrete Plane Curves and Discrete Burgers Hierarchy

We study deformations of plane curves in the similarity geometry. It is known that continuous deformations of smooth curves are described by the Burgers hierarchy. In this paper, we formulate the discrete deformation of discrete plane curves described by the discrete Burgers hierarchy as isogonal deformations. We also construct explicit formulas for the curve deformations by using the solution of linear diffusion differential/difference equations.


Introduction
Integrable deformations of curves play crucial roles in the differential geometry of space/plane curves [30]. Formulating the deformation of curves as the simultaneous system of the Frenet-Serret formula for the Frenet frame of curves and its deformation equation, it naturally gives rise to various integrable systems. This framework can be discretized so that it is consistent with the theory of discrete integrable systems, which is sometimes referred to as the discrete differential geometry [1]. Various deformations of discrete curves have been formulated in this context [6,7,8,15,16,17,18,19,26,27,28,29]. The theory of discrete differential geometry of curves is now making progress in explicit constructions of curves, by using the theory of τ functions [22,23,24,25].
When we change the geometric structure of space/plane in the framework of Klein geometry, the curve motions are governed by various integrable equations [3,4,5]. Therefore it may be an interesting and important problem to discretize such deformations of curves consistently with corresponding integrable structures.
In this paper, we consider deformation of the plane curves in the similarity geometry, which is a Klein geometry associated with the linear conformal group. In this setting, it is known that the Burgers hierarchy describes the deformations of similarity curvature of curves. We present discrete deformations of discrete plane curves in the similarity geometry described by the discrete Burgers hierarchy as the isogonal deformations in which each angle of adjascent segments is preserved. The lattice intervals of the hierarchy are generalized to arbitrary functions of corresponding independent variables. Using this formulation, we present explicit formulas of curves for both smooth and discrete cases. We note that the (complex) Burgers equation and its discrete analogue also arise in the curve deformations in complex hyperbola, where the Hamiltonian formulation of the deformation of smooth curves is discussed [15].
In Section 2, we give a brief summary of deformation of smooth plane curves in the similarity geometry, and we see that the Burgers hierarchy naturally arises as the equations for the similarity curvature. We also construct the explicit formula for the family of plane curves corresponding to the shock wave solutions to the Burgers equation. In Section 3, we discretize the whole theory described in Section 2 so that the deformations are governed by the discrete Burgers hierarchy. Formulations of the Burgers and the discrete Burgers hierarchies are discussed in detail in Appendix.
In [21,31], the deformation theory of plane curves in the similarity geometry can be applied to the construction and generalization of aesthetic curves in CAD. Also, in [9,10,11,12,13,14] discretizations for the class of nonlinear differential equations describing the motions of plane curves are constructed by using the geometric formulations, resulting in self-adaptive moving mesh discrete model of the original equation. This discretization enables to contruct highly accurate numerical scheme of given equation. The Burgers equation is widely used as the universal model describing one-dimensional nonlinear dissipative system after various transformations which are difficult to discretize. It may be possible to construct various useful discrete models by using the result in this paper. We hope that the results in this paper serves as the basis of such industry-based problems.

Deformation of smooth curves
Let γ = γ(s) be a smooth curve in R 2 , s be the arc-length, and κ be the curvature of γ. We denote by Sim(2) the similarity transformation group of R 2 , that is, Sim(2) = CO(2) R 2 where CO(2) is the linear conformal group CO(2) = A ∈ GL (2, R) t AA = c 2 id for some constant c .
The Sim(2)-invariant parameter x is given by the angle function and the Sim(2)-invariant curvature u is defined as The x and u are called the similarity arc-length parameter and the similarity curvature, respectively. If the similarity curvature is constant u = k 1 , then the inverse of Euclidean curvature is 1/κ = −k 1 s + k 2 for some constant k 2 . Thus γ is a log-spiral (if k 1 0) or a circle (if k 1 = 0, k 2 0). The Sim(2)-invariant frame φ = [T, N] is given by where the prime means differentiation with respect to the similarity arc-length parameter x. The SO(2)-invariant frame (the Frenet frame) φ E given by varies according to the Frenet formula Therefore, by using (2.1) and (2.2), we have We denote by γ(x, t) a deformation of a curve γ(x). We use the dot to indicate differentiation with respect to time t. Writingγ as the linear combination of T and N aṡ The compatibility condition of the linear system (2.4) and (2.5) is given by for some function a = a(t). Especially, choosing f = a − u, g = −1 and denoting t = t 2 , we have Equation (2.9) is called the Burgers equation, which is linearized to Further, the Burgers hierarchy naturally arises as follows [3,4,5]. Substituting (2.6) into (2.7), we have thatu = Ω 2 + 1 g + au , (2.12) where x is the recursion operator of the Burgers hierarchy (see Appendix A). Here, ∂ −1 x is the formal integration operator with respect to x, and in the following, the integration constant should be chosen to be 0. In view of this, we introduce an infinite number of time variables t = (t 2 , t 3 , t 4 , . . .), and choose g = Ω i−3 u (i ≥ 3). Then the higher flow with respect to the new time variable t i is given by The compatibility condition between (2.4) and (2.13) is the i-th Burgers equation which is linearized to via the Cole-Hopf transformation (2.11). Note that the case of i = 2 of (2.15) recovers (2.10).
It is possible to express the postion vector γ in terms of q as follows. The inverse of Euclidean curvature satisfies 1/κ = cq for some function c = c (t), because the similarity curvature is logarithmic differentiation of κ, that is, u satisfies that Since the similarity arclength parameter x is the angle function, we have where we have incorporated the ambiguity of the angle function x 0 = x 0 (t) explicitly. Hence We determine c and x 0 by the deformation equation (2.13). By differentiating T by t i (hereḋ enotes ∂ t i ), we have by substituting (2.16) into (2.13), Note that from (2.14) and (2.11) we have −∂ −1 x Ω i−1 + Ω i−3 + a u =q/q. Similarly for the case of i = 2 we also have −u + u 2 + 1 − au =q/q from (2.10) and (2.11). Then from (2.13) we obtaiṅ which implies c(t) = c(const.) and x 0 = A(t) whereȦ(t) = a(t). Therefore we obtain: . .) be a position vector of the plane curve in the similarity geometry satisfying (2.4), (2.8) and (2.13). Then γ admits the representation formula where c is a constant, and q(x, t) satisfies (2.15).
For a shock wave solution to the Burgers hierarchy, we can explictly construct the position vector. For a positive integer M, where λ 0 = ξ 0 = 0. Figure 1, 2 illustrate motion of plane curves corresponding to M-shock wave solutions (M = 1, 2, respectively) of the Burgers equation (2.9) with t i = 0 (i ≥ 3).
Remark 2.2. The parameter a originally arises as an integration constant in (2.6), and play a role of rotation in the deformation of smooth curves as seen in Proposition 2.1. This parameter can be formally absorbed by a suitable linear transformation of independent variables (see, for example, (2.14) and (2.15)). In the discrete case, however, such manipulation is not applicable since the chain rule does not work effectively. Actually the similar parameter appears in a nontrivial manner in the deformation of discrete curves as shown in Section 3.

Isogonal deformation of discrete curves
In this section, we consider the discrete deformation of discrete plane curves under the similarity geometry, which naturally gives rise to the discrete Burgers equation and its hierarchy. For the definition and fundamental properties of the discrete Burgers hierarchy, the readers may refer to Appendix B.

Discrete curve
For a map γ : Z → R 2 , n → γ n , if any consecutive three points γ n+1 , γ n , γ n−1 are not colinear, we call γ a discrete plane curve. For a discrete plane curve γ, we denote by q n the distance between the adjacent vertices q n = |γ n+1 − γ n | .
We introduce κ n as the angle between the two vectors γ n − γ n−1 , γ n+1 − γ n . More precisely, we define κ : where R is the rotation matrix Moreover, we put and introduce the map φ : Z → CO (2) by We call the map φ the similarity Frenet frame of the discrete plane curve γ.
Proposition 3.1. The similairity Frenet frame φ satisfies the linear difference equation Since the rotation matrix R (κ n+1 ) and the matrix φ n commute with each other, the statement is proved.

General settings
We next consider the deformation of the curves. We write the deformed curve as γ, and we also express the data associated with γ by putting . For instance, we define the function κ : Lemma 3.2. The necessary and sufficient condition for the deformation γ → γ being isogonal, namely, κ = κ, is that there exist a positive-valued function H and a constant a satisfying Proof. Since both T and T are planar vectors, it is obvious that there exist a positive-valued function H and an angle a such that T n = H n R (a n ) T n = H n φ n cos a n sin a n .
Therefore the equality κ = κ holds if and only if the angle a is independent of n.
Proposition 3.3. We fix δ ∈ R >0 , a, f 0 , g 0 ∈ R and a positive-valued function H. We introduce the functions f, g by the recursion relation

4)
and define the deformation γ → γ by γ n = γ n − δ ( f n T n + g n N n ) .

(3.5)
Then we have the following: (1) The deformation is isogonal. Namely, for the angle κ n defined by (3.2), we have κ n = κ n .
(2) The similarity Frenet frame φ of the discrete curve γ can be expresed in terms the frame φ of γ as Proof. We compute the difference of γ by using (3.5), (3.1) and (3.4) Then we have (3.3), which means κ = κ. The frame of γ satisfies which completes the proof.
(2) The similarity Frenet frames φ m , φ m+1 satisfy the system of linear difference equations (3.12) The length q m n = γ m n+1 − γ m n satisfy the linear difference equation

Discrete Burgers flow of higher order
Let us write down the deformation equation corresponding to (2.12). From (3.6) we have that We (3.14) Equation (3.13) or (3.14) is the general form of the deformation equation of the discrete curves in the framework of the similarity geometry, and is regarded as a discrete counterpart of (2.12).  (3) u m n satisfies (3.14). We note that (3.14) yields the discrete Burgers equation and its generalizations to that of higherorder by suitable specialization of g m n .
Autonomous case In the case of κ n = = const., (3.14) is reduced to where Ω (2) n is the recursion operator of the discrete Burgers hierarchy given in (B.6). Putting g n = sin 2 and a m = 0, (3.16) recovers the autonomous discrete Burgers equation (3.12). Equation (3.17) is a discrete counterpart of (2.12). Therefore, due to (B.5), by putting g m n as , which corresponds to (2.14).
Non-autonomous case For the case of generic κ n , we see that the recursion operator of the nonautonomous discrete Burgers hierarchy appears in the right hand side of (3.14). In fact, we have 1 sin κ n+1 u m n e ∂ n − by parametrizing sin κ n as which is a non-autonomous discrete analogue of the Burgers equation (2.9). If we set a m = 0, we obtain a simpler version of the non-autonomous discrete Burgers equation For i > 0, we put g m n = K (i) n [u m n ] and find that u m n satisfies a variant of non-autonomous higher-order discrete Burgers equation , (3.18) which is a non-autonomous discrete analogue of the higher-order Burgers equation (2.14). Note that q m n = |γ m n+1 − γ m n | satisfies the linear equation We now prove Theorem 3.6. The statement (2) and (3) are derived immediately by solving (3.6) and using the compatibility condition (3.10). For the statement (1), we have the following as a sufficient condition for the positivity of H m n : Lemma 3.7. We assume that κ n satisfies 0 < κ n < π or −π < κ n < 0 for all n. For each m, we choose δ m and a m in the following manner: Then (3.20) can be solved formally as Noticing that δ m , u m n > 0, it is sufficient for H m n > 0 that all of the following conditions sin κ n+1 1 + δ m U m n > 0, (3.22) n ν=0 − sin a m sin κ ν+1 sin(κ ν+1 − a m ) sin κ ν > 0, (3.23) are satisfied for all n. Then it is easy to see that (3.22) is satisfied by choosing δ m as (3.19). The conditions (3.21) and (3.22) imply sin κ n > 0, sin(κ n − a m ) > 0, sin a m < 0 for ∀ n, or sin κ n < 0, sin(κ n − a m ) < 0, sin a m > 0 for ∀ n, (3.24) from which we have 0 < κ n < π, κ max − a m < π, −π < a m < 0, or − π < κ n < 0, −π < κ min − a m , 0 < a m < π.

(3.25)
This is equivalent to the second condition in (3.19).

Explicit formula
An explicit representation formula for the curve γ m n is constructed in a similar manner to the smooth curves.

A Burgers hierarchy
The Burgers hierarchy is the family of nonlinear partial differential equations obtained from the linear partial differential equations the nonlinear equations in the hierachy are expressed [2] as ∂u Some of the flows of the hierarchy are given by An elementary calculation shows the following relation between K i [u] and K i−1 [u]: Here, Ω is called the recursion operator of the Burgers hierarchy, by which the equations in the hierarchy can be expressed as where Here ∆ is a central-difference operator in n defined as For instance, the first few K (i) are given by The discrete Burgers hierarchy admits the recursion operators which generate higher order flows from lower ones.

B.2 Non-autonomous discrete Burgers hierarchy
In order to formulate the non-autonomous discrete Burgers hierarchy, we first introduce the family of linear difference equations for q m n = q(x n , t m ), δ m = t m+1 − t m : The first few examples of L (i) n [q m n ] are given by We note that the following recursion relations hold: In particular, we have K (i+2) n u m n = Ω (2,i+2) n K (i) n u m n , where (B.14) Proof. The first half of the statement follows from the recursion relation of the divided differences.