Semi-discrete linear hyperbolic polyharmonic flows of closed polygons

TL;DR

This work defines and analyzes a semi-discrete, second-order in time linear hyperbolic flow for polygons: $\frac{d^2\vec{X}}{dt^2} + \beta \frac{d\vec{X}}{dt} = (-1)^{m+1} M^m\vec{X}$, with $M$ circulant and amenable to Fourier diagonalization. It proves that arbitrary polygons can be evolved to a prescribed target polygon (Yau-type problem) and, after suitable rescaling, converge to an affine image of a regular $n$-gon; with $\beta>0$ the flow decays exponentially to a point, with the asymptotic shape governed by dominant eigenmodes with eigenvalues $\lambda_{m,k} = -4^m[\sin^2(\pi k/n)]^m$. The paper provides explicit planar and higher-codimension self-similar solutions via the $P_k$ modes, and extends the analysis to higher codimension polygons, showing analogous convergence behavior. It also introduces a semi-discrete Yau-type difference flow between polygons, proving long-time existence and convergence to the target, and discusses handling mismatched vertex counts and moving targets, illustrating the results with representative cases. These results advance understanding of linear hyperbolic flows in discrete polygon geometry and suggest new discrete curvature-flow analogies with potential applications.

Abstract

We consider the damped hyperbolic motion of polygons by a linear semi-discrete analogue of polyharmonic curve diffusion. We show that such flows may transition any polygon to any other polygon, reminiscent of the Yau problem of evolving one curve to another by a curvature flow, before converging exponentially to a point that, under appropriate rescaling, is a planar basis polygon. We also consider a hyperbolic linear semi-discrete flow of the Yau curvature difference flow, where a polygonal curve is able to flow to any other such that we get convergence to the target polygon in infinite time.

Figures (4)

• Figure 1: Evolution of a pentagon under the semi-discrete hyperbolic polyharmonic flow for different values of $m$ and where we have a damping term $\beta = 4.$ All arbitrary constants $a_k$ are chosen to be zero. Distinct time steps of the evolution are shown superimposed over the initial polygon. The same time step values are used for each case of $m$. The polygon at time step $t=1.2$ is shown in red in each case which is to provide comparison to Figure \ref{['fig:general_n=5_intermediate']} where constants $a_k$ are chosen so that the polygon first flows to a intermediate polygon at time $t=1.2$ before shrinking to a point.
• Figure 2: Evolution of a pentagon under the semi-discrete hyperbolic polyharmonic flow for different values of $m$ and where we have a damping term $\beta = 4.$ The constants $a_k$ are chosen so that at time $t=1.2$ we have $\vec{X}(1.2) = \alpha_0^0P_0+ 3P_1$. Distinct time steps of the evolution are shown superimposed over the initial polygon with the intermediate prescribed polygon at time $t=1.2$ shown in red. The same time step values are used for each case of $m$.
• Figure 3: Different cases of pentagons flowing to regular pentagons under the semi-discrete Yau difference flow. In each case, selected time steps of the evolution are shown superimposed over the initial and target polygons. The initial polygon is given in blue and the target polygon in orange. The target polygon in this case is $5P_1.$ In (c), the arbitrary coefficients $a_k$ are prescribed such that the polygon flows to an intermediate polygon at a particular time, depicted in red. In this case the intermediate polygon is $3P_1$.
• Figure 4: Different cases of pentagons flowing under the semi-discrete Yau difference flow. In each case, selected time steps of the evolution are shown superimposed over the initial and target polygons. The initial polygon is given in blue, and the target polygon in orange. (a) depicts a quadrilateral flowing to a regular pentagon. (b) demonstrates a pentagon flowing to a triangle by duplicating excess vertices for the target polygon (c) demonstrates a pentagon flowing to a triangle by setting excess vertices to lie on the edges of the triangle.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• Lemma 3.4