Isogonal Deformation of Discrete Plane Curves and Discrete Burgers Hierarchy

TL;DR

This paper explores discrete deformations of plane curves within similarity geometry, linking them to the discrete Burgers hierarchy and providing explicit formulas via solutions to linear equations.

Contribution

It introduces a novel formulation of discrete curve deformations as isogonal transformations tied to the discrete Burgers hierarchy, with explicit solution formulas.

Findings

Discrete deformations are described by the discrete Burgers hierarchy.

Explicit formulas for curve deformations are constructed using linear diffusion equations.

The approach bridges discrete geometry with integrable systems.

Abstract

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.