# Discrete Models of Isoperimetric Deformation of Plane Curves

## Abstract

We consider the isoperimetric deformation of smooth curves on the Euclidean plane. It naturally gives rise to a nonlinear partial differential equation called the modified KdV(mKdV) equation as a deformation equation of the curvature, which is known as one of the most typical example of the soliton equations or the integrable systems. The Frenet equation and the deformation equation of the Frenet frame of the curve are the auxiliary linear problem or the Lax pair of the mKdV equation. Based on this formulation, we present two discrete models of isoperimetric deformation of plane curves preserving underlying integrable structure: the discrete deformation described by the discrete mKdV equation and the continuous deformation described by the semi-discrete mKdV equation.

### Keywords

Plane curve Discrete plane curve Isoperimetric deformation mKdV equation Semi-discrete mKdV equation Discrete mKdV equation### References

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