Discrete Isoperimetric Deformation of Discrete Curves

Abstract

We consider isoperimetric deformations of discrete plane/space curves. We first give a brief review of the theory of isoperimetric deformation of smooth curves, which naturally gives rise to the modified KdV (mKdV) equation as a deformation equation of the curvature. We then present its discrete model described by the discrete mKdV equation, which is formulated as the isoperimetric equidistant deformation of discrete curves. We next give a review of isoperimetric and torsion-preserving deformation of smooth space curves with constant torsion which is described by the mKdV equation. We formulate a discrete analogue of it as the isoperimetric, torsion-preserving and equidistant deformation on the osculating planes of space discrete curves. The deformation admits two discrete flows, namely by the discrete mKdV equation and by the discrete sine-Gordon equation. We also show that one can make an arbitrary choice of two flows at each step, which is controlled by tuning the deformation parameters appropriately.