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We consider integrable discretizations of some soliton equations associatedwith the motions of plane curves: the Wadati-Konno-Ichikawa elastic beamequation, the complex Dym equation, and the short pulse equation. They arerelated to the modified KdV or the sine-Gordon equations by the hodographtransformations. Based on the observation that the hodograph transformationsare regarded as the Euler-Lagrange transformations of the curve motions, weconstruct the discrete analogues of the hodograph transformations, which yieldintegrable discretizations of those soliton equations.

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves