Log-aesthetic Curves as Similarity Geometric Analogue of Euler's Elasticae

TL;DR

This paper explores log-aesthetic curves within similarity geometry, showing they are governed by the stationary Burgers equation and can be viewed as the geometric analogue of Euler's elasticae.

Contribution

It introduces a variational formulation linking log-aesthetic curves to integrable flows and characterizes them as similarity geometric analogues of Euler's elasticae.

Findings

Log-aesthetic curves are stationary solutions of the Burgers equation.

A variational principle for these curves is established.

They are identified as the similarity geometric analogue of Euler's elasticae.

Abstract

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler-Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.