Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which is used in CAGD. We consider these curves in the context of similarity geometry and characterize them in terms of a "stationary" integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formalism developed here, we propose a discretization of both the curves and the associated variational principle which preserves the underlying integrable structure. We finally present an algorithm for the generation of discrete log-aesthetic curves for given ${\rm G}^1$ data. The computation time to generate discrete log-aesthetic curves is much shorter than that for numerical discretizations of log-aesthetic curves due to the avoidance of fine numerical integration to calculate their shapes. Instead, only coarse summation is required.