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

Abstract

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 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.

Fairing of planar curves to log-aesthetic curves

“…From a different point of view, LAC have been characterized as curves that are obtained via a variational principle in the framework of similarity geometry; moreover, they can also be seen as invariant curves under the integrable flow on plane curves governed by the Burgers equation [9]. This fact was also shown to be useful in providing an integrable discretization of the LAC that preserves the underlying geometric structure [8]. These previous contributions are mainly focused on providing tools for curve generation with fixed boundary conditions.…”