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

TL;DR

This paper explores log-aesthetic curves within similarity geometry, linking them to integrable flows governed by the Burgers equation, and introduces a variational discretization method that preserves their geometric structure for high-quality curve generation.

Contribution

It introduces a novel variational principle and integrable discretization for log-aesthetic curves based on similarity geometry, enabling efficient and high-quality discrete curve generation.

Findings

Discrete curves can be generated with preserved integrable structure.

The method produces high-quality S-shaped and C-shaped curves.

Discrete curves are self-adaptive and effective with few points.

Abstract

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the 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 formulation developed here, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for the generation of discrete log-aesthetic curves for given ${\rm G}^1$ data based on the similarity geometry. Our method is able to generate $S$-shaped discrete curves with an inflection as well as $C$-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with a small number of points.