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. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given $G^1$ data based on 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 the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

在本文中，我们考虑一类称为对数美学曲线的平面曲线及其在计算机辅助几何设计中的推广。在相似几何的框架中，这些曲线被描述为由 Burgers 方程控制的平面曲线上可积流下的不变曲线。他们还承认导致稳态伯格斯方程为欧拉-拉格朗日方程的变分公式。作为该公式的应用，我们提出了这些曲线的离散化和相关的变分原理，保留了底层的可积结构。我们最后提出了为给定的情况生成离散对数美学曲线的算法$G^1$基于相似几何的数据。我们的方法能够根据边界条件生成带拐点的S形离散曲线以及C形曲线。得到的离散曲线被认为是自适应离散化，因此即使点数很少也能达到高质量。通过连续表示，这些离散曲线为生成高质量形状提供了灵活的工具。