  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.
