Self-affinities of planar curves: towards unified description of aesthetic curves

TL;DR

This paper explores the self-affinity properties of planar curves, providing a unified framework that characterizes various aesthetic and geometric curves, including log-aesthetic, parabolas, and quadratic curves, in different geometries.

Contribution

It reformulates existing self-affinities, proves their characterizations, and introduces a new self-affinity that unifies the description of constant curvature curves across geometries.

Findings

Self-affinity characterizes log-aesthetic curves.

A new self-affinity unifies description of quadratic curves.

Constant curvature curves are characterized in multiple geometries.

Abstract

In this paper, we consider the self-affinity of planar curves. It is regarded as an important property to characterize the log-aesthetic curves which have been studied as reference curves or guidelines for designing aesthetic shapes in CAD systems. We reformulate the two different self-affinities proposed in the development of log-aesthetic curves. We give rigorous proof that one self-affinity actually characterizes log-aesthetic curves, while another one characterizes parabolas. We then propose a new self-affinity which, in equiaffine geometry, characterizes the constant curvature curves (the quadratic curves). It integrates the two self-affinities, by which constant curvature curves in similarity and equiaffine geometries are characterized in a unified manner.