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<p>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 a 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). The family of curves determined by two self-affinities are captured by the proposed self-affinity.</p>

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年8月14日

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  • Shun Kumagai,
  • Kenji Kajiwara

摘要

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 a 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). The family of curves determined by two self-affinities are captured by the proposed self-affinity.