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arXiv · 2505.20713

The Klein-Lie W-curves as a new class of aesthetic curves based on self-affinity

Shun Kumagai·Kenji Kajiwara
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Abstract

In this paper, we consider planar curves and present a family of planar curves characterized by a symmetry called the extendable self-affinity (ESA). The ESA has been recognized through the investigation of the symmetry of the log-aesthetic curve (LAC), which has been studied as a reference for designing aesthetic shapes in CAGD and regarded as an analog of Euler's elastica in similarity geometry. We investigate the ESA and show that it gives rise to affine geometry and the Klein-Lie W-curve. Based on the observations on logarithmic curvature graphs, we propose the Klein-Lie W-curve as a new family of "aesthetic curves" in affine geometry, to be considered together with the LAC.

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Shun Kumagai, Kenji Kajiwara. 2026-08-24. The Klein-Lie W-curves as a new class of aesthetic curves based on self-affinity. https://arxiv.org/abs/2505.20713

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