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Generalization of log-aesthetic curves via similarity geometry

Abstract

The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as \(G^n\) Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation.

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Both pictures were adapted from [29]

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Acknowledgements

The first author was supported by JSPS KAKENHI, JP15K04834 grant. The authors would like to thank Prof. Rebecca Ramnauth of the Department of Computer Science at Long Island University (USA), who has generously volunteered her valuable time to substantively edit and review this paper. Her care, competence, and conscientiousness are much appreciated. Moreover, the issues, remarks, and very important suggestions of the anonymous reviewers which helped to improve the quality of this paper are appreciated.

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Correspondence to Jun-ichi Inoguchi.

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Inoguchi, Ji., Ziatdinov, R. & Miura, K.T. Generalization of log-aesthetic curves via similarity geometry. Japan J. Indust. Appl. Math. 36, 239–259 (2019). https://doi.org/10.1007/s13160-018-0335-7

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Keywords

  • Log-aesthetic curve
  • Superspiral
  • Similarity geometry
  • Similarity curvature
  • Riccati differential equation

Mathematics Subject Classification

  • 65D17
  • 68U07
  • 53A35
  • 53A04