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Gielis’ superformula and regular polygons

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Abstract

Circles, ellipses, squares, rectangles, rhombuses and cross shapes are all special cases of Lamé curves (also known as superellipses). As a further generalization of Lamé curves, the Belgian botanist Johan Gielis introduced the notion of the so-called superformula with a view to the application to modeling and understanding the shapes of plants and animals. Despite the fact that Gielis’ superformula is expressed by a single simple equation, it can describe a wide range of various shapes, including, for example, triangle-like shapes, star-like shapes, flower-like shapes and so on. So far, it seems that most of the studies about Gielis curves (the curves generated by Gielis’ superformula) are application-oriented. In this paper, we examine precisely and analytically the mathematical structure of Gielis curves from a theoretical point of view. The original equation of the superformula has six parameters, which is too many to deal with at once. Therefore, we focus on a restricted case where the number of the parameters is reduced to three. In particular, we analyze the curvature at the “corners” and the midpoint of the “sides” of Gielis curves. We also derive the limit curves of Gielis curves and compare them with regular polygons.

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Authors and Affiliations

  1. Department of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama, 790-8577, Japan

    Masaya Matsuura

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  1. Masaya Matsuura
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Correspondence to Masaya Matsuura.

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Matsuura, M. Gielis’ superformula and regular polygons. J. Geom. 106, 383–403 (2015). https://doi.org/10.1007/s00022-015-0269-z

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  • DOI: https://doi.org/10.1007/s00022-015-0269-z

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