Universal Natural Shapes: From Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems

Gielis, Johan; Caratelli, Diego; Fougerolle, Yohan; Ricci, Paolo Emilio; Tavkelidze, Ilia; Geräts, Tom

doi:10.1371/journal.pone.0029324

Cited by 20 publications

(13 citation statements)

References 33 publications

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“…For plants with nonsymmetric leaves, more complicated mathematical models are needed, such as elliptic Fourier analysis or summations of eq. into k ‐type functions (Gielis et al., ). For asymmetrical leaf bases, adjusting the condition

r (φ) = r (- φ)

could be considered.…”

Section: Discussionmentioning

confidence: 99%

A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation

Lin

Zhang

Reddy

et al. 2016

Ecology and Evolution

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The size and shape of plant leaves change with growth, and an accurate description of leaf shape is crucial for describing plant morphogenesis and development. Bilateral symmetry, which has been widely observed but poorly examined, occurs in both dicot and monocot leaves, including all nominated bamboo species (approximately 1,300 species), of which at least 500 are found in China. Although there are apparent differences in leaf size among bamboo species due to genetic and environmental profiles, bamboo leaves have bilateral symmetry with parallel venation and appear similar across species. Here, we investigate whether the shape of bamboo leaves can be accurately described by a simplified Gielis equation, which consists of only two parameters (leaf length and shape) and produces a perfect bilateral shape. To test the applicability of this equation and the occurrence of bilateral symmetry, we first measured the leaf length of 42 bamboo species, examining >500 leaves per species. We then scanned 30 leaves per species that had approximately the same length as the median leaf length for that species. The leaf‐shape data from scanned profiles were fitted to the simplified Gielis equation. Results confirmed that the equation fits the leaf‐shape data extremely well, with the coefficients of determination being 0.995 on average. We further demonstrated the bilateral symmetry of bamboo leaves, with a clearly defined leaf‐shape parameter of all 42 bamboo species investigated ranging from 0.02 to 0.1. This results in a simple and reliable tool for precise determination of bamboo species, with applications in forestry, ecology, and taxonomy.

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r (φ) = r (- φ)

could be considered.…”

Section: Discussionmentioning

confidence: 99%

A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation

Lin

Zhang

Reddy

et al. 2016

Ecology and Evolution

Self Cite

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“…We assume that such equations may comprise an equation of Gabriel Lame's curve, more widely known as a "superellipse" [9,10], and Johan Gielis' equation [11][12][13], known also a "superformula".…”

Section: Methodsmentioning

confidence: 99%

“…For evaluations of SPF technology transition from one stage to another is very often due to alternation of the system of coordinates, original assumptions, boundary conditions and the absence of interconnectivity of the indices comprising the equation (2). It makes the calculations more complicated, reducing their precision, it being in spite of all another prerequisite for application of Lame's [9,10] and Gielis' [11][12][13] formulas for solution of the problems of approximating the formed blanks' contours.…”

Section: Fig 2 the Design Scheme Of The Superplastic Forming Processmentioning

confidence: 99%

“…Nowadays an active search is going on for precise mathematical methods of describing bodies shapes and forming at a different stage of impression-free and dieless methods of sheet and bulk workpieces performing [4][5][6][7][8]. Possibilities of new mathematical discoveries in the field of approximate description of any existing natural contours make it possible to apply Lame's [9][10] and Gielis' [11][12][13] formulas, as they seem to be very perspective. Such investigations have not been carried out yet, it ensuring scientific novelty of the work.…”

Section: Introductionmentioning

confidence: 99%

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Application of G. Lame’s and J. Gielis’ formulas for description of shells superplastic forming

et al. 2018

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The paper shows that the contour of a sheet blank at all stages of superplastic forming can be described by application of the universal equations known as a "superformula" and "superellipse". The work provides information on the ranges of values of the coefficients entering these equations and shows that their magnitudes make it possible to additionally determine the level of superplastic properties of the workpiece metal, the shape of the shell being manufactured, the stage of superplastic forming, and the presence of additional operations to regulate the flow of the deformable metal. The paper has the results of approximation by the proposed equations of shell contours that manufactured by superplastic forming by different methods.

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“…However, none of the contributions already available in the scientific literature deals with the classical Fourier projection method [8] which has been extended in recent papers [9][10][11][12][13][14][15][16] in order to address boundary-value problems (BVPs) in simply connected starlike domains whose boundaries may be regarded as an anisotropically stretched unit circle or sphere centered at the origin.…”

Section: Introductionmentioning

confidence: 99%

Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

Caratelli¹,

Gielis²,

Tavkhelidze³

et al. 2013

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The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called "superformula" introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica © is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.

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Universal Natural Shapes: From Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems

Cited by 20 publications

References 33 publications

A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation

A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation

Application of G. Lame’s and J. Gielis’ formulas for description of shells superplastic forming

Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

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