Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
On Asymptotic Behavior of Blow-Up Solutions to Higher-Order Differential Equations with General Nonlinearity Read chapter Read first chapter

2018 | OriginalPaper | Chapter

The Common Descent of Biological Shape Description and Special Functions

Authors : J. Gielis, D. Caratelli, C. Moreno de Jong van Coevorden, P. E. Ricci

Published in: Differential and Difference Equations with Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Gielis transformations, with their origin in botany, are used to define square waves and trigonometric functions of higher order. They are rewritten in terms of Chebyshev polynomials. The origin of both, a uniform descriptor and the origin of orthogonal polynomials, can be traced back to a letter of Guido Grandi to Leibniz in 1713 on the mathematical description of the shape of flowers. In this way geometrical description and analytical tools are seamlessly combined.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter On Nonexistence of Solutions to Some Nonlinear Functional Differential Inequalities
next chapter Variational Iteration Method for Solving Problems with Integral Boundary Conditions
Literature
1.
Gielis, J.: A generic geometric transformation that unifies a wide range of natural and abstract shape. Am. J. Bot. (2003)
2.
Gielis, J., Haesen, S., Verstraelen, L.: Universal shapes: from the supereggs of Piet Hein to the cosmic egg of George Lemaitre. Kragujev. J. Math. 28, 55–67 (2005)MathSciNet
3.
Fougerolle, Y.D., Gribok, A., Foufou, S., et al.: Radial supershapes for solid modeling. J. Comput. Sci. Technol. 21(2), 238–243 (2006)CrossRef
4.
Natalini, P., Patrizi, R., Ricci, P.E.: The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface. Numer. Algorithms 49(1–4), 299–313 (2008)MathSciNetCrossRef
5.
Caratelli, D., Germano, B., Gielis, J., He, M.X., Natalini, P., Ricci, P.E.: Fourier solution of the Dirichlet problem for the Laplace and Helmholtz equations in starlike domains. Lecture Notes of Tbilisi International Centre of Mathematics and Informatics. Tbilisi University Press (2010)
6.
7.
8.
Caratelli, D., Natalini, P., Ricci, P.E.: Spherical harmonic solution of the robin problem for the Laplace equation in supershaped shells. In: Modeling in Mathematics, pp. 17–30. Atlantis Press, Paris (2017)CrossRef
9.
Gielis, J., Beirinckx, B., Bastiaens E.: Superquadrics with rational and irrational symmetries. In: Elber, G., Shapiro, V. (eds.) Proceedings of the 8\(^{th}\) ACM Symposium on Solid Modeling and Applications, Seattle, pp. 262-265, June 16–20, (2003)
10.
Tavkhelidze, I., Cassisa, C., Gielis, J., Ricci, P.E.: About bulky links, which are generated by generalized Mobius-Listing bodies. Rendiconti di Matematica dell’Accademia dei Lincei 24(1), 11–38 (2012)MATH
11.
Koiso, M., Palmer, B.: Anisotropic capillary surfaces. In: Dillen, F., Simon, U., Vrancken, L. (eds.) Symposium on the Differential Geometry of Submanifolds, Valenciennes, July 2007, pp. 185-196 (2007)
12.
Koiso, M., Palmer, B.: Equilibria for anisotropic energies and the Gielis Formula. FORMA Vol 22, Society for Science on Form, Japan Vol. 23 (No. 1) (2008)
13.
Koiso, M., Palmer, B.: Rolling constructions for anisotropic Delaunay surfaces. Pac. J. Math. 2008(2), 345–378 (2008)MathSciNetMATH
14.
Haesen, S., Nistor, A.-I., Verstraelen, L.: On growth and form and geometry I. Kragujev. J. Math. 36(1), 5–25 (2012)MathSciNetMATH
15.
16.
Shi, P.J., Xu, Q., Sandhu, H.S., Gielis, J., Ding, Y.L., Li, H.R., Dong, X.B.: Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant. In: Ecology and Evolution (2015). https://​doi.​org/​10.​1002/​ece3.​1728CrossRef
17.
De Tommasi, E., Gielis, J., Rogato, A.: Diatom frustule morphogenesis and function: a multidisciplinary survey. In: Marine Genomics (2017)
18.
Huclova, S., Erni, D., Frohlich, J.: Modeling effective dielectric properties of materials containing diverse types of biological cells. J. Phys. D Appl. Phys. 43(36) (2010)CrossRef
19.
20.
Richardson, J.S., et al.: RNA Backbone: Consensus all-angle conformers and modular string nomenclature. RNA 14, 465–481 (2008) . Published by Cold Spring Harbor Laboratory PressCrossRef
21.
Faisal, T.R., Abad, E.M.K., Hristozov, N., Pasini, D.: The impact of tissue morphology, cross-section and turgor pressure on the mechanical properties of the leaf petiole in plants. J. Bionic Eng. 7(1), S11–S23 (2010)CrossRef
22.
Faisal, T.R., Hristozov, N., Western, T.L., Rey, A., Pasini, D.: The twist-to-bend compliance of the Rheum rhabarbarum petiole: integrated computations and experiments. Comput. Methods Biomech. Biomed. Eng. 20(4), 343–354 (2017)CrossRef
23.
Lopes, D.S., Silva, M.T., Ambrosio, J.A., Flores, P.: A mathematical framework for rigid contact detection between quadric and superquadric surfaces. Multibody Sys. Dyn. 24(3), 255–280 (2010)MathSciNetCrossRef
24.
25.
Kuri-Morales, A., Bobadilla, Ea.: Clustering with an N-dimensional extension of gielis superformula. In: 7th WSEAS International Conference on Artificial Intelligence, Knowledge Engineering and Databases (AIKED’08), University of Cambridge, UK, Feb 20–22, 2008
26.
Acton, S.T., Ray, N.: Biomedical image analysis: segmentation. Synth. Lect. Image Video Multimed. Process. 4(1), 1–108 (2009)CrossRef
27.
Hadjidemetriou, S., Reichardt, W., Buechert, M., Hennig, J., Von Elverfeldt, D.: Analysis of MR Images of mice in preclinical treatment monitoring of polycystic kidney disease. In: Yang, et al., (eds.) MICCAI 2009, Part II, LCNS 5762, pp. 665–672 (2009)CrossRef
28.
29.
30.
31.
Preen, R.J., Bull, L.: Design mining interacting wind turbines: surrogate-assisted coevolution of rapid prototyped VAWT prototyped VAWT (2014). Arxiv
32.
Johnson, J.E., Starkey, R.P., Lewis, M.J.A.: Aerodynamic stability of reentry heat shield shapes for a crew exploration vehicle. J. Spacecr. Rocket. 43(4), 721–730 (2006)CrossRef
33.
Johnson, J., Lewis, M., Starkey, R.: Entry heat shield optimization for mars return. In: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition (2013)
34.
Vasie, M., Andre, L.A.: Noncircular gear design and generation by rack cutter. The Annals of Dunarea de Jos University of Galai. Fascicle V, Technologies in Machine Building (2012). ISSN 1221-4566, 2011
35.
Vasie, M., Andrei, L.: Analysis of noncircular gear meshing. Mechanical Testing and Diagnosis, pp. 70–78 (2012). ISSN 2247-9635
36.
Demasi, L., Dipace, A., Monegato, G., Cavallaro, R.: Invariant formulation for the minimum induced drag conditions of nonplanar systems. AIAA J. (2014). https://​arc.​aiaa.​org
37.
Vinogradov, S., Wilson, C.: Scattering of an E-polarized plane wave by elongated cylinder of arbitrary cross-section. In: 6th International Conference on Antenna Theory and Techniques, pp. 110–112. 17–21 Sept. 2007
38.
Caratelli, D., Simeoni, M.: Time-domain radiation properties of supershaped dielectric resonator antennas. In: 2011 IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting Will Be Held Jointly July 3–8, 2011, at the Spokane Convention Center and the Davenport Hotel in Spokane, Washington, USA (2011)
39.
40.
Paraforou, V., Tran, D., Caratelli, D.: A dual-band supershaped annular slotted patch antenna for WLAN systems. In: 8th European Conference on Antennas and Propagation (EuCAP), pp. 2365–2367, 6–11 April 2014
41.
42.
Silva, P.F., Freire, R.C.S., Serres, A.J.R., Silva, P.D.F., Silva, J.C.: Wearable textile bioinspired antenna for 2G, 3G, and 4G systems. Microw. Opt. Technol. Lett. 58(12), 2818–2823 (2016)CrossRef
43.
Silva Júnior, P.F., Freire, R.C.S., Serres, A.J.R., Catunda, S.Y., Silva, P.D.F.: Bioinspired transparent antenna for WLAN application in 5 GHz. Microw. Opt. Technol. Lett. 59(11), 2879–2884 (2017)CrossRef
44.
Rodrigo, J.A.: Fast optoelectric printing of plasmonic nanoparticles into tailored circuits. Sci. Rep. 7 (2017)CrossRef
45.
Codemard, C.A., Malinowski, A., Zervas, M.N.: Numerical optimisation of pump absorption in doped double-clad fiber with transverse and longitudinal perturbation. In: Proceedings of SPIE Vol, vol. 10083, pp. 1008315-1 (2017)
46.
47.
D’Arcy Thompson: On Growth and Form. Cambridge (1917)
48.
Grandi, G.: Letter to Leibniz. In: Leibnizens Mathematische Schriften Band IV, C.I. Gerhardt, pp. 221–224 (1713)
49.
Grandi, G.: Flores geometrici ex Rhodonearum et Cloeliarum curvarum descriptionibus resultantes. Florence (1728)
50.
Jean, R.V.: Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge University Press, Cambridge (2009)
51.
Janner, A.: Symmetry-adapted digital modeling II. The double-helix B-DNA. Acta Crystallogr. Sect. A: Found. Adv.72(3), 312–323 (2016)MathSciNetCrossRef
52.
Zaslavsky, G.M., Sagdeev, R.Z., Usikov, D.A., Chernikov, A.A.: Weak Chaos and Quasi-Regular Patterns. Cambridge University Press, Cambridge (1992)CrossRef
53.
Gielis, J., Caratelli, D., Fougerolle, Y., Ricci, P.E., Gerats, T.: A biogeometrical model for corolla fusion in asclepiad flowers. Atlantis Transactions in Geometry, vol. 2, pp. 83–106. Atlantis-Springer (2017)
54.
MacLellan, B.: \(L_{p}\) Circular Functions. University of Tennessee (1992)
55.
Lindqvist, P.: The remarkable sine and cosine functions. Ricerce Matematica Vol XLIV, fasc \(2^{\circ }\), 269–290 (1995)
56.
Lenjou, K.: Krommen en Oppervlakken van Lamé en Gielis: van de formule van Pythagoras tot de superformule. Master’s Thesis, University of Louvain, Departement of Mathematics (2005)
57.
Dattoli, G., Di Palma, E., Nguyen, F., Sabia, E.: Generalized trigonometric functions and elementary applications. Int. J. Appl. Comput. Math. 3(2), 445–458 (2017)MathSciNetCrossRef
58.
Gielis, J., Natalini, P., Ricci, P.E.: A note about generalized forms of the Gielis formula. In: Modeling in Mathematics, pp. 107–116. Atlantis Press, Paris (2017)CrossRef
59.
Rivlin, T.J.: The Chebyshev polynomials. Pure and Applied Mathematics. A Wiley Interscience Series of texts, monographs and tracts (1974)
60.
Belingeri, C., Ben, Cheikh Y., Ricci, P.E.: A set of multi-variable polynomials generalizing the Gegenbauer polynomials. Lecture notes of the Seminario Interdisciplinare di Matematica. 9, 173–186 (2010)
61.
Tavkhelidze, I., Cassisa, C., Ricci, P.E.: About connection of the generalized Möbius Listing’s surfaces with sets of knots and links. Lecture Notes of Seminario Interdisciplinare di Matematica 9, 187–200 (2010)
62.
Tavkhelidze, I., Caratelli, D., Gielis, J., Ricci, P.E., Rogava, M., Transirico, M.: On a geometric model of bodies with “Complex” configuration and some movements. In: Modeling in Mathematics, pp. 129–158. Atlantis Press, Paris (2017)CrossRef
63.
Metadata
Title
The Common Descent of Biological Shape Description and Special Functions
Authors
J. Gielis
D. Caratelli
C. Moreno de Jong van Coevorden
P. E. Ricci
Copyright Year
2018
Book
Differential and Difference Equations with Applications

Print ISBN: 978-3-319-75646-2

Electronic ISBN: 978-3-319-75647-9

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-75647-9_10

Premium Partner

    Image Credits