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The Common Descent of Biological Shape Description and Special Functions

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Differential and Difference Equations with Applications (ICDDEA 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 230))

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.

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References

  1. Gielis, J.: A generic geometric transformation that unifies a wide range of natural and abstract shape. Am. J. Bot. (2003)

    Google Scholar 

  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  Google Scholar 

  3. Fougerolle, Y.D., Gribok, A., Foufou, S., et al.: Radial supershapes for solid modeling. J. Comput. Sci. Technol. 21(2), 238–243 (2006)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  6. Caratelli, D., Ricci, P.E., Gielis, J.: The Robin problem for the Laplace equation in a three-dimensional starlike domain. Appl. Math. Comput. (2011). https://doi.org/10.1016/j.amc.2011.03.146.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gielis, J., Caratelli, D., Fougerolle, Y., Ricci, P.E., Gerats, T.: Universal Natural Shapes: From unifying shape description to simple methods for shape analysis and boundary value problems PlosONE-D-11-01115R2 (2012). https://doi.org/10.1371/journal.pone.0029324

  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)

    Chapter  Google Scholar 

  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)

    Google Scholar 

  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  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  13. Koiso, M., Palmer, B.: Rolling constructions for anisotropic Delaunay surfaces. Pac. J. Math. 2008(2), 345–378 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Haesen, S., Nistor, A.-I., Verstraelen, L.: On growth and form and geometry I. Kragujev. J. Math. 36(1), 5–25 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Shi, P.J., et al.: Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape. Front. Plant Sci. 6, 856 (2015). https://doi.org/10.3389/fpls.2015.00856

  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/ece3.1728

    Article  Google Scholar 

  17. De Tommasi, E., Gielis, J., Rogato, A.: Diatom frustule morphogenesis and function: a multidisciplinary survey. In: Marine Genomics (2017)

    Google Scholar 

  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)

    Article  Google Scholar 

  19. Huclova, S, Erni, D., Frohlich, J.: Modelling and validation of dielectric properties of human skin in the MHz region focusing on skin layer morphology and material composition. J. Phys. D: Appl. Phys. 45, 025301 (2012). https://doi.org/10.1088/0022-3727/45/2/025301

    Article  Google Scholar 

  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 Press

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  24. Fougerolle, Y.D., Gielis, J., Truchetet, F.: A robust evolutionary algorithm for the recovery of rational Gielis curves. Pattern Recognit. 46(8), 2078–2091 (2013). https://doi.org/10.1016/j.patcog.2013.01.024

    Article  Google Scholar 

  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

    Google Scholar 

  26. Acton, S.T., Ray, N.: Biomedical image analysis: segmentation. Synth. Lect. Image Video Multimed. Process. 4(1), 1–108 (2009)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  28. Zhou, S., Huang, X., Li, Q., Xie, Y.M.: A study of shape optimization on the metallic particles for thin-film solar cells. Nanoscale Res. Lett. 8, 447 (2013). https://doi.org/10.1186/1556-276X-8-447

  29. Qusba, A., Ramrakhyani, A., So, J., Hayes, G., Dickey, M.: On the design of microfluidic implant coil for flexible telemetry system. IEEE Sens. J. 14(4), 1074–1080 (2014). www.ieeexplore.ieee.org

    Article  Google Scholar 

  30. Rodriguez-Oliveiros, R., Sanchez-Gil, Ja.: Gold nanostars as thermoplasmonic nanoparticles for optical heating. Opt. Express 20(1), 621–626 (2012). https://doi.org/10.1364/OE.20.000621

  31. Preen, R.J., Bull, L.: Design mining interacting wind turbines: surrogate-assisted coevolution of rapid prototyped VAWT prototyped VAWT (2014). Arxiv

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Google Scholar 

  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

    Google Scholar 

  35. Vasie, M., Andrei, L.: Analysis of noncircular gear meshing. Mechanical Testing and Diagnosis, pp. 70–78 (2012). ISSN 2247-9635

    Google Scholar 

  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

    Google Scholar 

  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)

    Google Scholar 

  39. Simeoni, M., Cichetti, R., Yarovoy, A., Caratelli, D.: Plastic supershapes dielectric resonator antennaes for wide-band applications. IEEE Trans. Antennas Propag. (2011). https://doi.org/10.1109/TAP.2011.2165477

  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

    Google Scholar 

  41. Bia, P., Caratelli, D., Mescia, L.: Analysis and synthesis of supershaped dielectric lens. In: IET Microwaves Antennas Propagation (2015). https://doi.org/10.1049/iet-map.2015.0091

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  44. Rodrigo, J.A.: Fast optoelectric printing of plasmonic nanoparticles into tailored circuits. Sci. Rep. 7 (2017)

    Article  Google Scholar 

  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)

    Google Scholar 

  46. Gielis, J.: The Geometrical Beauty of Plants. Atlantis-Springer (2017)

    Book  Google Scholar 

  47. D’Arcy Thompson: On Growth and Form. Cambridge (1917)

    Google Scholar 

  48. Grandi, G.: Letter to Leibniz. In: Leibnizens Mathematische Schriften Band IV, C.I. Gerhardt, pp. 221–224 (1713)

    Google Scholar 

  49. Grandi, G.: Flores geometrici ex Rhodonearum et Cloeliarum curvarum descriptionibus resultantes. Florence (1728)

    Google Scholar 

  50. Jean, R.V.: Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  51. Janner, A.: Symmetry-adapted digital modeling II. The double-helix B-DNA. Acta Crystallogr. Sect. A: Found. Adv.72(3), 312–323 (2016)

    Article  MathSciNet  Google Scholar 

  52. Zaslavsky, G.M., Sagdeev, R.Z., Usikov, D.A., Chernikov, A.A.: Weak Chaos and Quasi-Regular Patterns. Cambridge University Press, Cambridge (1992)

    Article  Google Scholar 

  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)

    Google Scholar 

  54. MacLellan, B.: \(L_{p}\) Circular Functions. University of Tennessee (1992)

    Google Scholar 

  55. Lindqvist, P.: The remarkable sine and cosine functions. Ricerce Matematica Vol XLIV, fasc \(2^{\circ }\), 269–290 (1995)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  59. Rivlin, T.J.: The Chebyshev polynomials. Pure and Applied Mathematics. A Wiley Interscience Series of texts, monographs and tracts (1974)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  63. Koseleff, P.V., Pecker, D.: Chebyshev knots. J. Knot Theory Ramif. 20(04), 575–593 (2011)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Antwerp, Groenenborgerlaan 171, 2020, Antwerp, Belgium

    J. Gielis

  2. The Antenna Company, High Tech Campus, 5656 AE, Eindhoven, The Netherlands

    J. Gielis, D. Caratelli & C. Moreno de Jong van Coevorden

  3. Tomsk Polytechnic University, 84/3 Sovetskaya Street, 634050, Tomsk, Russia

    D. Caratelli & C. Moreno de Jong van Coevorden

  4. International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186, Rome, Italy

    P. E. Ricci

Authors
  1. J. Gielis
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  2. D. Caratelli
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  3. C. Moreno de Jong van Coevorden
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  4. P. E. Ricci
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Corresponding author

Correspondence to J. Gielis .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Engenharia, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Tomás Caraballo

  3. Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  4. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA

    John R. Graef

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Gielis, J., Caratelli, D., Moreno de Jong van Coevorden, C., Ricci, P.E. (2018). The Common Descent of Biological Shape Description and Special Functions. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_10

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