Journal of Mathematical Biology

, Volume 68, Issue 1–2, pp 81–108 | Cite as

Surface growth kinematics via local curve evolution

  • Derek E. MoultonEmail author
  • Alain Goriely


A mathematical framework is developed to model the kinematics of surface growth for objects that can be generated by evolving a curve in space, such as seashells and horns. Growth is dictated by a growth velocity vector field defined at every point on a generating curve. A local orthonormal basis is attached to each point of the generating curve and the velocity field is given in terms of the local coordinate directions, leading to a fully local and elegant mathematical structure. Several examples of increasing complexity are provided, and we demonstrate how biologically relevant structures such as logarithmic shells and horns emerge as analytical solutions of the kinematics equations with a small number of parameters that can be linked to the underlying growth process. Direct access to cell tracks and local orientation enables for connections to be made to the underlying growth process.


Biological growth Morphology Seashell Mathematical model 

Mathematics Subject Classification

92B99 74K99 53A04 



This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST), and based in part upon work supported by the National Science Foundation under grant DMS-0907773 (AG). AG is a Wolfson/Royal Society Merit Award Holder.


  1. Ackerly S (1989) Kinematics of accretionary shell growth, with examples from brachiopods and molluscs. Paleobiology 15(2):147–164Google Scholar
  2. Antman SS (1995) Nonlinear problems of elasticity. Springer, New YorkCrossRefzbMATHGoogle Scholar
  3. Ateshian G (2007) On the theory of reactive mixtures for modeling biological growth. Biomech Model Mechanobiol 6(6):423–445CrossRefGoogle Scholar
  4. Bishop RL (1975) There is more than one way to frame a curve. Am Math Month 82:246–251CrossRefzbMATHGoogle Scholar
  5. Black R, Turner S, Johnson M (1994) The early life history of Bembicium vittatum Philippi, 1846 (gastropoda: Littorinidae). Veliger 37(4):393–399Google Scholar
  6. Bobenko A, Suris Y (2008) Discrete differential geometry: integrable structure, vol 98. American Mathematical SocietyGoogle Scholar
  7. Boettiger A, Ermentrout B, Oster G (2009) The neural origins of shell structure and pattern in aquatic mollusks. Proc Natl Acad Sci 106(16):6837CrossRefGoogle Scholar
  8. Cook S (1979) The curves of life: being an account of spiral formations and their application to growth in nature, to science, and to art: with special reference to the manuscripts of Leonardo da Vinci. DoverGoogle Scholar
  9. Dera G, Eble G, Neige P, David B (2008) The flourishing diversity of models in theoretical morphology: from current practices to future macroevolutionary and bioenvironmental challenges. Paleobiology 34(3):301CrossRefGoogle Scholar
  10. Epstein M (2010) Kinetics of boundary growth. Mech Res Commun 37(5):453–457CrossRefzbMATHGoogle Scholar
  11. Fournier M, Bailleres H, Chanson B (1994) Tree biomechanics: growth, cumulative prestresses, and reorientations. Biomimetics 2:229–251Google Scholar
  12. Fowler D, Meinhardt H (1992) Modeling seashells. In: Proc SIGGRAPH, pp 379–387Google Scholar
  13. Garikipati K (2009) The kinematics of biological growth. Appl Mech Rev 62:030801CrossRefGoogle Scholar
  14. Hammer Ø, Bucher H (2005) Models for the morphogenesis of the molluscan shell. Lethaia 38(2):111–122CrossRefGoogle Scholar
  15. Hodge N, Papadopoulos P (2010) A continuum theory of surface growth. Proc R Soc A: Math Phys Eng Sci 466(2123):3135CrossRefzbMATHMathSciNetGoogle Scholar
  16. Hodge N, Papadopoulos P (2012) Continuum modeling and numerical simulation of cell motility. J Math Biol 64(7):1253–1279CrossRefMathSciNetGoogle Scholar
  17. Iijima A (2001) Growth of the intertidal snail, Monodonta labio (gastropoda, prosobranchia) on the Pacific coast of central Japan. Bull Mar Sci 68(1):27–36Google Scholar
  18. Meinhardt H (2009) The algorithmic beauty of sea shells. Springer, BerlinCrossRefGoogle Scholar
  19. Moseley H (1838) On the geometrical forms of turbinated and discoid shells. Phil Trans R Soc Lond 128:351–370CrossRefGoogle Scholar
  20. Moulton DE, Goriely A, Chirat R (2012) Mechanical growth and morphogenesis of seashells. J Theor Biol 311:69–79Google Scholar
  21. Okamoto T (1988) Analysis of heteromorph ammonoids by differential geometry. Palaeontology 31(pt 1):35–52Google Scholar
  22. Okamoto T (1988) Developmental regulation and morphological saltation in the heteromorph ammonite Nipponites. Paleobiology 14(3):272–286Google Scholar
  23. Pollack J, Hubickyj O, Bodenheimer P, Lissauer J, Podolak M, Greenzweig Y (1996) Formation of the giant planets by concurrent accretion of solids and gas. ICARUS 124(1):62–85CrossRefGoogle Scholar
  24. Raup D (1961) The geometry of coiling in gastropods. Proc Natl Acad Sci USA 47(4):602CrossRefGoogle Scholar
  25. Raup D, Michelson A (1965) Theoretical morphology of the coiled shell. Science 147(3663):1294CrossRefGoogle Scholar
  26. Rice S (1998) The bio-geometry of mollusc shells. Paleobiology 24(1):133–149Google Scholar
  27. Savazzi E (1987) Geometric and functional constraints on bivalve shell morphology. Lethaia 20(4):293–306CrossRefGoogle Scholar
  28. Savazzi E (1990) Biological aspects of theoretical shell morphology. Lethaia 23(2):195–212CrossRefGoogle Scholar
  29. Schöne B, Rodland D, Wehrmann A, Heidel B, Oschmann W, Zhang Z, Fiebig J, Beck L (2007) Combined sclerochronologic and oxygen isotope analysis of gastropod shells (Gibbula cineraria, North Sea): life-history traits and utility as a high-resolution environmental archive for kelp forests. Mar Biol 150(6):1237–1252CrossRefGoogle Scholar
  30. Skalak R, Farrow D, Hoger A (1997) Kinematics of surface growth. J Math Biol 35(8):869–907CrossRefzbMATHMathSciNetGoogle Scholar
  31. Stone J (1996) The evolution of ideas: a phylogeny of shell models. Am Nat 148(5):904–929CrossRefGoogle Scholar
  32. Thompson D (1942) On growth and form. Cambridge University Press, LondonzbMATHGoogle Scholar
  33. Tsui Y, Clyne T (1997) An analytical model for predicting residual stresses in progressively deposited coatings. Part 1: planar geometry. Thin Solid Films 306(1):23–33CrossRefGoogle Scholar
  34. Tyszka J, Topa P (2005) A new approach to modeling of foraminiferal shells. Paleobiology 31(3):522CrossRefGoogle Scholar
  35. Urdy S, Goudemand N, Bucher H, Chirat R (2010) Allometries and the morphogenesis of the molluscan shell: a quantitative and theoretical model. J Exp Zool B: Mol Dev Evol 314(4):280–302CrossRefGoogle Scholar
  36. van der Helm A, Ebell P, Bronsvoort W (1998) Modelling mollusc shells with generalized cylinders. Comput Graph 22(4):505–513CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.OCCAM, Mathematical InstituteUniversity of OxfordOxfordUK

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