Find out how to access previewonly content
Date:
18 Nov 2012
Surface growth kinematics via local curve evolution
 Derek E. Moulton,
 Alain Goriely
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
References
Ackerly S (1989) Kinematics of accretionary shell growth, with examples from brachiopods and molluscs. Paleobiology 15(2):147–164
Ateshian G (2007) On the theory of reactive mixtures for modeling biological growth. Biomech Model Mechanobiol 6(6):423–445CrossRef
Black R, Turner S, Johnson M (1994) The early life history of Bembicium vittatum Philippi, 1846 (gastropoda: Littorinidae). Veliger 37(4):393–399
Bobenko A, Suris Y (2008) Discrete differential geometry: integrable structure, vol 98. American Mathematical Society
Boettiger A, Ermentrout B, Oster G (2009) The neural origins of shell structure and pattern in aquatic mollusks. Proc Natl Acad Sci 106(16):6837CrossRef
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. Dover
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):301CrossRef
Fournier M, Bailleres H, Chanson B (1994) Tree biomechanics: growth, cumulative prestresses, and reorientations. Biomimetics 2:229–251
Fowler D, Meinhardt H (1992) Modeling seashells. In: Proc SIGGRAPH, pp 379–387
Garikipati K (2009) The kinematics of biological growth. Appl Mech Rev 62:030801CrossRef
Hammer Ø, Bucher H (2005) Models for the morphogenesis of the molluscan shell. Lethaia 38(2):111–122CrossRef
Hodge N, Papadopoulos P (2010) A continuum theory of surface growth. Proc R Soc A: Math Phys Eng Sci 466(2123):3135CrossRefMATHMathSciNet
Hodge N, Papadopoulos P (2012) Continuum modeling and numerical simulation of cell motility. J Math Biol 64(7):1253–1279CrossRefMathSciNet
Iijima A (2001) Growth of the intertidal snail, Monodonta labio (gastropoda, prosobranchia) on the Pacific coast of central Japan. Bull Mar Sci 68(1):27–36
Meinhardt H (2009) The algorithmic beauty of sea shells. Springer, BerlinCrossRef
Moseley H (1838) On the geometrical forms of turbinated and discoid shells. Phil Trans R Soc Lond 128:351–370CrossRef
Moulton DE, Goriely A, Chirat R (2012) Mechanical growth and morphogenesis of seashells. J Theor Biol 311:69–79
Okamoto T (1988) Analysis of heteromorph ammonoids by differential geometry. Palaeontology 31(pt 1):35–52
Okamoto T (1988) Developmental regulation and morphological saltation in the heteromorph ammonite Nipponites. Paleobiology 14(3):272–286
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):62–85CrossRef
Raup D (1961) The geometry of coiling in gastropods. Proc Natl Acad Sci USA 47(4):602CrossRef
Raup D, Michelson A (1965) Theoretical morphology of the coiled shell. Science 147(3663):1294CrossRef
Rice S (1998) The biogeometry of mollusc shells. Paleobiology 24(1):133–149
Savazzi E (1987) Geometric and functional constraints on bivalve shell morphology. Lethaia 20(4):293–306CrossRef
Savazzi E (1990) Biological aspects of theoretical shell morphology. Lethaia 23(2):195–212CrossRef
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): lifehistory traits and utility as a highresolution environmental archive for kelp forests. Mar Biol 150(6):1237–1252CrossRef
Skalak R, Farrow D, Hoger A (1997) Kinematics of surface growth. J Math Biol 35(8):869–907CrossRefMATHMathSciNet
Stone J (1996) The evolution of ideas: a phylogeny of shell models. Am Nat 148(5):904–929CrossRef
Thompson D (1942) On growth and form. Cambridge University Press, LondonMATH
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):23–33CrossRef
Tyszka J, Topa P (2005) A new approach to modeling of foraminiferal shells. Paleobiology 31(3):522CrossRef
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):280–302CrossRef
van der Helm A, Ebell P, Bronsvoort W (1998) Modelling mollusc shells with generalized cylinders. Comput Graph 22(4):505–513CrossRef
 Title
 Surface growth kinematics via local curve evolution
 Journal

Journal of Mathematical Biology
Volume 68, Issue 12 , pp 81108
 Cover Date
 20140101
 DOI
 10.1007/s0028501206257
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Biological growth
 Morphology
 Seashell
 Mathematical model
 92B99
 74K99
 53A04
 Industry Sectors
 Authors

 Derek E. Moulton ^{(1)}
 Alain Goriely ^{(1)}
 Author Affiliations

 1. OCCAM, Mathematical Institute, University of Oxford, Oxford, UK