Surface growth kinematics via local curve evolution
 Derek E. Moulton,
 Alain Goriely
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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.
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 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
 Authors

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

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