Surface growth kinematics via local curve evolution

Abstract

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.

Theoretical Modelling of the Molluscan Shell: What has been Learned From the Comparison Among Molluscan Taxa?

“…Also, a number of other mathematical formulations exist that are described as generalized models of molluscan growth (Fowler et al 1992;Pappas and Miller 2013;Moulton and Goriely 2014). These formulations are stemming from the study of 3D surface kinematics (Skalak et al 1997) and can provide analytical solutions when the generating curve is planar and maintains a fixed shape throughout growth (Moulton & Goriely 2014). Although mathematically convenient, these formulations do not bring further insight with regards to the study of allometry and phenotypic variation at lower phylogenetic levels or with regards to morphospace studies at high taxonomic levels.…”

“…However, given that there is almost a one-to-one cor-respondence between the parameters of these two models (C = alpha; D = delta, T = gamma, R corresponds to an angle not considered as relevant by Ackerly), Moulton et al's (2012) model won't be discussed in details. Also, a number of other mathematical formulations exist that are described as generalized models of molluscan growth (Fowler et al 1992;Pappas and Miller 2013;Moulton and Goriely 2014). These formulations are stemming from the study of 3D surface kinematics (Skalak et al 1997) and can provide analytical solutions when the generating curve is planar and maintains a fixed shape throughout growth (Moulton & Goriely 2014).…”

Surface‐dependent growth kinematics in euclidean 3‐space

“…The velocity vector can be a function of the local geometry as well as possible physical, chemical, or biological fields (rate of accretion, morphogen gradient, temperature, pH, etc.). 9 In order to describe the velocity locally, we attach to the curve r(s, 0) a local orthonormal Darboux frame…”

“…Later, Moulton et al developed an appropriate mathematical framework, which is not bound to computer algorithms to generate surfaces, to solve this problem. 9 Tug et al 10 consider a space curve instead of a planar curve, and they define the growth vector field in terms of an alternative moving frame {N, C, W} on the generating curve. This frame ideally describes the growth in the direction of the Darboux vector, 11 and it is investigated that the time dimension in the method affects the growth of the surfaces.…”

Accretive growth kinematics via null evolution curve

“…Furthermore, it is shown that biological processes of some beings, such as logarithmic crusts and horns, can be expressed by few parameters of the growth process. 14 In this context, choice of the geometry is very important while modeling the growth process of biological beings. Many biological structures are modeled by using the Euclidean geometry, but it is seen that mathematical modeling of these structures can be expressed by simpler mathematical equations by using more suitable geometries.…”