A Mathematical Model for Mollusc Shells Based on Parametric Surfaces and the Construction of Theoretical Morphospaces

Abstract

In this study, we propose a mathematical model based on parametric surfaces for the shell morphology of the phylum Mollusca. Since David Raup’s pioneering works, many mathematical models have been proposed for different contexts to describe general shell morphology; however, there has been a gap in the practicality of models that allow the estimation of their parameter values in real specimens. Our model collects ideas from previous pioneering studies; it rests on the equation of the logarithmic spiral, uses a fixed coordinate system (coiling axis), and defines the position of the generating curve with a local moving system using the Frenet frame. However, it improves upon previous models by applying apex formation, rotations, and substantially different parameter definitions. Furthermore, the most conspicuous improvement is the development of a simple and standardized methodology to obtain the six theoretical parameters from shell images from different mollusc classes and to generate useful theoretical morphospaces. The model was applied to reproduce the shape of real mollusc-shell specimens from Gasteropoda, Cephaloda and Bivalvia, which represent important classes in geological time. We propose a specific methodology to obtain the parameters in four morphological groups: helicoidal, planispiral, conic, and valve-like shells, thereby demonstrating that the model offers an adequate representation of real shells. Finally, possible improvements to the model are discussed along with further work. Based on the above considerations, the capacity of the model to allow the construction of theoretical morphospaces, the methodology to estimate parameters and from the comparison between several existing models for shells, we believe that our model can contribute to future research on the development, diversity and evolutionary processes that generated the diversity in mollusc shells.