Geometric Analysis and Quantification of Coiling-Patterns of Gastropod Shells based on 3D and 2D Image Data

Gastropods have various forms of shells, and a coiling-pattern is a one of the most important traits. “Growing tube model” have been regarded as a general method to parameterize coiling-patterns of molluscan shells because this model is formulated by differential geometry (though Okamoto (1988) introduced it to describe shell forms of heteromorph ammonoids). Growing tube model, however, hardly used in empirical researches because it is too difficult to measure the morphological trait to estimate model’s parameters.
Hence, I introduce two practical methods to estimate growing tube model’s parameters. If available 3D CT data, we can get time series of growing tube model’s parameters by a spline curve approximation of a growing trajectory of a shell. This method makes us possible to estimate any coiling-patterns. If only 2D digital camera images can be used, we might “translate” to growing tube model’s parameters through a “measurable model” (in this talk, I adapt the Raup’s model (Raup 1962; Raup and Michelson 1965) to get growing tube model’s parameters). If growing tube model’s parameters are constant, we can formulate a dependency of growing tube model’s parameters on Raup’s model’s parameters.