Article: Analysis of heteromorph ammonoids by differential geometry

The complicated shell form of heteromorph ammonoids can be considered simply as an integration of ad hoc accretional growth of the aperture, without defining any coordinate system. This 'growing tube model' is newly developed herein, using differential geometry, and is applicable to the growth pattern of any coiled shell. Using the model, any coiled shell with a circular cross-section can be analysed and described by three differential parameters: E, radius enlarging ratio; C, standardized curvature; and T, standardized torsion. I applied the growing tube model to eight nostoceratid species of heteromorph ammonoid and analysed growth patterns and ontogenetic change; stable stages and transitional intervals are clearly recognizable during growth. Morphologies of real specimens can be produced using computer graphics. A particular advantage of the model is that perfect similitude is kept at any growth stage because growth patterns are described relative to whorl radius. This is the most appropriate model for the study of comparative morphology and function of free tubular shells.