Smart and smarter: improving on a classic egg shape model.

Smart's model (SM) describing the geometry of avian eggs is, uniquely, based on physiological characteristics of eggs formation in oviduct walls transforming a sphere to an ellipsoid, to an ovoid. The purpose of this study was to revisit and perform a more in-depth examination of SM, providing a possible improvement in terms of reducing the number of initial parameters and compliance with geometric principles fundamental for bodies of revolution. SM requires measuring five egg parameters: length (L), maximum breadth (B), displacement of the central axis to the level of maximum breadth (w), and two radii of the egg at a point shifted by ¼L from the pointed (r) and blunt (R) ends, respectively. A practical test for the reproduction degree of three egg shape varieties using five-parameter model confirmed its maximum accuracy compared to all others. Modifications using four parameters (L, B, w and r or B₀, which is egg diameter at ½L) were also relatively accurate, and only slightly inferior. Using three parameters (L, B and w) was clearly insufficient; however, one of our three-parameter models met the requirements of the "Main Axiom of the mathematical formula of the bird's egg". In our opinion, two of Smart's postulates, the point of applying an oviduct force to provide the appropriate egg shape and the equality of L and the length of original ellipsoid, were used as fixed initial premises, which allowed to exclude many other possible options and to derive a mathematical model. Such an assumption arose according to the theoretical studies presented herein. Nevertheless, Smart's formula derivation based on physiology of egg formation is a pioneering approach to the development of egg-shape models.