A Parsimonious, Accurate, Predictive Model for Avian Egg Geometry.

Over the years, many attempts have been made to mathematically describe an ovoid, both from a purely mathematical standpoint as well as to inform the underlying biological process of egg formation. Most existing formulations are complex and non-predictive, requiring each individual shape to be fitted separately. The aim of this study was to develop a parsimonious, elegant, and predictive model for the full range of avian egg geometry-one based on simple geometric components and a minimal number of parameters. This approach constructs an ovoid from two ellipses and a straight line, governed by only three adjustable parameters. The resulting model is predictive: it can accurately reconstruct the full contour of an avian egg using only the coordinate of maximum breadth and the breadth at one-quarter of the egg's length from the pointed end. The mean error of this method is 0.047 (RMSE), which is competitive with other comparable methods. This accuracy arises because the geometric invariance of the model closely matches the allometric invariance observed in real avian eggs. Moreover, the ovoids generated by this method mostly conform closely to the Main Axiom, derived from Fermat's extremum theory, making the model particularly suitable for engineers, designers, and ornithologists who require a straightforward means of constructing realistic ovoid shapes.