A novel model for eggs like pears: How to quantify them geometrically with a mathematical formula of two parameters?

Of the variety of bird egg shapes, perhaps the most fascinating and unusual are pyriform (pear-shaped, or conical) eggs. Among oologists, there is still no consensus on what exactly caused this evolutionary and ecological adaptation. To address this, our research was aimed to develop a minimalistic mathematical model for an accurate description of the pyriform egg contour, using the minimum number of measurements. As such, egg length (L) and its maximum breadth (B) were found to be an optimal set of parameters that were easy enough to measure with the required accuracy. We tested four analytical premises that can be used for successful pyriform egg shape modelling. To validate these four model premises, images of pyriform eggs characteristic of 32 species were used. As a result, we derived a novel mathematical dependence that we called the 'pyriform model with two parameters'. Based on this model, it is feasible to geometrically reconstruct any pyriform egg profile under study using only two egg measurements, i.e., L and B. Since pyriform eggs are characteristic of wild bird species that are most often investigated in the field, the measurement of only two parameters minimizes the time spent and, accordingly, the stress factor on the animals. The least error estimate for the new model was 3.9%, which turned out to be even more accurate than that of the previously developed model with three parameters.