Comparison of two polar equations in describing the geometries of domestic pigeon (Columba livia domestica) eggs

Two-dimensional (2D) egg-shape equations are potent mathematical tools, facilitating the description of avian egg geometries in their applied mathematical modelling and poultry science implementations. They aid in the precise quantification of avian egg sizes, including traits such as volume (V) and surface area (S). Despite their potential, however, polar coordinate egg-shape equations have received relatively little attention for practical applications in oomorphology. This may be attributed to their complex model structure and the absence of explicit geometric interpretations for the equation parameters. In the present study, 2 distinct polar equations, namely the Carter-Morley Jones equation (CMJE) and simplified Gielis equation (SGE), were used to fit the profile geometries of 415 domestic pigeon (Columba livia domestica) eggs based on nonlinear least squares regression methods. The adequacy of goodness-of-fit for each nonlinear egg-shape equation was evaluated through the adjusted root-mean-square error (RMSE_adj), while relative curvature measures of nonlinearity were utilized to assess the nonlinear behavior of equations. All of the RMSE_adj values of the 2 polar equations were lower than 0.05, which demonstrated the validity of CMJE and SGE in depicting the shapes of C. livia egg profiles. Moreover, the 2 egg-shape equations showed good nonlinear behavior across all 415 C. livia eggs. Wilcoxon signed rank tests relative to RMSE_adj values between CMJE and SGE revealed that CMJE displayed inferior fits to empirical data when compared to SGE. CMJE, however, had a better linear approximation performance than SGE at the global level. At the individual parameter level, all of the parameters of CMJE or SGE exhibited good close-to-linear behavior. This study provides an instrumental mathematical tool for the practical application of polar egg-shape equations, such as nondestructively estimating V and S of avian eggs. Additionally, it offers valuable insights into assessing nonlinear regression models for accurately describing the geometries of 2D egg profiles.