We digitized the side profiles of 2221 eggs of six species of poultry, two species of Anatidae (Anas platyrhynchos domesticus and Anser cygnoides domesticus) and four species of Phasianidae (Alectoris chukar domesticus, Coturnix japonica domesticus, Gallus gallus domesticus, and Phasianus colchicus domesticus), and used an explicit re-expression of Preston’s equation (referred to as EPE hereinafter) to fit the planar coordinates of each egg profile. By using the estimated parameters, we calculated the volume and surface area of each egg. The black_white.rar file saves the black and white .bmp images at 600 dpi for the 2221 egg profiles. The edge_data.zip file includes the planar coordinates (in cm) of each of the 2221 egg profiles, and the EPE.csv file includes the estimated parameters and goodness-of-fit of the EPE, and the egg surface area and volume calculated by the surface area and volume equations of the solid of revolution based on the EPE for each egg. In addition, the EPE.csv file also provided the measured volumes of 120 A. cygnoides eggs, and 366 P. colchicus eggs using the graduated cylinders. For each species of poultry, we boiled 120 eggs that were randomly sampled, and then measured the mass of albumen, yolk, and shell of boiled eggs. The results were saved in the EPE.csv file.

We selected six species of poultry for study because of the availability in large numbers of their eggs: two species of Anatidae (Anas platyrhynchos domesticus and Anser cygnoides domesticus), and four species of Phasianidae (Alectoris chukar domesticus, Coturnix japonica domesticus, Gallus gallus domesticus, and Phasianus colchicus domesticus). For each species, 351−390 eggs were selected for detailed study. We used an adjustable tabletop phone-mount to hold a smartphone (Huawei P30Pro, Huawei, Dongguan, China) to photograph eggs of A. platyrhynchos, A. cygnoides, C. japonica, and G. gallus, and another smartphone (Redmi K40S, Xiaomi, Kunshan, China) to photograph eggs of A. chukar and P. colchicus. Over 2200 eggs (2221 in total, >360 of each of the six species on average) were photographed at a constant scale to determine the representative 2-D egg-profiles of each species. To focus the camera on the center of each egg, we estimated the midpoint of the length of each profile. In addition, we prepared a test tube rack and a small beaker as a concave base to support each egg to make the mid-line of each profile orthogonal to the tabletop phone mount. In addition, we measured each egg’s length to provide a correction factor for the actual size from its image size.

The egg images were converted to black and white .bmp files with Photoshop (version 13.0; Adobe, San Jose, CA, USA). The Matlab (version ≥ 2009a; MathWorks, Natick, MA, USA) procedures developed by Shi et al. (2018) and Su et al. (2019) were used to extract the planar coordinates of each egg profile. Each egg profile was characterized by 2000 approximately equidistantly spaced coordinates using the ‘adjdata’ function of the ‘biogeom’ package (version 1.3.5; Shi et al., 2022) in R (version 4.2.0; R Core Team, 2022). And then we used a re-expression of Preston’s equation (referred to as EPE hereinafter; Shi et al., 2023) to fit the planar coordinate data of the egg profiles. The ‘fitEPE’ function in the ‘biogeom’ package (version 1.3.5) in R (version 4.2.0) was used to fit the data points to estimate the values of a, b, c₁, c₂, and c₃ (i.e., the parameters of the EPE) by minimizing the residual sum of squares (RSS) between the observed and predicted y-values using the Nelder-Mead optimization method. After obtaining the estimated parameters of the EPE, we used the surface area and volume equations for the solid of revolution (Narushin et al., 2022) based on the EPE to calculate the egg surface area and volume for each egg of the 2221 eggs.

We measured the V of 120 A. cygnoides eggs using a 1000 mL glass graduated cylinder with a diameter of 6.7 cm, and the V of 366 P. colchicus eggs using a 250 mL glass graduated cylinder with a diameter of 4 cm. We photographed 360 A. cygnoides eggs and 367 P. colchicus eggs. Because it is time-consuming to measure the V of an egg using the graduated cylinder method, we randomly sampled 120 A. cygnoides eggs and all P. colchicus eggs (apart from a broken egg in the experiment), and measured the volumes using the graduated cylinders. The predicted volumes were compared to those measured using graduated cylinders for a total of 120 + 366 = 486 eggs, comprising 120 of A. cygnoides and 366 of P. colchicus.

In addition, for each of the six species of poultry, we boiled 120 eggs, and then measured the mass of albumen, yolk, and shell of boiled eggs using an electronic balance (Type: ML 204; Mettler Toledo Company, Greifensee, Switzerland; measurement accuracy 0.0001 g). The results were saved in the EPE.csv file.

The above contents were cited from Shi et al. (2023).

References

• Narushin, V. G., Romanov, M. N., Mishra, B., & Griffin, D. K. (2022). Mathematical progression of avian egg shape with associated area and volume determinations. Annals of the New York Academy of Sciences, 1513: 65–78.
• R Core Team. (2022). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
• Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., & Niklas, K.J. (2023). A simple way to calculate the volume and surface area of avian eggs. Annals of the New York Academy of Sciences, in press.
• Shi, P., Gielis, J., Quinn, B. K., Niklas, K. J., Ratkowsky, D. A., Schrader, J., Ruan, H., Wang, L., & Niinemets, Ü. (2022). ‘biogeom’: An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences, 1516: 123–134.
• Shi, P., Ratkowsky, D. A., Li, Y., Zhang, L., Lin, S., & Gielis, J. (2018). A general leaf area geometric formula exists for plants—Evidence from the simplified Gielis equation. Forests, 9: 714.
• Su, J., Niklas, K. J., Huang, W., Yu, X., Yang, Y., & Shi, P. (2019). Lamina shape does not correlate with lamina surface area: An analysis based on the simplified Gielis equation. Global Ecology and Conservation, 19: e00666.