We digitized the side profiles of 574 fruits of Canarium album (Lour.) DC., and used an explicit re-expression of Preston’s equation (referred to as EPE hereinafter) to fit the planar coordinates of each fruit profile. By using the estimated parameters, we calculated the volume and surface area of each fruit. The black_white.zip file saves the black and white .bmp images at 600 dpi for the 574 fruit profiles. The edge_data.zip file includes the planar coordinates of each of the 574 fruits, and the EPE_results.csv file includes the estimated parameters and goodness-of-fit of the EPE, and the fruit surface area and volume calculated by the surface area and volume formulas of the solid of revolution based on the EPE for each fruit. In addition, the EPE_results.csv file also provided the measured volumes of 574 fruits using graduated cylinders.

574 Canarium album (Lour.) DC. fruits were selected for detailed study. Each fruit was photographed using a smartphone (Huawei Nova5Pro, Guangzhou, China), mounted on an adjustable tabletop phone mount, while placing the fruit on a metal rack to ensure that the mid-line (i.e., the line through the two endpoints on the fruit profile) was parallel to the desktop and the surface of the smartphone. To calibrate the deviation of the image size of each fruit from its actual size, we measured the maximum length of the fruit using a vernier caliper.

To obtain the planar coordinates of fruit 2D profiles, the photographs were converted into black-white images and saved as .bmp format using Adobe Photoshop CS2 (version 9.0; Adobe, San Jose, CA, USA). A program developed by Shi et al. (2018) and Su et al. (2019) developed in Matlab (version >= 2009a; MathWorks, Natick, MA, USA) was then used to extract the planar coordinates of each fruit profile from the corresponding .bmp black-white image. The function ‘fitEPE’ in the package ‘biogeom’ (version 1.3.5) (Shi et al., 2022) in R (version 4.2.1) (R Core Team, 2022) was used to estimate the numerical values of a, b, c₁, c₂, and c₃. The Nelder-Mead optimization method was used to minimize the residual sum of squares (RSS) between the observed and predicted Y-values of C. album fruit profiles. 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 fruit surface area and volume for each fruit of the 574 fruits.

To compare the predicted volume of a fruit with its observed volume, we measured volume by submerging each fruit in water in a 250 mL graduated cylinder with a diameter of 4 cm and reading the observed volume of displaced water. The results were saved in the EPE_results.csv file.

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

References

• Narushin VG, Romanov MN, Mishra B, Griffin DK. 2022. Mathematical progression of avian egg shape with associated area and volume determinations. Annals of the New York Academy of Sciences 1513(1): 65–78.
• R Core Team. 2022. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Website: https://www.r-project.org
• Shi P, Gielis J, Quinn BK, Niklas KJ, Ratkowsky DA, Schrader J, Ruan H, Wang L,  Niinemets Ü. 2022b. ‘biogeom’: An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516(1): 123–134.
• Shi P, Ratkowsky D, 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(11):714.
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• Wang L, Shi P, Chen L, Gielis J, Niklas KJ. 2023. Evidence that Chinese white olive (Canarium album (Lour.) DC.) fruits are solids of revolution. Botany Letters In press. https://doi.org/10.1080/23818107.2023.2238020