# Explicit Preston’s Equation Describes the Geometries of Egg-Shaped Tomato Cultivars and Its Potential for Estimating the Volume and Surface Area

## TL;DR

This study uses a mathematical model to describe the shape of egg-shaped tomatoes and estimate their volume and surface area from 2D photos.

## Contribution

The study introduces a low-cost, non-destructive method using explicit Preston’s equation to estimate tomato volume and surface area.

## Key findings

- EPE effectively fits the 2D profiles of tomatoes, with truss tomatoes fitting best, followed by cherry and Qianxi.
- For cherry and Qianxi tomatoes, surface area is proportional to observed volume with exponents of 0.62–0.63.
- The method reveals that biological evolution shapes organisms for functional optimization rather than geometric similarity.

## Abstract

In nature, some tomato (Solanum lycopersicum) shapes appear to be ellipsoidal. This study aims to fit the ellipsoid tomato profile using explicit Preston’s equation (EPE), and calculate its volume (Vpred) and surface area (S) based on the estimated EPE’s parameters. This method offers low-cost and non-destructive advantages compared to three-dimensional (3D) scanning. A total of 917 tomatoes from three cultivars were photographed, and the two-dimensional (2D) boundary coordinates of each fruit profile were digitized and then fitted using EPE. The results demonstrated that the EPE effectively fitted the tomato 2D-profile, with truss tomato ranking highest, followed by cherry, and then Qianxi. A significant relationship was found between Vpred and observed volume (Vobs) at the cultivar level. The 95% confidence intervals for the slopes for cherry tomatoes include 1.0, and for Qianxi were close to 1.0, which confirmed that these two cultivars were solids of revolution. Additionally, for cherry and Qianxi tomato, S is proportional to the Vobs (i.e., S∝Vobs0.62~0.63), Vpred is proportional to (LW2)0.73~0.74, and S is proportional to (LW2)0.49 (L is the length and W is the maximum width). For any isometrically scaling solid of revolution, the theoretical exponent of surface area to volume is exactly 2/3. The observed exponent of 0.62–0.63 is a biological reality, which reveals that evolution has shaped organisms not for geometric similarity, but for functional optimization. This study can be extended to a geometry study on other egg-shaped fruits and provides a potentially simple method for calculating volume and surface area based on photographed 2D fruit profiles.