# A New Program to Estimate the Parameters of Preston’s Equation, a General Formula for Describing the Egg Shape of Birds

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{P}and y

_{P}are the abscissa and ordinate coordinates in the plane of an arbitrary point on an egg’s boundary; a, b, c

_{1}, c

_{2}, and c

_{3}are constants to be estimated, among which a and b represent half the egg’s length and approximately half the egg’s maximum breadth, respectively; and ζ is an angle ranging from 0 to 2π. We refer to Equation (1) as Preston’s equation (PE) for convenience hereinafter. There is a need to note that the mid-line of a simulated egg by the PE is aligned to the y-axis on which the egg base is uppermost and the egg tip is lowermost (Figure 1). The PE is flexible enough to produce curves representing a wide variety of egg shapes by setting different combinations of values to its parameters (Figure 1), and the x and y coordinates on the curve plotted by the PE can be known at a given value of ζ. Nevertheless, ζ is not a polar angle, and this means that $\mathrm{tan}\mathsf{\zeta}\ne {y}_{P}/{x}_{P}$. To estimate the parameters of the PE, Todd and Smart [7] re-expressed the PE as:

_{P}and y

_{P}can be obtained by digitizing the image, and the maximum distance between two points on the egg’s profile is regarded as the egg’s length. In this case, the numerical value of a is known, as it then equals half of the maximum distance between the two points. This then allows the variables z

_{0}to z

_{3}to be calculated. The parameters d

_{0}to d

_{3}can be estimated using a multiple linear regression procedure based on ordinary (or weighted) least-squares, with the intercept set to 0 [5,6]. We refer to Equation (3) as the Todd-Smart equation (TSE).

## 2. Materials and Methods

#### 2.1. Egg Samples and Image Processing

#### 2.2. Models and Data-Fitting Approaches

_{0}, y

_{0}), and the angle between the mid-line and the x-axis as θ, which then formed the three location parameters for curve-fitting [3,8]. There are then five model parameters to be estimated for the EPE: a, b, c

_{1}, c

_{2}, and c

_{3}. The Nelder–Mead optimization method [14] was used to minimize the residual sum of squares (RSS) between the observed and predicted y values on the egg’s profile. We refer to this method as optimPE hereinafter.

## 3. Results

## 4. Discussion

#### 4.1. The Optimization Versus Multiple Linear Regression Method for Estimating Parameters of Preston’s Equation

#### 4.2. The Potential Extension of Preston’s Equation to Other Egg Shapes

#### 4.3. Recommendations for Field Biologists

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Egg shapes simulated using Preston’s equation (PE). Here, a = 9, b = 6, c

_{2}= −0.04, c

_{3}= 0.02, and c

_{1}ranges from 0.1 to 0.5, in 0.1 increments. The x and y coordinates of the Preston curves are obtained when ζ ranges from 0 to 2π.

**Figure 2.**Results of fitting the egg boundaries of nine species of birds using the explicit Preston equation (EPE) with the Nelder–Mead optimization method (optimPE). Panels (

**A**–

**I**) represent the fitted results for the nine representative species of birds. In each panel, the gray curve represents the observed egg perimeter, and the red curve represents the egg perimeter predicted by the explicit Preston equation fit using the optimization method. RMSE is the root-mean-square error between the observed and predicted y values.

**Figure 3.**Results of fitting the egg boundaries of nine species of birds using the Todd-Smart equation (TSE) (i.e., the re-expression of Preston’s equation) with the multiple linear regression method (lmPE). Panels (

**A**–

**I**) represent the fitted results for the nine representative species of birds. In each panel, the gray curve represents the observed egg perimeter, and the red curve represents the egg perimeter predicted by the Todd-Smart equation (re-expressed Preston’s equation) based on the multiple linear regression method. Note that the coordinates were scaled by a to compare the goodness of fit achieved using this method with the goodness of fit using the explicit Preston equation and the optimization method (see Figure 2). RMSE is the root-mean-square error between the observed and predicted y values.

**Figure 4.**Boxplot of the root-mean-square errors (RMSEs, adjusted by dividing by half the egg’s maximum breadth) compared between two methods (optimPE, and lmPE) for the 50 egg-shape data sets from [5]. Here, optimPE represents the explicit Preston equation fit based on the Nelder–Mead optimization method; and lmPE represents the Todd-Smart equation (i.e., the re-expression of the Preston equation) fit based on the multiple linear regression method. The 50 eggs were divided into two types: pyriform and other shapes (n = 25 each), as in [5]. Significant differences between the two methods using the paired t-test at the 0.05 significance level were found for both types of egg shape. The vertical solid line in each box represents the median; the whiskers extend to the most extreme data point, which is no more than 1.5 times the interquartile range from the box. The three asterisks represent p < 0.001.

**Figure 5.**An insect egg shape simulated using the explicit Preston equation and its deformed shapes generated with the parabolic deformation function. In panel (

**A**), the parameters in the explicit Preston equation are θ = π/2, a = 9, b = 6, c

_{1}= 0.4, c

_{2}= −0.04, and c

_{3}= 0. In panel (

**B**), the parameters are the same as those in panel (A), but there is an additional parabolic deformation function as follows: ${x}^{\prime}=x+0.05{\left(7-x\right)}^{2}$, and ${y}^{\prime}=y$, where x′ and y′ are the coordinates of the deformed egg shape.

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**MDPI and ACS Style**

Shi, P.; Wang, L.; Quinn, B.K.; Gielis, J.
A New Program to Estimate the Parameters of Preston’s Equation, a General Formula for Describing the Egg Shape of Birds. *Symmetry* **2023**, *15*, 231.
https://doi.org/10.3390/sym15010231

**AMA Style**

Shi P, Wang L, Quinn BK, Gielis J.
A New Program to Estimate the Parameters of Preston’s Equation, a General Formula for Describing the Egg Shape of Birds. *Symmetry*. 2023; 15(1):231.
https://doi.org/10.3390/sym15010231

**Chicago/Turabian Style**

Shi, Peijian, Lin Wang, Brady K. Quinn, and Johan Gielis.
2023. "A New Program to Estimate the Parameters of Preston’s Equation, a General Formula for Describing the Egg Shape of Birds" *Symmetry* 15, no. 1: 231.
https://doi.org/10.3390/sym15010231