Comparison of the performance of two polar equations in describing the geometries of elliptical fruits

ABSTRACT

In nature, the two-dimensional (2D) profiles of fruits from many plants often resemble ellipses. However, it remains unclear whether these profiles strictly adhere to the ellipse equation, as many natural shapes resembling ellipses are actually better described as superellipses. The superellipse equation, which includes an additional parameter n compared to the ellipse equation, can generate a broader range of shapes, with the ellipse being just a special case of the superellipse. To investigate whether the 2D profiles of fruits are better described by ellipses or superellipses, we collected a total of 751 mature and undamaged fruits from 31 naturally growing plants of Cucumis melo L. var. agrestis Naud. Our analysis revealed that most adjusted root-mean-square errors (> 92% of the 751 fruits) for fitting the superellipse equation to the fruit profiles were consistently less than 0.0165. Furthermore, there were 638 of the 751 fruits (ca. 85%) with the 95% confidence intervals of the estimated parameter n in the superellipse equation not including 2. These findings suggest that the profiles of C. melo var. agrestis fruits align more closely with the superellipse equation than with the ellipse equation. This study provides evidence for the existence of the superellipse in fruit profiles, which has significant implications for studying fruit geometries and estimating fruit volumes using the solid of revolution formula. Furthermore, this discovery may contribute to a deeper understanding of the mechanisms driving the evolution of fruit shapes.

KEYWORDS:

• Close-to-linear behavior
• curve fitting
• field muskmelon
• fruit geometry
• superellipse

Acknowledgments

We thank Qiying Li, Zhihao Sun and Yan Zhou for their valuable assistance in the preparation of this work. We also thank the expert reviewer for providing constructive comments on the early version of this manuscript.

Author contributions

Weihao Yao: Formal analysis (equal); Investigation (equal); writing – original draft (equal). Cang Hui: Formal analysis (equal); Writing – review & editing (leading); Lin Wang: Formal analysis (equal); Writing – review & editing (equal); Jinfeng Wang: Investigation (equal); Johan Gielis: Formal analysis (equal); Methodology (equal). Peijian Shi: Methodology (equal); Supervision (leading); Writing – review & editing (equal).

Data availability statement

The raw data used in this study can be accessible on Dryad, a public repository (see He et al. 2024), https://doi.org/10.5061/dryad.mgqnk995f.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Weihao Yao

Mr. Weihao Yao is a graduate student of forestry under the supervision of Dr. Peijian Shi, whose primary area of interest is leafing intensity theory.

Cang Hui

Prof. Cang Hui is a renowned mathematical biologist, whose primary area of interest is biological invasion ecology and theoretical biology.

Lin Wang

Mr. Lin Wang is a graduate student of applied statistics under the supervision of Dr. Peijian Shi, whose primary area of interest is the linear approximation of nonlinear models.

Jinfeng Wang

Ms. Jinfeng Wang is a graduate student of ecology under the supervision of Dr. Peijian Shi, whose primary area of interest is the scaling relationships of floral parts.

Johan Gielis

Prof. Johan Gielis is a vigorous mathematician and plant taxonomist, and proposed the superformula that can describe a variety of natural geometries.

Peijian Shi

Dr. Peijian Shi is a naturalist, whose primary area of interest is bio-geometry and scaling theory.