Abstract
Many cross-sectional shapes of plants have been found to approximate a superellipse rather than an ellipse. Square bamboos, belonging to the genus Chimonobambusa (Poaceae), are a group of plants with round-edged square-like culm cross sections. The initial application of superellipses to model these culm cross sections has focused on Chimonobambusa quadrangularis (Franceschi) Makino. However, there is a need for large scale empirical data to confirm this hypothesis. In this study, approximately 750 cross sections from 30 culms of C. utilis were scanned to obtain cross-sectional boundary coordinates. A superellipse exhibits a centrosymmetry, but in nature the cross sections of culms usually deviate from a standard circle, ellipse, or superellipse because of the influences of the environment and terrain, resulting in different bending and torsion forces during growth. Thus, more natural cross-sectional shapes appear to have the form of a deformed superellipse. The superellipse equation with a deformation parameter (SEDP) was used to fit boundary data. We find that the cross-sectional shapes (including outer and inner rings) of C. utilis can be well described by SEDP. The adjusted root-mean-square error of SEDP is smaller than that of the superellipse equation without a deformation parameter. A major finding is that the cross-sectional shapes can be divided into two types of superellipse curves: hyperellipses and hypoellipses, even for cross sections from the same culm. There are two proportional relationships between ring area and the product of ring length and width for both the outer and inner rings. The proportionality coefficients are significantly different, as a consequence of the two different superellipse types (i.e., hyperellipses and hypoellipses). The difference in the proportionality coefficients between hyperellipses and hypoellipses for outer rings is greater than that for inner rings. This work informs our understanding and quantifying of the longitudinal deformation of plant stems for future studies to assess the influences of the environment on stem development. This work is also informative for understanding the deviation of natural shapes from a strict rotational symmetry.
Highlights
The superellipse equation is a generalized form of the equation of an ellipse
The reduction in RMSEadj using superellipse equation with a deformation parameter (SEDP) achieved 18.8% of the RMSEadj of SE, indicating that the introduction of the deformation parameter w significantly improved the goodness of fit
Using cross sections through the culms of C. utilis, we demonstrate that the superellipse equation with a deformation parameter (SEDP) is empirically valid in describing the shapes of the outer and inner rings of culm cross sections
Summary
The superellipse equation is a generalized form of the equation of an ellipse. It can generate the shapes of diamonds, ellipses, rectangles, and their transition shapes. The first systematic study of Symmetry 2020, 12, 2073; doi:10.3390/sym12122073 www.mdpi.com/journal/symmetry. Symmetry 2020, 12, 2073 these curves was done by Gabriel Lamé to describe crystal shapes [1]. The superellipse is defined analytically by the formula n (1). |x/α|n + y/β = 1, where x and y are the Cartesian coordinates; α and β are semidiameters of the superellipse; and n is an arbitrary positive number. For n = 1, the superellipse equation produces a diamond (or an inscribed square when α = β). For n = 2, the equation gives an ellipse (or a circle when α = β). For n < 2, the shapes are called hypoellipses, and, when n > 2, hyperellipses result [2,3] (see Figure 1 for details)
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