# A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Cross-Section Sampling and Preparation

#### 2.2. Image Processing and Boundary Data Acquisition

#### 2.3. Fitting the Superellipse Equation and the Introduction of a Deformation Parameter

_{0}, y

_{0}and θ need to be fitted, where (x

_{0}, y

_{0}) is the translational pole and θ is the angle between the straight line where semidiameter α lies and the x-axis. For a standard superellipse, the pole’s coordinates are (0, 0), and θ = 0.

**M**:

**N**represent the matrix that preserves the x- and y-coordinates of a superellipse curve generated by Equation (2):

**M**by

**N**:

**G**=

**MN**,

**G**saves the x- and y-coordinates of the deformed superellipse curve. In the polar coordinate system, the deformed superellipse is:

#### 2.4. Analysis of the Fitted Results

_{adj}) [12,13]:

_{norm}is the normalized |w| and c is a base that can render all or most groups of the w

_{norm}values to pass the test of normality. To find this base, we set c to range between 1.1 to 200 in 0.1 increments to calculate the p values of the Shapiro—Wilk test of normality [16]. We selected the c value such that all or most groups of the w

_{norm}values passed the test of normality. The Tukey’s HSD test was then used to test whether there were significant differences in the normalized |w| values among the eight groups of the combinations of n and k for the outer and inner rings.

## 3. Results

_{adj}) ranged from 0.0026 to 0.0299 with a median of 0.0078 (Figure 4A; Table S1). All RMSE values were less than 5% of the effective radius (=$\sqrt{A/\pi}$). A comparison of the normalized data showed that SEDP had a smaller normalized RMSE

_{adj}value than the superellipse equation without the deformation parameter (SE) for each cross section (Figure 4B). On average, the reduction in RMSE

_{adj}using SEDP achieved 18.8% of the RMSE

_{adj}of SE, indicating that the introduction of the deformation parameter w significantly improved the goodness of fit. Figure 5 shows the fitted results for the six cross-sectional examples in Figure 2 for the outer and inner rings. The fitted results for other cross sections are provided in Table S1.

## 4. Discussion

**M**is more than sufficient to describe the deviation of actual cross sections of C. utilis from a standard superellipse. To add additional parameters appears to be unnecessary and can decrease the close-to-linear performance of nonlinear regression [14]. However, for other objects, whether it is necessary to add additional deformation parameters requires further study.

_{1}, n

_{2}, n

_{3}are constants to be fitted, and m should be a positive integer that controls the number of angles of the supercurve (which was generated by the superformula). This was used to study the mechanical properties of petioles of Philodendron melinonii Brongn. ex Regel and Rheum rhabarbarum L. [21,22], and the current study provides a simpler model for the study of square bamboos. Equation (12) can be rewritten as:

_{e}as:

_{e}, which can be described as a linear relationship on a log-log plot. Equation (13) has been demonstrated to be valid in describing many natural shapes including the leaves of many plants [3,9,24,25,26], and the seeds of Ginkgo biloba L. [23]. However, it was found to be invalid in describing several shapes of starfish. Thus, Shi et al. [13] put forward another superformula based on Equation (13) by changing the linear relationship between r and r

_{e}on a log-log plot to a hyperbolic relationship on a log-log plot, i.e.,

_{0}, γ

_{1}and γ

_{2}are constants to be fitted. Equation (15) was found to fit the shapes of eight actual starfish specimens well [13]. In actual data fit, Equations (13) and (15) appear to be more valid for the symmetrical shapes including bilateral symmetry and centrosymmetry than those asymmetrical shapes. For instance, some leaves tend to exhibit a degree of bilateral asymmetry because of the influence of environments and the aboveground architecture of plants [27,28,29]. The deviation from a standard symmetrical shape largely limits the applicability of a supercurve to describe natural shapes. The deviation in nature is usually not anisotropic, and it tends to be a linear transformation from a standard symmetrical shape (e.g., the leaf shape of Sapium sebiferum (L.) Roxb.), which has been earlier noted by Thompson [30]. However, such a linear transformation in natural shapes has been little studied after Thompson. The present study demonstrated that the deformation in cross-sectional shapes of C. utilis stalks could be quantified by introducing a parameter w as a linear transformation from a standard centrosymmetrical superellipse.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of the Superellipse Equation with the Deformation Parameter w

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**Figure 1.**Simulated superellipses. α = 1, β = 0.85, and n ranges from 0.01 to 10 in 0.01 increments. Five simulated superellipses with n = 0.5, 1, 1.5, 2 and 2.5 are shown with isolines.

**Figure 2.**Six examples of the culm cross sections of C. utilis (Keng) Keng. The samples (

**A**–

**F**) were sampled from the same site (Tongzi County, Guizhou Province, P.R. China) in September of 2020. The aboveground heights of sections (

**A**–

**F**) were 26.50, 40.85, 30.85, 20.00, 36.60 and 24.45 cm, respectively.

**Figure 3.**Four examples of the deformation in superellipses. (

**A**) n = 1.7; α = 1; k = 0.85; w = −0.2 and 0.2; (

**B**) n = 2.5; α = 1; k = 0.85; w = −0.2 and 0.2. Panel (

**A**) exhibits two deformed hypoellipses, and panel (

**B**) exhibits two deformed hyperellipses. The sign of w does not affect the degree of deformation; it merely designates the direction of deformation.

**Figure 4.**(

**A**) The frequency distribution of the adjusted root-mean-square errors using the superellipse equation with a deformation parameter (SEDP) and (

**B**) a comparison in the log-transformed RMSE

_{adj}between the superellipse equation without a deformation parameter (SE) and with a deformation parameter (SEDP). In panel B, a and b were used to show the significance of the difference in the mean log-transformed RMSE

_{adj}between the two equations based on the Tukey’s HSD test. Group SE has a larger mean log-transformed RMSE

_{adj}than group SEDP.

**Figure 5.**Fitted curves for the six actual cross-sectional examples of C. utilis using the superellipse equation with a deformation parameter. The gray curves are the actual outer and inner rings; the red curves are fitted curves. The panels (

**A**–

**F**) correspond to the actual scanned images in Figure 2.

**Figure 6.**Boxplot for the normalized deformation parameters for the four superellipse types (S1: n ≥ 2 and k ≥ 1; S2: n ≥ 2 and k < 1; S3: n < 2 and k ≥ 1; S4: n < 2 and k < 1) for outer and inner rings. The dark solids in the box represent the median of data; the red snowflakes represent means. The letters of a, b, c and d in the upper whiskers denote the significance of the difference in the mean normalized deformation parameters between any two groups. The same letter indicates an insignificant difference between the two groups. Here, a represents the group with the largest mean normalized deformation parameter, and d represents the group with the smallest mean normalized deformation parameter.

**Figure 7.**Bimodal distribution of the n values and their influence on the proportional relationship between the ring area and the product of the ring length and width. (

**A**,

**B**) The frequencies of the n values in outer and inner rings, and (

**C**) their comparison using a boxplot. There was a significant difference in the mean n value between the outer and inner rings, the mean n of the group marked by a is significantly larger than that marked by b. (

**D**) For the outer rings, the data apparently exhibited two separate linear trends according to the n values, and (

**E**) the similar separation in the data occurred for the inner rings. However, the extent of the data separation for outer rings is larger than that for inner rings (

**D**vs.

**E**). The red small open circles are observations; the straight lines were not fitted by linear regressions, and they were derived from the superellipse area formula (i.e., Equation (11)). The slopes of the four straight lines in (

**D**,

**E**) are both equal to 1.

**Table 1.**Comparison between the theoretical intercept based on the superellipse area formula and the estimated intercept based on the linear regression.

Ring Type | Data Range | Theoretical Intercept ^{1} | Estimated Intercept | LCI | UCI | RMSE |
---|---|---|---|---|---|---|

Outer | n ≥ 2 | 0.8421 | 0.8246 | 0.8235 | 0.8257 | 0.0134 |

Outer | n < 2 | 0.7282 | 0.7291 | 0.7279 | 0.7303 | 0.0158 |

Inner | n ≥ 2 | 0.8071 | 0.7994 | 0.7986 | 0.8001 | 0.0092 |

Inner | n < 2 | 0.7637 | 0.7600 | 0.7593 | 0.7607 | 0.0087 |

^{1}The theoretical intercept represents the value derived from the superellipse area formula (Equation (11)); the estimated intercept represents the value based on the linear regression; LCI and UCI represent the lower and upper bounds of the 95% confidence intervals of the estimated intercept; and RMSE represents the root-mean-square error of the linear regression.

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**MDPI and ACS Style**

Huang, W.; Li, Y.; Niklas, K.J.; Gielis, J.; Ding, Y.; Cao, L.; Shi, P.
A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo. *Symmetry* **2020**, *12*, 2073.
https://doi.org/10.3390/sym12122073

**AMA Style**

Huang W, Li Y, Niklas KJ, Gielis J, Ding Y, Cao L, Shi P.
A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo. *Symmetry*. 2020; 12(12):2073.
https://doi.org/10.3390/sym12122073

**Chicago/Turabian Style**

Huang, Weiwei, Yueyi Li, Karl J. Niklas, Johan Gielis, Yongyan Ding, Li Cao, and Peijian Shi.
2020. "A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo" *Symmetry* 12, no. 12: 2073.
https://doi.org/10.3390/sym12122073