# The Generalized Gielis Geometric Equation and Its Application

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, n

_{2}and n

_{3}are constants (both $\in \mathbb{R}$); positive integer m was introduced to make the curve generate arbitrary polygons (with m angles) consequently enhancing the flexibility of Lamé curves. We refer to Equation (5) as the original Gielis equation (OGE) in the following text for simplicity. OGE has been used to simulate many natural shapes, e.g., diatoms, eggs, cross sections of plants, snowflakes and starfish [1,2]. OGE also has shown its validity in describing several actual natural shapes, e.g., leaf shapes of Hydrocotyle vulgaris L., Polygonum perfoliatum L. and seed planar projections of Ginkgo biloba L. [6,7]. Furthermore, OGE can produce regular, or at least very approximately regular polygons [8,9]. When m = 1, A = B, n

_{1}= n, and n

_{2}= n

_{3}= 1, OGE has a special case:

## 2. The Generalized Gielis Equation (GGE) and Its Two New Special Cases

_{e}. We refer to this relationship as the link function, f. As this link function can take on other forms, we use the following more general expression to replace OGE:

_{e}is the polar radius of the elementary Gielis curve generated by EGE, and r is the polar radius of the generalized Gielis curve generated by GGE. Therefore, OGE is actually a special case of GGE with a power-law link function.

_{2}in Equation (13) be zero. Of course, in nature, there are forms other than the above link functions; however, these other forms also belong to the scope of GGE if there exists a clear functional expression between r and r

_{e}. Figure 1 provides a simulation example for Equation (10).

## 3. Application of the Generalized Gielis Equation

_{adj}) that can reduce the influence of the object’s size [13]:

^{2}) as an indicator because it has been considered to be problematic for reflecting the goodness of fit of a nonlinear regression [14,15].

_{e}of the eight starfish on a log–log plot. Figure 4 exhibits the fitted leaf shapes and corresponding link functions for the four leaves on a log–log plot. Table A2 in Appendix A tabulates the estimated parameters and indicators of goodness of fit.

^{2}vs. 43.22 cm

^{2}). It proves that the adjusted RMSE is more valid than RMSE in comparing the goodness of fit when there is a large difference in size between any two objects.

## 4. Discussion

_{2}≠ 0 did not improve the goodness of fit.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Sample Code | Scientific Name | Family | Locality | Sampling Time |
---|---|---|---|---|

1 | Anthenoides tenuis Liao & A.M. Clark | Goniasteridae | Philippines, Siquijor | 2019 |

2 | Culcita schmideliana Bruzelius | Oreasteridae | Philippines, Bohol. Cabulan Island | 2018 |

3 | Culcita schmideliana Bruzelius | Oreasteridae | Philippines, Bohol. Cabulan Island | 2018 |

4 | Culcita schmideliana Bruzelius | Oreasteridae | Philippines, Bohol. Cabulan Island | 2018 |

5 | Stellaster equestris Bruzelius | Goniasteridae | Philippines, Surigao | 2018 |

6 | Tosia australis Gray | Goniasteridae | Edithburgh, Australia | 1980 |

7 | Tosia magnifica Müller & Troschel | Goniasteridae | Edithburgh, Australia | 1980 |

8 | Tosia magnifica Müller & Troschel | Goniasteridae | Edithburgh, Australia | 1980 |

9 | Trachelospermum jasminoides (Lindl.) Lem. | Apocynaceae | Nanjing, China | 2019 |

10 | Vinca major L. | Apocynaceae | Nanjing, China | 2018 |

11 | Chimonanthus praecox (L.) Link | Calycanthaceae | Nanjing, China | 2017 |

12 | Phyllostachys incarnata T.H. Wen | Poaceae | Nanjing, China | 2016 |

Code | ${\widehat{x}}_{0}$ | ${\widehat{y}}_{0}$ | $\widehat{\theta}$ | $\widehat{n}$ | $\widehat{a}$ | $\widehat{b}$ | $\widehat{c}$ | Sample Size | Area (cm^{2}) | RSS | RMSE | RMSE_{adj} |

1 | 18.67 | 18.05 | 1.56 | 2006.90 | 1.1362 | 0.0599 | 0.8742 | 2488 | 32.00 | 235.22 | 0.3075 | 0.0963 |

2 | 7.75 | 7.47 | 2.85 | 8.39 | 2.0738 | 4.2943 | 0.4556 | 1482 | 13.16 | 3.8561 | 0.0510 | 0.0249 |

3 | 8.08 | 7.74 | 1.58 | 4.83 | 1.5802 | 3.7025 | 0.3311 | 1617 | 13.52 | 1.2516 | 0.0278 | 0.0134 |

4 | 14.51 | 13.94 | 0.30 | 139.04 | 2.8784 | 0.6546 | 1.1619 | 2684 | 43.22 | 8.4710 | 0.0562 | 0.0151 |

5 | 9.34 | 8.83 | 0.31 | 543.36 | 0.8644 | 0.1151 | −0.1390 | 2282 | 6.09 | 34.53 | 0.1230 | 0.0883 |

6 | 5.66 | 5.42 | 0.29 | 7.73 | 2.7866 | 6.8694 | 0.2415 | 1526 | 7.37 | 0.4804 | 0.0177 | 0.0116 |

7 | 5.11 | 4.85 | 0.29 | 4.28 | 2.3100 | 5.4630 | 0.0503 | 1263 | 6.07 | 0.5415 | 0.0207 | 0.0149 |

8 | 5.88 | 5.66 | 2.79 | 5.95 | 2.6437 | 7.5884 | 0.2371 | 1267 | 7.58 | 0.3502 | 0.0166 | 0.0107 |

Code | ${\widehat{x}}_{0}$ | ${\widehat{y}}_{0}$ | $\widehat{\theta}$ | $\widehat{n}$ | ${\widehat{\delta}}_{0}$ | ${\widehat{\delta}}_{1}$ | ${\widehat{\delta}}_{2}$ | Sample Size | Area (cm^{2}) | RSS | RMSE | RMSE_{adj} |

9 | 4.98 | 5.06 | 0.76 | 1.32 | 1.3485 | 18.8344 | 48.2690 | 1653 | 5.40 | 2.7591 | 0.0409 | 0.0312 |

10 | 7.16 | 6.73 | 0.76 | 1.14 | 1.1925 | 7.2615 | 18.3014 | 1379 | 11.65 | 1.1927 | 0.0294 | 0.0153 |

11 | 7.84 | 8.12 | 0.81 | 0.73 | 1.9303 | 1.9169 | −5.7404 | 1985 | 22.25 | 8.9448 | 0.0671 | 0.0252 |

12 | 14.67 | 15.22 | 0.81 | 1.23 | 2.8179 | 38.5217 | 0 | 2476 | 33.22 | 350.16 | 0.3761 | 0.1156 |

_{adj}is the adjusted root-mean-square error.

## References

- Gielis, J. A general geometric transformation that unifies a wide range of natural and abstract shapes. Am. J. Bot.
**2003**, 90, 333–338. [Google Scholar] [CrossRef] [PubMed] - Gielis, J. The Geometrical Beauty of Plants; Atlantis Press: Paris, France, 2017. [Google Scholar]
- Shi, P.J.; Huang, J.G.; Hui, C.; Grissino-Mayer, H.D.; Tardif, J.; Zhai, L.H.; Wang, F.S.; Li, B.L. Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape. Front. Plant Sci.
**2015**, 6, 856. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wei, Q.; Jiao, C.; Guo, L.; Ding, Y.L.; Cao, J.J.; Feng, J.Y.; Dong, X.B.; Mao, L.Y.; Sun, H.H.; Yu, F.; et al. Exploring key cellular processes and candidate genes regulating the primary thickening growth of Moso underground shoots. New Phyotol.
**2017**, 214, 81–96. [Google Scholar] [CrossRef] [PubMed] - Guo, L.; Sun, X.P.; Li, Z.R.; Wang, Y.J.; Fei, Z.J.; Chen, J.; Feng, J.Y.; Cui, D.F.; Feng, X.Y.; Ding, Y.L.; et al. Morphological dissection and cellular and transcriptome characterizations of bamboo pith cavity formation reveal a pivotal role of genes related to programmed cell death. Plant Biotechnol. J.
**2019**, 17, 982–997. [Google Scholar] [CrossRef] [PubMed] - Tian, F.; Wang, Y.J.; Sandhu, H.S.; Gielis, J.; Shi, P.J. Comparison of seed morphology of two ginkgo cultivars. J. Forest Res.
**2018**. [Google Scholar] [CrossRef] - Shi, P.J.; Liu, M.D.; Yu, X.J.; Gielis, J.; Ratkowsky, D.A. Proportional relationship between leaf area and the product of leaf length and width of four types of special leaf shapes. Forests
**2019**, 10, 178. [Google Scholar] [CrossRef] [Green Version] - Caratelli, D.; Gielis, J.; Tavkhelidze, I.; Ricci, P.E. Fourier-Hankel solution of the Robin problem for the Helmholtz equation in supershaped annular domains. Bound. Value Probl.
**2013**, 2013, 253. [Google Scholar] [CrossRef] [Green Version] - Matsuura, M. Gielis’ superformula and regular polygons. J. Geom.
**2015**, 106, 383–403. [Google Scholar] [CrossRef] - Shi, P.J.; Xu, Q.; Sandhu, H.S.; Gielis, J.; Ding, Y.L.; Li, H.R.; Dong, X.B. Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant. Ecol. Evol.
**2015**, 5, 4578–4589. [Google Scholar] [CrossRef] [PubMed] - Lin, S.Y.; Zhang, L.; Reddy, G.V.P.; Hui, C.; Gielis, J.; Ding, Y.L.; Shi, P.J. A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation. Ecol. Evol.
**2016**, 6, 6798–6806. [Google Scholar] [CrossRef] [PubMed] - Koiso, M.; Palmer, B. Rolling construction for anisotropic Delaunay surfaces. Pac. J. Math.
**2008**, 234, 345–378. [Google Scholar] - Wei, H.L.; Li, X.M.; Huang, H. Leaf shape simulation of castor bean and its application in nondestructive leaf area estimation. Int. J. Agric. Biol. Eng.
**2019**, 12, 135–140. [Google Scholar] - Ratkowsky, D.A. Nonlinear Regression Modeling: A Unified Practical Approach; Marcel Dekker: New York, NY, USA, 1983. [Google Scholar]
- Spiess, A.-N.; Neumeyer, N. An evaluation of R squared as an inadequate measure for nonlinear models in pharmacological and biochemical research: A Monte Carlo approach. BMC Pharmacol.
**2010**, 10, 6. [Google Scholar] [CrossRef] [PubMed] [Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2019; Available online: https://www.R-project.org/ (accessed on 1 January 2020).
- Giometto, A.; Formentin, M.; Rinaldo, A.; Cohen, J.E.; Maritan, A. Sample and population exponents of generalized Taylor’s law. Proc. Natl. Acad. Sci. USA
**2015**, 112, 7755–7760. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shi, P.J.; Ratkowsky, D.A.; Wang, N.T.; Li, Y.; Reddy, G.V.P.; Zhao, L.; Li, B.L. Comparison of five methods for parameter estimation under Taylor’s power law. Ecol. Compl.
**2017**, 32, 121–130. [Google Scholar] [CrossRef] - Lin, S.Y.; Niklas, K.J.; Wan, Y.W.; Hölscher, D.; Hui, C.; Ding, Y.L.; Shi, P.J. Leaf shape influences the scaling of leaf dry mass vs. area: A test case using bamboos. Ann. Forest Sci.
**2020**, 77, 11. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**A simulation example of the generalized Gielis curve. In panel (

**a**), the red curve represents the elementary Gielis curve, and the black curve represents the generalized Gielis curve (m = 5, k = 1, n

_{2}= n

_{3}= 5, a = 2.0, b = 1.5 and c = 0.2); in panel (

**b**), the red curve represents the link function between r and r

_{e}according to Equation (10); in panel (

**c**), the red curve represents the link function on the log–log plot according to Equation (12).

**Figure 2.**Original images of eight starfish (codes 1−8 in Table A1) and fitted generalized Gielis curves. The number in the upper left corner for each black background image panel represents its sample code. The panel below each black background image panel shows the scanned edge (represented by a gray curve) and the fitted edge using GGE (represented by a red curve). The intersection between the blue vertical and horizontal dashed lines represents the polar point; the black inclined dashed line represents the previously used horizontal line of a standard GGE without an angle transformation.

**Figure 3.**The scatter plots of ln(r) vs. ln(r

_{e}) and the fitted link functions for each of the eight starfish. In each panel, the small open circles represent the actual values, and the red curve represents the fitted link function based on Equation (12). RMSE values shown here were calculated based on the log-transformed data, while RMSE values in Table A1 were based on the untransformed data of r vs. r

_{e}.

**Figure 4.**Original images, scanned and predicted leaf edges, and fitted link functions on a log–log plot for four plant species (codes 9–12 in Table A1). The number in the upper left corner for each green image panel represents its sample code. The panel below each green image panel shows the scanned edge (represented by a gray curve) and the fitted edge using GGE (represented by a red curve). In each panel in the bottom row, the small open circles represent the actual values of ln(r) vs. ln(r

_{e}), and the red curve represents the fitted link function based on Equation (13).

**Figure 5.**Illustration for the comparison between a broad blade (

**a**) and a narrow blade (

**b**). In each panel, the red point represents the polar point; the blue points represent the data points on the edge of the blade; the gray curve represents the blade edge; the segments between the polar point and the data points on the edge represent radii. Each leaf shape was generated by GGE with a power-law relationship (which is equation [13] when δ

_{2}= 0). The horizontal axis of the generalized Gielis curve is rotated counterclockwise by π/4 to conveniently show the image. Here, when δ

_{1}decreases towards 0, the curve approximates a circle with radius exp(δ

_{0}) and the polar point becomes the center of this circle at the point (0, 0). In contrast, when δ

_{1}increases towards a large value, the curve approximates a line segment with length exp(δ

_{0}) and the polar point approaches the left endpoint of the segment.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shi, P.; Ratkowsky, D.A.; Gielis, J.
The Generalized Gielis Geometric Equation and Its Application. *Symmetry* **2020**, *12*, 645.
https://doi.org/10.3390/sym12040645

**AMA Style**

Shi P, Ratkowsky DA, Gielis J.
The Generalized Gielis Geometric Equation and Its Application. *Symmetry*. 2020; 12(4):645.
https://doi.org/10.3390/sym12040645

**Chicago/Turabian Style**

Shi, Peijian, David A. Ratkowsky, and Johan Gielis.
2020. "The Generalized Gielis Geometric Equation and Its Application" *Symmetry* 12, no. 4: 645.
https://doi.org/10.3390/sym12040645