S S symmetry Article The Generalized Gielis Geometric Equation and Its Application Peijian Shi 1 , David A. Ratkowsky 2 and Johan Gielis 3,* 1 Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China; [email protected] 2 Tasmanian Institute of Agriculture, University of Tasmania, Private Bag 98, Hobart 7001, Australia; [email protected] 3 Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium * Correspondence: [email protected] Received: 28 March 2020; Accepted: 15 April 2020; Published: 17 April 2020 Abstract: Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves. Keywords: data fitting; hyperbolic functions; leaf shape; polar coordinates; power-law functions; starfish 1. Introduction In geometry, the equation of a circle in the Euclidean plane is usually expressed as x2 y2 + = 1 (1) r r where x and y are the coordinates of the circle on the x- and y-axes, respectively, with r the radius. The circle is a special case of that of an ellipse: x 2 y 2 + = 1 (2) A B where A and B (A B > 0) represent the major and minor axis semi-diameters, respectively. Interestingly, ≥ circles and ellipses, as well as squares and rectangles, can be regarded as special cases of Lamé curves [1,2], whose mathematical expression is listed as x n y n + = 1 (3) A B Symmetry 2020, 12, 645; doi:10.3390/sym12040645 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 645 2 of 10 where n is a real number. This equation has been shown to be valid for describing the actual cross-sections of tree rings and bamboo shoots [2–5]. Equation (3) in polar coordinates can be rewritten as n n! 1/n 1 1 − r(') = cos(') + sin(') (4) A B where r and φ are the polar radius and the angle between the straight line where the polar radius lies and the x-axis, respectively. Gielis proposed a more general polar equation that can reflect more complex natural shapes [1,2]: n n ! 1/n1 1 m 2 1 m 3 − r(') = cos ' + sin ' (5) A 4 B 4 where n1, n2 and n3 are constants (both R); positive integer m was introduced to make the curve 2 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, n1 = n, and n2 = n3 = 1, OGE has a special case: 1/n 1 1 − r(') = l cos ' + sin ' (6) 4 4 where l = A1/n. This simplified version has been used to describe the actual leaf shapes of 46 bamboo species [2,10,11]. Although OGE is rather flexible in fitting the edge data of many natural shapes, it sometimes fails to describe accurately some symmetrical natural shapes. To further strengthen its flexibility of data fitting, we attempt to build a more generalized equation based on OGE, motivated by the study of starfish, which display a wide diversity of shapes, not only the archetypical shapes. In the Plateau problem of minimal surfaces, one of the constant mean curvature solutions for a soap film is a sphere, but these solutions are for isotropic energy distributions only. In crystallography, Wulff shapes describe anisotropic distributions of energy, and can take many forms, with their corresponding constant anisotropic mean curvature surfaces [12]. Extending this principle to biological species, starfish can be considered as spheres for specific anisotropic energy distributions. Remarkably, pincushion starfish of the genus Culcita are close to spherical, intermediate between classical spheres and the archetypical shapes of five-armed starfish. Another notable group of starfish are biscuit starfish, almost pentagonal and flat. They belong to the genus Tosia. In order to apply the above methods to these groups, a modification of the original Gielis equation is necessary. 2. The Generalized Gielis Equation (GGE) and Its Two New Special Cases OGE can be rewritten as [6] 0 1β B C B 1 C r(') = α B C (7) B n2 n3 C @B m + m AC cos 4 ' k sin 4 ' Symmetry 2020, 12, 645 3 of 10 n /n n n where α = A 2 1 , k = A 2 /B 3 and β = 1/n1. Consider the formula inside the parentheses of Equation (7), which we define as follows: 1 r (') = (8) e n2 n3 m + m cos 4 ' k sin 4 ' We refer to this as the elementary Gielis equation (EGE) for convenience. OGE hypothesizes the existence of a power-law relationship between r and re. 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 toSymmetry replace 20 OGE:20, 12, x FOR PEER REVIEW 3 of 9 r(') = f [re(')] (9) which we refer to as the generalized Gielis equation (GGE). In actuality, re is the polar radius of the whichelementary we refer Gielis to ascurve the generalizedgenerated by Gielis EGE, equation and r is (GGE).the polar In radius actuality, of there is generalized the polar radius Gielis of curve the elementarygenerated by Gielis GGE. curve Therefore, generated OGE by is EGE,actually and a rspecialis the polarcase of radius GGE ofwith the a generalizedpower-law link Gielis functi curveon. generatedIn the by current GGE. Therefore,study, we propose OGE is actually the following a special two case candidate of GGE forms with a of power-law the link function: link function. In the current study, we propose the following two candidate forms of the link function: 1 푟 = exp" [ + 푐#⁡] (10) 푎 + 1푏⁡ln(푟푒⁡) r = exp + c (10) and a + b ln(re ) and 푟 = exp[훿 + 훿 ⁡ln(푟 ) + 훿 (ln(푟 ))2⁡] (11) h 0 1 푒 2 푒 i r = exp δ + δ ln(r ) + δ (ln(r ))2 (11) If we use the log transformation for the0 left1 -ande right2-hande sides of the above two equations, we obtain,If we respectively: use the log transformation for the left-and right-hand sides of the above two equations, we obtain, respectively: 1 푌 = 1 + 푐 (12) Y = 푎 + 푏⁡푋+ c (12) a + b X and and 22 Y푌==δ훿0 0++δ훿1 1X⁡푋++δ훿22X푋 ((13)13) wherewhereY 푌==lnln(r()푟)and andX 푋==lnln(r(e푟)푒.) The. The first first equation equation is is actually actually a a hyperbolic hyperbolic equation, equation, and and the the second second oneone isis aa quadratic equation. If If we we also also use use the the log log transformation transformation for for both both sides sides of the of thepower power-law-law link linkfunction function in OGE, in OGE, a linear a linear equation equation is obtained. is obtained. We We can can express express that that by by letting letting the the coef coeficientfficient δ2δ in2 inEquation Equation (13) (13) be be zero. zero. Of Of course, course, in in nature, nature, there there are are forms forms other other than the aboveabove linklink functions;functions; however,however, these these other other forms forms also also belong belong to the to scope the ofscope GGE of if thereGGE existsif there a clear exists functional a clear expressionfunctional betweenexpressionr and betweenre. Figure r and1 provides re. Figure a 1 simulation provides a example simulation for example Equation for (10). Equation (10). FigureFigure 1. 1.A A simulationsimulation exampleexample ofof thethe generalizedgeneralized Gielis Gielis curve. curve. InIn panelpanel ((aa),),the the red red curve curve represents represents thethe elementary elementary Gielis Gielis curve, curve, and and the the black black curve curve represents represents the the generalized generalized Gielis Gielis curve curve ( m(m= =5, 5,k k= =1, 1, nn22= = nn33 = 5,5, aa = 2.0,2.0, bb == 1.51.5 and and cc == 0.2);0.2); in in panel panel ( (bb)),, the the red red curve represents thethe linklink functionfunction betweenbetween rrand andr ereaccording according to to Equation Equation (10); (10); in in panel panel (c ),(c the), the red red curve curve represents represent thes the link link function function on theon log–logthe log– plotlog accordingplot according to Equation to Equation (12). (12). 3. Application of the Generalized Gielis Equation In this section, we mainly illustrate the application of GGE to several starfish species of the families Goniasteridae and Oreasteridae. Culcita and Tosia are the target genera, and Anthenoides tenuis and Stellaster equestris are used as long-armed reference species.