The Generalized Gielis Geometric Equation and Its Application

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.

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

“…To compare the fit between the superellipse equation with a deformation parameter w (SEDP) and Equation (1) without a deformation parameter (SE), we calculated adjusted root-mean-square errors (RMSE adj ) [12,13]:…”

“…Equation ( 13) can be regarded as a power-law relationship between r and r 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.…”

“…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., r = exp γ 0 + 1…”

Cross-Sectional Performance of Hollow Square Prisms with Rounded Edges

“…Note that the model possesses the tangent continuity at any point on the closed curve. In addition, the model is essentially different from super-ellipses [14,15], being expressed in terms of the p-norm and the vector notation by r − r 0 p = const, or its generalization called a Gielis curve [16][17][18], while both of them are known as powerful tools for describing the natural shape of plants. Furthermore, a method of drawing a rounded triangle by using a complex function has been proposed quite recently [19], but again, our method is essentially different from it.…”

Rate of Entropy Production in Evolving Interfaces and Membranes under Astigmatic Kinematics: Shape Evolution in Geometric-Dissipation Landscapes

“…Likewise, surface curvedness ( C ) or deviation from planarity, can be characterized by a positive scalar (the condition reduces to a plane if ), with natural units of inverse length. Shape parameter S and curvedness C are essential features and determinants of surface properties and functionalities that can be leveraged in many engineering [ 2 , 3 ], biological and biomimetic applications [ 4 , 5 , 6 , 7 , 8 ].…”