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.
Highlights
In geometry, the equation of a circle in the Euclidean plane is usually expressed as 2 y 2 x + =1 r r (1)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: =1 A B (2)where A and B (A ≥ B > 0) represent the major and minor axis semi-diameters, respectively
We developed a group of R scripts for extracting planar coordinates of shapes of interests and fit generalized Gielis equation (GGE) based on R [16]
In Appendix S2, we minimize the residual sum of squares (RSS) between the observed and predicted polar radii of a starfish or a leaf described by GGE to estimate the parameters in GGE
Summary
The equation of a circle in the Euclidean plane is usually expressed as. Gielis proposed a more general polar equation that can reflect more complex natural shapes [1,2]: r(φ) =. 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. OGE hypothesizes the existence of a power-law relationship between r and re We refer to this relationship as the link function, f. Generated by OGE is a special case of GGE with a power-law link function. IfIfwe wealso alsouse usethe the log transformation both sides of the power-law function in OGE, a linear equation is obtained. We can express that by letting the coefficient δ link function in OGE, a linear equation is obtained.
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