A generic geometric transformation that unifies a wide range of natural and abstract shapes

Abstract

To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.

Broadband negative refractive index obtained by plasmonic hybridization in metamaterials

“…Broadband negative permeability and negative refractive index materials can also be obtained using geometric optimization based on Gielis' superformula. 18,19 However, the optimal structures have complex shapes, which make them difficult to manufacture. A more convenient way to create a broadband negative permeability and pave the way for practical implementations is using the hybridization of interacting identical resonators.…”

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

“…It should be noted that the matrix A is identical regardless of the shape of the inhomogeneity and the expansion point r. The latter affect only the right-hand side F. Therefore one can immediately write the explicit form 35) and the only computation in order to form the polynomial approximation are the integrals in F. In the next section several examples are considered with various eigenstrains and a number of inhomogeneities of distinct shape, in order to illustrate the efficacy of this scheme.…”

Road Sign Detection and Shape Reconstruction Using Gielis Curves

“…Introduced by (Gielis, 2003), the superformula extends the superellipses by introducing variable rotational symmetry and asymmetric shape coefficients. The angle φ is replaced by mφ 4 to obtain m rotational symmetries, and the unique shape coefficient n for superellipses is replaced by a triplet (n 1 , n 2 , n 3 ), leading to the following radial polar parameterization:…”

“…In this paper, we propose a robust road sign detection method which uses the color information to localize potential road signs and then, based on shape representation using Gielis curves (Gielis, 2003), identifies the shape of the detected signs. The major advantage of our approach is that it does not need a multi-layer architecture as used in machine learning approaches (neural network or support vector machines).…”