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Superformula

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From Wikipedia, the free encyclopedia

Equation in polar coordinates

This article is about a generalization of the superellipse. For the Japanese formula racing series, see Super Formula Championship.

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003.[1] Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10.[2]

In polar coordinates, with

{\displaystyle r}

the radius and

{\displaystyle \varphi }

the angle, the superformula is:

cos

sin

{\displaystyle r\left(\varphi \right)=\left(\left|{\frac {\cos \left({\frac {m\varphi }{4}}\right)}{a}}\right|^{n_{2}}+\left|{\frac {\sin \left({\frac {m\varphi }{4}}\right)}{b}}\right|^{n_{3}}\right)^{-{\frac {1}{n_{1}}}}.}

By choosing different values for the parameters

{\displaystyle a,b,m,n_{1},n_{2},}

and

{\displaystyle n_{3},}

different shapes can be generated.

The formula was obtained by generalizing the superellipse, named and popularized by Piet Hein, a Danish mathematician.

2D plots<br>[edit]

In the following examples the values shown above each figure should be m, n1, n2 and n3.

A GNU Octave program for generating these figures

function sf2d(n, a)<br>u = [0:.001:2 * pi];<br>raux = abs(1 / a(1) .* abs(cos(n(1) * u / 4))) .^ n(3) + abs(1 / a(2) .* abs(sin(n(1) * u / 4))) .^ n(4);<br>r = abs(raux) .^ (- 1 / n(2));<br>x = r .* cos(u);<br>y = r .* sin(u);<br>plot(x, y);<br>end

Extension to higher dimensions<br>[edit]

It is possible to extend the formula to 3, 4, or n dimensions, by means of the spherical product of superformulas. For example, the 3D parametric surface is obtained by multiplying two superformulas r1 and r2. The coordinates are defined by the relations:

cos

cos

{\displaystyle x=r_{1}(\theta )\cos \theta \cdot r_{2}(\phi )\cos \phi ,}

sin

cos

{\displaystyle y=r_{1}(\theta )\sin \theta \cdot r_{2}(\phi )\cos \phi ,}

sin

{\displaystyle z=r_{2}(\phi )\sin \phi ,}

where

{\displaystyle \phi }

(latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π.

3D plots<br>[edit]

3D superformula: a = b = 1; m, n1, n2 and n3 are shown in the pictures.

A GNU Octave program for generating these figures:

function sf3d(n, a)<br>u = [- pi:.05:pi];<br>v = [- pi / 2:.05:pi / 2];<br>nu = length(u);<br>nv = length(v);<br>for i = 1:nu<br>for j = 1:nv<br>raux1 = abs(1 / a(1) * abs(cos(n(1) .* u(i) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * u(i) / 4))) .^ n(4);<br>r1 = abs(raux1) .^ (- 1 / n(2));<br>raux2 = abs(1 / a(1) * abs(cos(n(1) * v(j) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * v(j) / 4))) .^ n(4);<br>r2 = abs(raux2) .^ (- 1 / n(2));<br>x(i, j) = r1 * cos(u(i)) * r2 * cos(v(j));<br>y(i, j) = r1 * sin(u(i)) * r2 * cos(v(j));<br>z(i, j) = r2 * sin(v(j));<br>endfor;<br>endfor;<br>mesh(x, y, z);<br>endfunction;

Generalization<br>[edit]

The superformula can be generalized by allowing distinct m parameters in the two terms of the superformula. By replacing the first parameter

{\displaystyle m}

with y and second parameter

{\displaystyle m}

with z:[3]

cos

sin

{\displaystyle r\left(\varphi \right)=\left(\left|{\frac {\cos \left({\frac {y\varphi }{4}}\right)}{a}}\right|^{n_{2}}+\left|{\frac {\sin \left({\frac {z\varphi }{4}}\right)}{b}}\right|^{n_{3}}\right)^{-{\frac {1}{n_{1}}}}}

This allows the creation of rotationally asymmetric and nested structures. In the following examples a, b,

{\displaystyle {n_{2}}}

and

{\displaystyle {n_{3}}}

are 1:

References<br>[edit]

↑ Gielis, Johan (2003), "A generic geometric transformation that unifies a wide range of natural and abstract shapes", American Journal of Botany, 90 (3): 333–338, doi:10.3732/ajb.90.3.333, ISSN 0002-9122, PMID 21659124

↑ EP patent 1177529, Gielis, Johan, "Method and apparatus for synthesizing patterns", issued 2005-02-02

↑ Stöhr, Uwe (2004), SuperformulaU (PDF), archived from the original (PDF) on December 8, 2017

External links<br>[edit]

Wikimedia Commons has media related to Superformula.

Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization

Least Squares Fitting of Chacón-Gielis Curves By the Particle Swarm Method of Optimization

Superformula 2D Plotter & SVG Generator

Interactive example using JSXGraph

SuperShaper: An OpenSource, OpenCL accelerated, interactive 3D SuperShape generator with shader based visualisation (OpenGL3)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Superformula&oldid=1305278571"

Categories: Geometric shapes<br>Curves<br>Surfaces

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