Superformula

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 around 2000.^[1] Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis holds a patent related to the synthesis of patterns generated by the superformula.^[2]

In polar coordinates, with $r$ the radius and $\varphi$ the angle, the superformula is:

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

By choosing different values for the parameters $a,b,m_{1},m_{2},n_{1},n_{2},$ and $n_{3},$ different shapes can be generated.

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

Extension to higher dimensions

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:

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

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

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

where $\phi$ (latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π.

Generalization

The superformula can be generalized by replacing the parameter $m_{1}$ with y and $m_{2}$ with z:^[3]

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, ${n_{2}}$ and ${n_{3}}$ are 1:

Plots

A GNU Octave program for generating these figures:

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

3D superformula: a = b = 1; $m_{1}$ = $m_{2}$ = m, n₁, n₂ and n₃ are shown in the pictures.

A GNU Octave program for generating these figures:

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

References

↑ 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
↑ EP patent 1177529, Gielis, Johan, "Method and apparatus for synthesizing patterns", issued 2005-02-02
↑
- Stöhr, Uwe (2004), SuperformulaU (PDF)

External links

Wikimedia Commons has media related to Superformula.

Website with information about the superformula and Johan Gielis
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
3D Superdupershape Explorer using Processing
Interactive 3D Superformula plotter using Processing (with code)
SuperShaper: An OpenSource, OpenCL accelerated, interactive 3D SuperShape generator with shader based visualisation (OpenGL3)
Simpel, WebGL based SuperShape implementation
The Gielis Supershape Formula, Provides an interactive Java applet for exploring the variety of different shapes, natural or mathematical, that can be formed with the formula.

This article is issued from Wikipedia - version of the 11/29/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.