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. 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.

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:

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

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

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

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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;

External links

• 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.