Superformula

The superformula is a generalization of the superellipse and was first 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.

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

$r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{m\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{m\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}}.$

By choosing different values for the parameters $a, b, m, n_1, n_2, n_3$, 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 with two new parameters y and z:^[2]

$r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{y\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{z\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}}$

This allows to create 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, 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;

References

1. • 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
2. • Stöhr, Uwe (2004), SuperformulaU (PDF)

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.