Superformula

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 the radius and the angle, the superformula is:

By choosing different values for the parameters and 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[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:

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

Generalization[edit]

The superformula can be generalized by replacing the parameter with y and with z:[3]

This allows the creation of rotationally asymmetric and nested structures. In the following examples a, b, and are 1:

SuperformulaU-several-structures.svg

Plots[edit]

Sf2d.png

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, 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[edit]

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

External links[edit]