Superformula

2D plots

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)
  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

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.

3D plots

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

Image:Sf3d_3257.svg Image:Sf3d_3.5.5.5.svg Image:Sf3d_3301515.svg Image:Sf3d_7284.svg Image:Sf3d_5111.svg Image:sf3d_4.5.54.svg Image:sf3d_8.5.58.svg Image:Sf3d_4121515.svg

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;

Generalization

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

References

References

1. (2003). "A generic geometric transformation that unifies a wide range of natural and abstract shapes". [[American Journal of Botany]].
2. "Method and apparatus for synthesizing patterns".
3. (2004). "SuperformulaU".