Superformula

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 has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10.^[2]

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

$${\displaystyle r\left(\varphi \right)=\left(\left|{\frac {\cos \left({\frac {m\varphi }{4}}\right)}{a}}\right|^{n_{2}}+\left|{\frac {\sin \left({\frac {m\varphi }{4}}\right)}{b}}\right|^{n_{3}}\right)^{-{\frac {1}{n_{1}}}}.}$$ By choosing different values for the parameters ${\displaystyle a,b,m,n_{1},n_{2},}$ and ${\displaystyle n_{3},}$ different shapes can be generated.

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

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

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:

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

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

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;

The superformula can be generalized by allowing distinct m parameters in the two terms of the superformula. By replacing the first parameter ${\displaystyle m}$ with y and second parameter ${\displaystyle m}$ with z:^[3] $${\displaystyle r\left(\varphi \right)=\left(\left|{\frac {\cos \left({\frac {y\varphi }{4}}\right)}{a}}\right|^{n_{2}}+\left|{\frac {\sin \left({\frac {z\varphi }{4}}\right)}{b}}\right|^{n_{3}}\right)^{-{\frac {1}{n_{1}}}}}$$

This allows the creation of rotationally asymmetric and nested structures. In the following examples a, b, ${\displaystyle {n_{2}}}$ and ${\displaystyle {n_{3}}}$ are 1:

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