The superformula is a generalization of the superellipse and was 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.
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
the radius and
the angle, the superformula is:
sin
m φ
By choosing different values for the parameters
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 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:
( θ ) cos θ ⋅
( ϕ ) cos ϕ ,
( θ ) sin θ ⋅
( ϕ ) cos ϕ ,
( ϕ ) sin ϕ ,
ϕ
(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: The superformula can be generalized by allowing distinct m parameters in the two terms of the superformula.
By replacing the first parameter
This allows the creation of rotationally asymmetric and nested structures.
In the following examples a, b,