Superformula

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 ${\displaystyle r}$ 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.