Superformula

The superformula is a generalization of the superellipse and was proposed by Johan Gielis around 2000. 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.

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

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 $a,b,m,n_{1},n_{2},$ and $n_{3},$ different shapes can be generated.

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

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