Super formula

Various curves generated with the super formula. a = b = 1; the values ​​under the curves indicate m, n ₁ , n ₂ and n ₃ .

The super formula is a geometric figure (curve) published in 1997 by the Belgian scientist Johan Gielis . It is a generalization of Lamé's curve .

Using the super formula, it is possible to describe outlines of different symmetry (see picture on the right) uniformly with a few parameters. In polar coordinates and the formula is: ${\displaystyle r}$ ${\displaystyle \varphi }$

${\displaystyle {\frac {1}{r}}={\sqrt[{n_{1}}]{\,\left|{\frac {1}{a}}\cos \left({\frac {m}{4}}\varphi \right)\right|^{n_{2}}+\left|{\frac {1}{b}}\sin \left({\frac {m}{4}}\varphi \right)\right|^{n_{3}}}}}$

${\displaystyle r}$: Distance from the center

${\displaystyle \varphi }$: Angle to the x-axis

${\displaystyle m}$: Symmetry

${\displaystyle n_{1}}$, , : Form ${\displaystyle n_{2}}$ ${\displaystyle n_{3}}$

${\displaystyle a}$, : Expansion (semi-axes) ${\displaystyle b}$

The term “super formula” refers to the connection with the super ellipse of the Danish scientist, inventor and writer Piet Hein (1905–1996).

Web links

• Genicap, Johan Gieli's homepage
• P. Bourke: Supershapes (Superformula)
• Eric W. Weisstein: Superellipse. (MathWorld - A Wolfram Web Resource)
• Graphic "super formula": A circle is a square is a fish is a starfish (heise)
• Simple 3D modeller with Supershape support (reference to relevant documentation)
• P. Bourke: Three-dimensional super formula (English)