Superformula

Generalized version of the superellipse formula proposed by Johan Giellis around 2000, capable of far more variety than the original superellipse. Unfortunately, Johan has patented use of the formula (via his company Genicap) in both the US and the EU, which means you should avoid using it for any kind of commercial work, or work that could be commercialized in some way later.

The superformula can be used to generate both 2D and 3D forms. To create 2D forms, use the general form equation to obtain polar coordinates that can be converted into Cartesian coordinates for drawing on a screen. To create 3D forms, compute the polar coordinates for two 2D supershapes, then "mix" them together using the 3D equations below.

Equations:

General form	Where r is a radius and φ (phi) is an angle
3D equations	Where φ (latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π.

Articles:

• Superformula on Wikipedia
• Supershapes / Superformula by Paul Bourke

Code projects:

• SuperformulaSVG-for-web by Jason Webb (Javascript, p5.js)
• SuperformulaSVG by Jason Webb (Processing)
• Visualize: Superformula from FORM+CODE book
• glsl-superformula by JC Leyba (Software)
• supershape.js by Sebastian Sadowski (ahoiin)

Videos:

• Coding Challenge #26: 3D Supershapes by Daniel Shiffman (Processing sketch on Github)