An interactive online tool to explore complex natural and abstract shapes in 3D.

Try the live 3D supershapes generator in your browser now!

The Superformula is a powerful mathematical equation capable of describing a wide variety of natural and abstract shapes using only a few parameters. It extends both the sphere and ellipse equations, providing a geometrical framework to model and explore 3D shapes, from simple circles to complex organic and artificial forms.

Supershapes are 3D shapes generated from the Superformula, ranging from simple circles to complex organic and abstract forms.

Supershapes are versatile 3D forms generated from the Superformula and have a wide range of applications across different fields:

• Generative art: Artists and designers use supershapes to create abstract, organic, and visually striking 3D forms.
• Scientific visualization: Supershapes help model natural structures, such as shells, flowers, and biological forms, providing insight into patterns in nature.
• Procedural modeling: In computer graphics and game development, supershapes are used to generate complex 3D assets algorithmically, saving time compared to manual modeling.
• Education and research: Supershapes demonstrate parametric equations, geometry, and mathematical transformations in a visual and interactive way, making them useful for teaching math, physics, and computational modeling.

The combination of mathematical elegance and visual appeal makes supershapes a powerful tool for both technical and creative projects.

Yes, the tool runs in your browser using WebAssembly and OpenGL, compiled with Emscripten from the original C++ code, while desktop versions are available in two flavors: one using OpenGL (with optional OpenCL support) to compute the Superformula and update the GPU buffers efficiently, and another using Vulkan for high-performance rendering.

You can try the Shaper Superformula online tool directly in your browser.

The formula was developed around 2000 by Johan Gielis by generalizing the superellipse originally introduced by Piet Hein, a Danish mathematician.

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

$$r(\phi) := \frac{1}{\left\{ \left[ \left( \left| \frac{1}{a}\cos(\phi\frac{m}{4}) \right| \right)^{n_2} + \left( \left| \frac{1}{b}\sin(\phi\frac{m}{4}) \right| \right)^{n_3} \right]^{-\frac{1}{n_1}} \right\}} \qquad \qquad (1)$$

It’s possible to generate different shapes choosing distinct values where the parameters $m$, $n_{1}$, $n_{2}$ and $n_{3}$ are real numbers and $a$, $b$ are real numbers excluding zero. When $n_{1}=n_{2}=n_{3} = 2$, $m = 4$ in equation 1, an ellipse is obtained. A circle is obtained when additionally, $a=b$.

The equation 1 is defined in 2 dimensions, but is possible to extend the superformula to 3, 4 or $n$ dimensions via the spherical product of superformulas, so whether we focused on a 3D space, it’s possible to obtain a 3D parametric surface multiplying two superformulas $r_{1}$ and $r_{2}$, the 3D coordinates are described with the following relations:

$$\begin{aligned}x = r_{1}(\theta)\cos(\theta)r_{2}(\phi)\cos(\phi)\\y = r_{1}(\theta)\sin(\theta)r_{2}(\phi)\cos(\phi)\\z = r_{2}(\phi)\sin(\phi)\end{aligned}$$

Where latitude is represented between $-\pi/2\le\phi\le\pi/2$ and longitude between $\pi\le\theta\le\pi$.

With these relations shown above, we can map it onto a spherical topology, but could be possible to map it onto different topologies such as a toroidal mapping, for example. To reach it, in 3D coordinates would be described with the following relations:

$$\begin{aligned}x = \cos(\theta)\left[r_{1}(\theta)+r_{2}(\phi)\cos(\phi)\right]\\y = \sin(\theta)\left[r_{1}(\theta)+r_{2}(\phi)\cos(\phi)\right]\\z = r_{2}(\phi)\sin(\phi)\end{aligned}$$

Shaper supershapes superformula tool

The Shaper superformula is a tool developed over Flow render engine written in C++, that implements the superformula as described above mapping onto a spherical and toroidal topology and modulated by a logarithmic and Archimedean spiral within a 3D space, it being possible to explore different shapes with this tool.

The tool published here, is the Webassembly version build via Emscripten from original code written in C++ so, it exists a desktop version of the tool that runs on Windows and Linux, but for the moment is not available.

Main Features

• Exploring different 3D shapes by either single parameter ranges or simply choosing a random parameter set.
• Spherical and toroidal 3D mapping with logarithmic and Archimedean modulation.
• One shape can be morphed to another modifying the parameters manually.
• Different rasterization modes such as wireframe, point and fill.
• The shapes can be rendered with Phong, Blinn, Lambert and Gouraud shading models with several colour ramps.
• Shapes in low, medium and high poly can be generated and rendered, modifying the resolution defined by longitude x latitude.
• The shapes can be translated, rotated, scaled and zoomed during the exploration.
• Screenshots can be taken and downloaded in PNG format.

Next improvements

• Waveform OBJ Exporter/Importer to save and retrieve the supershapes.
• Improvements in background, it’s only able to change the background colour for now.
• New shading models implementation.
• GUI improvements.
• Improvements in morphing transition control.

The following images depicts some abstract and nature shapes generated with the interactive tool.

A well-known real-time example of the Superformula in action is the San Angeles Observation demo by Jetro Lauha. This OpenGL-based visualization uses supershapes generated from the Superformula, following the classic implementation described by Paul Bourke, to procedurally create complex 3D structures and city-like forms in real time.

Superformula

Gielis, Johan(2003), A generic geometric transformation that unifies a wide range of natural and abstract shapes

Paul Bourke Supershapes