Superformula: The Mathematical Shape That Models Nature

Superformula Applications: From 3D Modeling to Procedural ArtThe superformula — a compact mathematical expression introduced by Johan Gielis in the early 2000s — is a remarkably flexible tool for generating a vast family of shapes. By adjusting just a few parameters, the superformula can reproduce circles, polygons, star shapes, flowers, and many organic forms that resemble leaves, shells, and biological cross-sections. This flexibility has made it an attractive tool across disciplines: from computational geometry and 3D modeling to procedural art, architecture, and scientific visualization. This article explores the superformula’s mathematical roots, implementation strategies, and a variety of practical applications, with examples and tips for artists, designers, and developers.

What is the Superformula?

At its core, the superformula generalizes the superellipse and provides a polar equation that defines radius r as a function of angle θ:

r(θ) = [ |(cos(m θ / 4) / a)|^n2 + |(sin(m θ / 4) / b)|^n3 ]^(−1/n1)

Key parameters:

• a, b — scale parameters along cosine and sine components (usually set to 1).
• m — symmetry parameter controlling the number of lobes or repetitions.
• n1, n2, n3 — shape parameters that determine curvature and angular sharpness.

By varying these five parameters you can continuously morph between smooth, flower-like curves and sharp, polygonal forms. The formula’s compactness and expressive range explain its popularity in procedural generation.

Implementing the Superformula

Implementing the superformula is straightforward in most programming environments. Typical steps:

1. Iterate θ from 0 to 2π (or a chosen range).
2. Compute r(θ) using the formula.
3. Convert polar coordinates (r, θ) to Cartesian (x = r cos θ, y = r sin θ).
4. Optionally apply scaling, rotation, or noise for variation.
5. For 3D, extrude the 2D contour, sweep along a path, revolve around an axis, or use r as a radius in spherical coordinates.

Example (pseudocode):

for θ from 0 to 2π step Δθ:     r = ( abs(cos(m*θ/4)/a)^n2 + abs(sin(m*θ/4)/b)^n3 )^( -1/n1 )     x = r * cos(θ)     y = r * sin(θ)     add vertex (x, y) 

Tips:

• Use a fine Δθ for smooth curves (e.g., 0.005–0.01 radians).
• Clamp or handle zero/near-zero values in exponents to avoid numerical instability.
• Interpolate parameters over time to animate morphing shapes.

3D Modeling Applications

1. Extrusion and Revolve: Create a 2D superformula profile and revolve it around an axis to form shells, vases, or organic solids. Extruding the profile along a path produces ornamental columns, fronds, or procedural furniture legs.

2. Lofting Multiple Profiles: Generate multiple superformula contours with varying parameters and loft between them to create smooth transitional forms — useful for architectural components, lamp shades, or biomorphic sculptures.

3. Spherical and Radial Mapping: Use the superformula as a radial function over latitude/longitude to create complex, radially symmetric 3D surfaces (similar to superquadrics). For each spherical angle pair, compute a radius by combining multiple superformula evaluations.

4. Mesh Detail and Displacement: Apply the superformula as a displacement map to base meshes (planes, spheres). This enables creation of complex surface textures: ridges, petals, and crater-like features without hand-sculpting.

5. Procedural Asset Generation: Integrate superformula parameter sets into asset pipelines to auto-generate variation sets for games and films — shells, plants, alien artifacts — ensuring stylistic cohesion while producing many unique assets.

Example uses in tools:

• Blender: Use a Python script or Geometry Nodes to compute (x, y) points for a curve, then convert to mesh and use modifiers (Solidify, Subdivision).
• Houdini: Build VOPs or wrangles to generate contours and sweep/extrude nodes for volumetric forms.
• CAD/Parametric Modeling: Use superformula curves in Rhino/Grasshopper for concept forms and then refine for production.

Procedural Art and Generative Design

Artists and generative designers leverage the superformula for visually rich, parameter-driven works:

• Animated Morphing: Smoothly interpolate parameters (m, n1–n3) over time to create organic animations where shapes bloom, fold, and transform.
• Pattern Generation: Tile or repeat superformula shapes with variations in scale, rotation, or color to produce complex wallpaper patterns, mandalas, or textile prints.
• Shader-Based Rendering: Implement the superformula directly in fragment or vertex shaders to render real-time, highly parameterized visuals for interactive installations and VJing.
• Hybrid Systems: Combine the superformula with noise functions (Perlin, Simplex) or with other parametric shapes for hybrid, less-regular structures that feel more natural.

Practical art example: generate a base superformula silhouette, sample points along the contour, and use those as control points for generative strokes, particle emitters, or L-systems to grow structures that follow the silhouette.

Scientific Visualization and Modeling

The superformula isn’t only aesthetic — it’s useful in science and engineering:

• Botanical Modeling: Approximate leaves, petals, and cross-sections of biological organisms for study or realistic rendering.
• Morphometrics: Use the formula to fit and compare biological outlines (e.g., leaf shapes, shells) in quantitative morphology.
• Acoustics and Antenna Design: Create aperture shapes that tailor diffraction and radiation patterns.
• Data Visualization: Map data variables to superformula parameters, turning multidimensional data into interpretable shapes.

Procedural Workflow Examples

1. Creating a decorative vase:

   • Choose m to set symmetry (e.g., m = 6 for sixfold petals).
   • Generate 2D contour, adjust n values for rim sharpness.
   • Revolve and apply solidify and subdivision modifiers.
   • Add displacement texture derived from another superformula for surface detail.

2. Real-time interactive art in GLSL:

   • Implement r(θ) in a fragment shader.
   • Use time to modulate m and n parameters for animation.
   • Use color mapping based on curvature or radial distance.

3. Game asset variety:

   • Parameterize an “organism” template using arrays of m/n values.
   • Randomize within artist-defined ranges to produce hundreds of distinct but coherent assets.

Tips, Pitfalls, and Parameter Intuition

• m controls symmetry: integer m yields m-fold symmetry. Non-integer yields rotated/asymmetric effects.
• n2 and n3 affect lobe sharpness; lower values produce sharper points, higher values smooth them.
• n1 controls overall rounding and interpolation between the cosine and sine contributions.
• Watch for extreme exponents: they can produce extremely large or tiny r values—use clamps or normalization.
• For animation, interpolate parameters smoothly (use easing curves) to avoid abrupt geometry changes.

Conclusion

The superformula is a small, mathematically elegant tool with outsized creative power. Its compact parameter set can produce both precise geometric forms and richly organic shapes, making it ideal for 3D modeling, procedural asset generation, interactive art, and scientific visualization. Whether you’re an artist, developer, or researcher, the superformula offers a concise way to explore a broad shape-space and rapidly iterate on forms that would otherwise require extensive manual modeling.

If you’d like, I can provide code snippets for Blender (Python/Geometry Nodes), GLSL shader versions, or a small parameter set library to generate specific types of shapes (flowers, stars, shells).