Mathematical Curves for Design
Table of Contents
Explorations of parametric mathematical curves for favicon and logo design.
The 7:4:3 Hypotrochoid
From parametric equations to pixel-perfect assets
The 7:4:3 hypotrochoid is the signature curve for this project. With 7 lobes and elegant symmetry, it scales beautifully from 16x16 favicons to full vector logos. The parameters (R=7, r=4, d=3) produce a curve that closes after exactly one revolution.
Resources
- Full Documentation - Complete technical reference with code
- Presentation Slides - Overview for talks (org-reveal compatible)
- Generation Script - Python code to generate all curve images
Other Curves
Spirograph Variants
Lissajous
Superellipse
Rose
Lemniscate
Curves to Explore
Additional mathematical curves well-suited for favicon and logo design:
Rose Curves
r = cos(nθ) — Petal patterns, visually distinctive at small sizes. The number of petals depends on whether n is odd or even.
Hypotrochoids/Epitrochoids
Spirograph-like curves, very recognizable. Special cases:
- Deltoid (n=3) — Three-cusped hypocycloid
- Astroid (n=4) — Four-cusped hypocycloid
Cardioid
r = 1 + cos(θ) — Iconic single-cusp heart-like shape. Special case of an epicycloid.
Reuleaux Triangle
Constant-width curve with a distinctive rounded-triangle appearance. Think Wankel rotary engine rotors.
Logarithmic Spiral
r = ae^{bθ} — The golden spiral variant (b = ln(φ)/(π/2)) is immediately recognizable even at 64×64 pixels. Appears throughout nature.