Mathematical Curves for Design
Table of Contents
Explorations of parametric mathematical curves for favicon and logo design.
1. The 7:4:3 Hypotrochoid
Re-derived the curve x=(R-r)cos(t)+d*cos((R-r)t/r), y=(R-r)sin(t)-d*sin((R-r)t/r) for R=7, r=4, d=3 and closed it numerically: it does not close after one revolution of t -- it closes after exactly 4 revolutions, matching the period r/gcd(R,r)=4. Corrected "one revolution" to "4 revolutions" below. The 7-lobe count checked out correct.
The 7:4:3 hypotrochoid has 7 lobes. The parameters (R=7, r=4, d=3) produce a curve that closes after exactly 4 revolutions.
2. 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
4. Curves to Explore
Additional mathematical curves well-suited for favicon and logo design:
4.1. Rose Curves
r = cos(nθ) – Petal patterns, visually distinctive at small sizes. The number of petals depends on whether n is odd or even.
4.2. Hypotrochoids/Epitrochoids
Spirograph-like curves, very recognizable. Special cases:
- Deltoid (n=3) – Three-cusped hypocycloid
- Astroid (n=4) – Four-cusped hypocycloid
4.3. Cardioid
r = 1 + cos(θ) – Iconic single-cusp heart-like shape. Special case of an epicycloid.
4.4. Reuleaux Triangle
Constant-width curve with a distinctive rounded-triangle appearance. Think Wankel rotary engine rotors.
4.5. 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.