🌹 Spirograph — Hypotrochoids & Epitrochoids

A hypotrochoid is traced by a circle rolling inside a fixed ring, using x(t) = (R−r)cos(t) + d·cos((R−r)/r·t) and y(t) = (R−r)sin(t) − d·sin((R−r)/r·t). An epitrochoid is traced by a circle rolling outside a fixed circle, using the same form with plus signs: x(t) = (R+r)cos(t) − d·cos((R+r)/r·t). Switching between the two changes which side of R the rolling circle sits on.

Switch to Epitrochoid mode, then set the rolling radius r equal to the outer radius R and set the pen offset d equal to r as well. With r = R and d = r the epitrochoid equations simplify to the classic cardioid, the heart-shaped curve also seen in microphone pickup patterns and the main bulb of the Mandelbrot set.

The drawing repeats after a period of 2π·r/gcd(R, r) in the parameter t. If R and r share a large common factor the curve closes quickly with few loops; if their greatest common divisor is small, the pen must sweep round many more times before the pattern joins back onto itself.

Rose-like hypotrochoids appear when the pen offset d equals R − r and r divides evenly into R. The petal count then equals R/r when that ratio is odd, or twice R/r when it is even. Try R = 120, r = 40 and d = 80 for a clean three-petal rose.

When d exceeds r the pen sits beyond the edge of the rolling circle, so the traced curve grows extra loops instead of staying smooth. When d equals r the curve passes cleanly through the centre of the rolling path, and when d is smaller than r the whole curve stays within a narrow ring around radius R − r.