A roulette is the curve traced by a point attached to a circle that rolls without slipping along another curve. Here the rolling circle (radius r) rolls along a fixed circle (radius R).

Hypocycloid (rolling inside)

x = (R−r)cosθ + d·cos((R−r)/r·θ)
y = (R−r)sinθ − d·sin((R−r)/r·θ)

Epicycloid (rolling outside)

x = (R+r)cosθ − d·cos((R+r)/r·θ)
y = (R+r)sinθ − d·sin((R+r)/r·θ)

The pen offset d equals r for the true cusped curve. Smaller d gives a curtate roulette (loops smoothed); larger gives a prolate roulette (extra loops).

Counting cusps

The number of cusps is R / gcd(R, r). The curve closes after r / gcd(R, r) trips around the fixed circle. When R and r share no common factor the cusp count is simply R.

Named special cases

• R/r = 2 — straight line (Tusi couple / Cardano circles): the point oscillates on a diameter.
• R/r = 3 — deltoid (3 cusps).
• R/r = 4 — astroid (4 cusps), x²ᐟ³+y²ᐟ³=R²ᐟ³.
• Epicycloid r = R — cardioid (1 cusp).
• Epicycloid R/r = 2 — nephroid (2 cusps).