Hypocycloids & Epicycloids — Rolling Circle Constraint (2D)

Rolling Mode

Radii (rolling-without-slip)

Fixed circle R 5

Rolling circle r 2

Trace-point offset d (×r) 1.00

Roll speed 1.0×

Named Presets

Live Readout

R / r ratio

2.50

Cusps (R/gcd)

Orbit angle φ

0.0°

Rolling spin ψ

0.0°

Contact arc rolled (no-slip check: R·φ = r·Δψ)

0.00 / 0.00

How the rolling is computed

The small circle's centre revolves around the fixed circle's centre at orbit angle φ. Rolling without slipping means the contact point has zero velocity in the fixed frame, which forces the small circle's own spin angle ψ to satisfy the no-slip constraint every frame:

hypocycloid:  dψ/dt = -((R-r)/r) · dφ/dt
epicycloid:   dψ/dt =  ((R+r)/r) · dφ/dt

This is integrated frame by frame (not evaluated from the closed-form curve) — the readout panel shows the contact arc-length identity R·φ = r·Δψ holding at every instant, which is exactly the rolling constraint. The pen point, offset d from the rolling circle's centre and fixed to its rim, then lands at:

x(t) = (R∓r)cos φ ± d·cos ψ
y(t) = (R∓r)sin φ − d·sin ψ

Because the trace is a consequence of the rolling motion rather than a plotted formula, an irrational R/r ratio visibly never closes — the pen keeps finding new gaps forever, while any rational R/r closes after exactly R/gcd(R,r) cusps and r/gcd(R,r) full revolutions of the rolling circle.