Epicycloid & Hypocycloid

Interactive Mathematical Visualization

Classic Curves

Custom Parameters

Ratio (R:r) 2:1
Curve Color
Nephroid
Epicycloid · Ratio 2:1 · 2 cusps
Speed 1.0×

Display Options

Complete Curve
Fixed Circle
Rolling Circle
Trace Point

Parametric Equations

x = (R + r)cos(t) - r·cos((R+r)t/r) y = (R + r)sin(t) - r·sin((R+r)t/r)
2
Ratio (k)
2
Cusps

An epicycloid with ratio 2:1 has 2 cusps. The rolling circle moves around the outside of the fixed circle.

Where
R = Fixed circle radius
r = Rolling circle radius
t = Parameter (angle)
k = R/r (ratio)