Epicycloid
Rolling curves (roulettes)The flower of cusps traced by a circle rolling outside a fixed circle; k = R/r sets the cusp count.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Epicycloid?
Roll a circle of radius r around the outside of a fixed circle of radius R, and a rim point traces the epicycloid. When the ratio k = R/r is an integer the curve closes after one lap with exactly k outward-pointing cusps: k = 1 gives the cardioid, k = 2 the nephroid.
Epicycles — circles rolling on circles — carried planetary astronomy from Ptolemy to Copernicus, and the same geometry now shapes the flanks of cycloidal gears in clocks and reducers. The cusps are where the tracing point momentarily touches the base circle and reverses its radial motion.
Key properties
- Integer k = R/r → k cusps and closure after one revolution.
- One arch has length 8r(R+r)/R; total length 8(R+r) for integer k.
- Enclosed area: π(R+r)(R+2r).
- k = 1 is the cardioid; k = 2 is the nephroid.
- Rational k = p/q closes after q revolutions; irrational k never closes.
Where you'll see it
Cycloidal gear teeth, historical planetary epicycles, spirograph toys, and rotary engine geometry.