Hypocycloid
Rolling curves (roulettes)The star traced by a circle rolling inside a fixed circle; k = 3 gives the deltoid, k = 4 the astroid.
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 Hypocycloid?
Let a circle roll without slipping around the inside of a larger circle: a point on its rim traces the hypocycloid, a star with k = R/r inward-curving sides when the ratio is an integer. The family contains famous members — k = 3 is the deltoid, k = 4 the astroid.
The case k = 2 is a genuine surprise: the traced “curve” degenerates to a straight diameter. This Tusi couple, described by Nasir al-Din al-Tusi in 1247, converts rotation into exact straight-line motion and reappeared centuries later in Copernican astronomy and in mechanical linkages.
Key properties
- Integer k = R/r → k cusps; the curve stays inside the fixed circle.
- k = 2 degenerates to a straight diameter (the Tusi couple).
- One arch has length 8r(R−r)/R; enclosed area is π(R−r)(R−2r).
- k = 3 is the deltoid; k = 4 is the astroid.
- The cusps touch the fixed circle where the tracing point meets it.
Where you'll see it
The Tusi couple in astronomy, cycloidal speed reducers, spirograph patterns, and star-shaped engineering logos.