Trochoid
A pen fixed at distance d from the centre of a rolling wheel: waves for d < r, the cycloid for d = r, loops for d > r.
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Formulas and key values
- Parametric equations
- Height range
- Period · along the lineIn the graph ≈ 6.283
r is the wheel’s radius (r = 1 in the graph), d the pen’s distance from its centre, and t the angle the wheel has turned.
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Start the video01 · Definition
What is the Trochoid?
Roll a wheel of radius r along a straight line and follow a point fixed to it at distance d from the centre. On the rim (d = r) the point traces the cycloid, with cusps where it touches the ground. Inside the rim (d < r, a curtate trochoid) the cusps soften into a smooth wave; outside it (d > r, a prolate trochoid, like a point on a train’s flanged wheel) they open into loops.
All three come from the same two lines, x = rt − d·sin t and y = r − d·cos t. The point’s height swings between r − d and r + d, and the pattern repeats every 2πr along the line, once per turn of the wheel.
02 · Properties
Key properties
- d = r gives the cycloid; d < r a curtate (wavy) trochoid; d > r a prolate (looped) trochoid.
- Height ranges from r − d to r + d.
- Period 2πr along the line: one turn of the wheel.
- A prolate trochoid moves backwards for part of each loop, like the flange of a train wheel.
- Rolling the wheel on a circle instead of a line gives the epitrochoid and hypotrochoid.
03 · Questions
Trochoid: common questions
- What is the equation of the Trochoid?
- Parametric equations: x = rt - d sin t, y = r - d cos t. r is the wheel’s radius (r = 1 in the graph), d the pen’s distance from its centre, and t the angle the wheel has turned.
04 · Keep exploring