Cycloid
Rolling curves (roulettes)The arch traced by a point on a rolling wheel — solution of both the brachistochrone and tautochrone problems.
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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 Cycloid?
Mark a point on the rim of a wheel and roll the wheel along a straight line: the point rises and falls in a chain of arches called the cycloid. Each arch spans 2πr horizontally and reaches height 2r, with a cusp where the point touches the ground. Galileo named the curve and tried to measure its area by weighing cut-outs of it.
The cycloid earned the nickname “the Helen of geometers” for the quarrels it caused. It answers two famous physics questions at once: an upside-down cycloid is the brachistochrone (the fastest slide between two points under gravity, proved via the calculus of variations after Johann Bernoulli’s 1696 challenge) and the tautochrone (from any starting point, a bead reaches the bottom in the same time — Huygens’ route to an ideal pendulum clock).
Key properties
- One arch has length 8r — exactly four wheel diameters (Wren, 1658).
- The area under one arch is 3πr² — three times the rolling circle’s area.
- Cusps occur at every ground contact, where speed is momentarily zero.
- Inverted, it is both the brachistochrone and the tautochrone.
- Its evolute is another cycloid of the same size, shifted half an arch.
Where you'll see it
Skate-park transition profiles, Huygens’ pendulum clock cheeks, gear design ancestors, and every calculus-of-variations course.