Brachistochrone curve

Uncommon card · No. 97642

In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead from rest slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696 and famously solved in one night by Isaac Newton in 1697, though Bernoulli and several others had already found solutions of their own months earlier. The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. But the portion of the cycloid used for each varies. Specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp, while the tautochrone can use only up to the first half rotation and always ends at the horizontal...

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