![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Brachistochrone_curve.gif/640px-Brachistochrone_curve.gif&w=640&q=50)
Brachistochrone curve
Fastest curve descent without friction / From Wikipedia, the free encyclopedia
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In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'),[1] 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 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.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Brachistochrone.gif/320px-Brachistochrone.gif)
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More 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. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.[2] The problem can be solved using tools from the calculus of variations[3] and optimal control.[4]
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Brachistochrone_curve.gif/640px-Brachistochrone_curve.gif)
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.[5] If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.