Brachistochrone curve

The brachistochrone curve is the cycloidal path along which a particle sliding without friction under uniform gravity reaches a lower point in the least time.

Cycloid: A curve traced by a point on a circle rolling along a straight line. The brachistochrone is a cycloid arc, usually rotated and scaled to join the endpoints.
Johann Bernoulli: A Swiss mathematician who made major contributions to calculus, mechanics, and the calculus of variations. He posed the brachistochrone problem publicly in 1696 and supplied a solution.
Cycloidal pendulum: A pendulum constrained by cycloidal guides so its oscillation period is independent of amplitude. The same cycloid geometry gives both the fastest descent and an isochronous pendulum.
Brachistochrone problem: The problem of finding the path between two points that minimizes descent time under gravity. The curve is the solution to this optimization problem under ideal assumptions.
Straight line: The shortest Euclidean path between two points in a plane. It minimizes distance, but usually not descent time under gravity.
Calculus of variations: A mathematical framework for finding functions that extremize quantities such as time, distance, or energy. It turns the least-time path into an optimization problem over possible curves.
Jakob Bernoulli: A Swiss mathematician whose work helped establish calculus and probability theory. He independently solved his brother Johann’s challenge and developed a general method for such problems.
Tautochrone curve: A curve on which a particle reaches its lowest point in the same time from every starting position. The cycloid’s equal-time descent property is distinct from, but closely related to, its brachistochrone property.
Cycloidal gear tooth: A gear-tooth profile formed from cycloidal arcs to transmit motion between gears. Cycloidal geometry has practical uses beyond fastest-descent tracks, though gear profiles solve a different design problem.
Catenary: The curve formed by a uniform flexible chain hanging under gravity between supports. It resembles a cycloid but describes static balance rather than minimum travel time.

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