 a brachistochrone curve (gr. Î²ÏÎ¬ÏÎ¹ÏÏÎ¿Ï, brachistos &ndash; the shortest, ÏÏÏÎ½Î¿Ï, chronos &ndash; time) or curve of fastest descent, is the path that will carry a point-like body from one place to another in the least amount of time. the body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under the action of constant gravity. the brachistochrone curve is the same as the tautochrone curve for a given starting point. == the brachistochrone is the cycloid == given two points a and b, with a not lower than b, only one upside down cycloid passes through both points, has a vertical tangent line at a, and has no maximum points between a and b: the brachistochrone curve. the curve does not depend on the body's mass or on the strength of the gravitational constant. the problem can be solved with the tools from the calculus of variations and optimal control.ross, i. m. the brachistochrone paridgm, in a primer on pontryagin's principle in optimal control, collegiate publishers, 2009. isbn 978-0-9843571-0-9. if the body is given an initial velocity at a, or if friction is taken into account, then the curve that minimizes time will differ from the one described above. ===johann bernoulli's solution=== according to fermatâs principle: the actual path between two points taken by a beam of light is the one which is traversed in the least time. in 1697 johann bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g). the conservation of energy can be used to express the speed of a body in a constant gravitational field as: :v=\sqrt{2gy}, where y represents the vertical distance the body has fallen. the speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement. johann bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density: :\frac{\sin{\theta}}{v}=\frac{1}{v}\frac{dx}{ds}=\frac{1}{v_m}, where vm is the constant and \theta represents the angle of the trajectory with respect to the vertical. the equations above allow us to draw two conclusions: # at the onset, the angle must be zero when the particle speed is zero. hence, the brachistochrone curv
Curves