[font=Trebuchet MS][size=150][b][color=#ff7700]Brachistochrone curve[/color][/b] is the curve that would carry an idealized point-like body, starting at rest and moving along the curve, without friction, under constant gravity, to a given end point in the shortest time.[/size][/font]

[font=Trebuchet MS][size=150][color=#ff7700][b]Tautochrone curve[/b][/color] is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point.[/size][/font]

[b][color=#ff7700][size=150]Cycloid is both the brachistochrone curve and tautochrone curve.[/size][/color][/b]