Cycloids ( Cabri JAVA )

Cycloid is the locus of points traced out by a point on a circle which rolls without slipping along a straight line.

Parametric representation

i a t - b sint, a t - b cost )

Cycloid ( a = b )


Prolate Cycloid ( a < b )


Curate Cycloid ( a > b )


Cycloid as Isochrone
When the cycloid is put downward, the time when the particle goes and slides and meets the bottom doesn't depend on the initial position of P from the position P in place of the uniform gravity. But, it ignore friction. (Huygens)

Cycloid as Brachistochrone (line of steepest)
Around the biginning of the 18c J Bernoulli discoverd the brachistochrone property of the cycloid,namely that give two points A,B in a vertical plane, the curve along which a particle takes the least time to slide from A to B is cycloid. The observation represents the genesis of the area of mathematics now known as the theory of variations. iJohannDBernouliij