**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 )**

**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