The Cycloid
Let the parameter measure the central angle of rotation of a rolling wheel when
the point of interest on the wheel has rotated from its initial position Po = ( 0 , 0 ) to the
point with location P = ( x ( ) , y ( ) ). Then it is evident from the geometry of the
figure below that:
x ( ) = | OQ | ‒ | P R |
= arclength (P Q ) ‒ | P R |
⟹ x ( ) = r · ‒ r· sin (1)
and
y ( ) = | C Q | ‒ | C R |
⟹ y ( ) = r ‒ r · cos (2)
From (2) it follows that r · cos = r ‒ y ( )
r y ( )
⟹ cos ;
r
r ( ‒ ")
r y ( )
⟹ arccos (3)
r
" – ( − ")" = "" − ""
provided that ∈ [ 0 , ].
Now substitute for from (3) into (1) to find that
ry ry
x r arccos r sin arccos
r r
Let the parameter measure the central angle of rotation of a rolling wheel when
the point of interest on the wheel has rotated from its initial position Po = ( 0 , 0 ) to the
point with location P = ( x ( ) , y ( ) ). Then it is evident from the geometry of the
figure below that:
x ( ) = | OQ | ‒ | P R |
= arclength (P Q ) ‒ | P R |
⟹ x ( ) = r · ‒ r· sin (1)
and
y ( ) = | C Q | ‒ | C R |
⟹ y ( ) = r ‒ r · cos (2)
From (2) it follows that r · cos = r ‒ y ( )
r y ( )
⟹ cos ;
r
r ( ‒ ")
r y ( )
⟹ arccos (3)
r
" – ( − ")" = "" − ""
provided that ∈ [ 0 , ].
Now substitute for from (3) into (1) to find that
ry ry
x r arccos r sin arccos
r r
