General expression for the cycloid and the centred cycloids (epi- and hypocycloids).

They can be defined in a general way as the trajectories of the movements composed of two uniform motions, circular and linear, at the same speed; or also by their intrinsic equation:
 

Intrinsic equation 1: ; Intrinsic equation 2: ; w  = 1: cycloid (A = 4 times the radius of the rolling circle) 0 < w < 1: epicycloid (,  where a is the radius of the base circle, b that of the rolling one) w > 1: hypocycloid (,  where a is the radius of the base circle, b that of the rolling one).

The polar differential equations of epi- and hypocycloids are:
 

; k < 0: hypocycloid, k > 0: epicycloid.

The evolute, and even any evolutoid of a cycloidal curve is a cycloidal curve.

By analogy, the curves with intrinsic equation  are called pseudo-cycloidal curves.
For a generalisation to the space, see spherical cycloid.
 

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© Robert FERRÉOL  2017