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CYCLOIDAL CURVE

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