CYCLOID

Definition

The cycloid is the curve traced by a point fixed on a circle as it rolls without slipping along a straight line. It is one of the fundamental curves of rolling geometry, studied since the 17th century for its remarkable mathematical properties.

Context in watchmaking

In watchmaking, the cycloid and its variants determine the tooth profile of gears. Two derived curves are used: the epicycloid, traced by a point on a circle rolling on the outside of another circle, and the hypocycloid, traced by a point on a circle rolling on the inside of another circle. Unlike involute gearing, the most widespread in general mechanics, cycloidal gearing remains the historical and current reference in watchmaking.

Applications

The epicycloidal profile shapes the tips of the wheel teeth, while the hypocycloidal profile shapes their flanks. A special case of the hypocycloid, obtained when the diameter of the generating circle is equal to the radius of the directing circle, gives a perfectly straight flank: this is the profile chosen for the flanks of the teeth of the wheels and pinions of a watch’s train (wheel train), as it facilitates meshing with pinions having a very small number of leaves, which are common in micromechanics for watchmaking.

Good to know

This historical choice of cycloidal gearing meets a constraint specific to watchmaking: pinions there often have fewer than ten leaves (teeth), a number for which involute gearing produces less regular meshing. The cycloidal profile thus remains the standard in the construction of mechanical movement trains (wheel trains).