EPICYCLOID

Definition

The epicycloid is the curve traced by a point fixed on a generating circle as it rolls without slipping on the outside of a second circle, called the directing circle. It is one of the curves derived from the cycloid, obtained when the rolling takes place no longer on a straight line, but on the circumference of a circle.

Context in watchmaking

In watchmaking, the epicycloidal profile shapes the tip of the teeth of gear trains: this part of the tooth that extends beyond the pitch circle and engages the complementary tooth of the wheel or pinion in mesh. It is combined with the hypocycloidal profile, which shapes the flanks of the teeth, to form a complete and coherent cycloidal gearing system.

Applications

Unlike involute gearing, where the two meshing teeth share an identical profile, cycloidal gearing combines two distinct and complementary curves: the epicycloidal tip of one tooth must correspond precisely to the hypocycloidal root of the tooth it engages with. This complementarity requirement makes cycloidal gearing more sensitive to center distance errors than involute gearing, but it remains preferred in watchmaking for its ability to ensure smooth meshing with pinions having a very small number of leaves.

Good to know

The radius of the generating circle used to trace the epicycloid directly influences the shape of the resulting tooth tip: a larger generating circle produces a more rounded tooth tip, while a smaller circle produces a more pointed tip.