Trochoidal

Trochoidal is an adjective related to trochoids, a family of curves generated by a point attached to a circle as it rolls without slipping along a straight line or another circle. The term derives from Greek roots referring to a wheel or wheel-like motion.

In the standard case of a circle of radius r rolling along a straight line, a point at distance d from the circle’s center traces a trochoid with parametric equations x = r t − d sin t and y = r − d cos t, where t is the rotation parameter. When d equals r, the trochoid reduces to the cycloid, the classical curve traced by a point on the circumference. If d < r, the curve is a curtate trochoid, remaining above the rolling line and lacking cusps. If d > r, the curve is a prolate trochoid, which can exhibit loops below the base line.

Trochoidal curves appear in various contexts beyond pure geometry, including mechanical and computer-aided design applications. In milling and material removal, trochoidal milling describes tool paths that follow trochoid-like trajectories to distribute cutting forces more evenly and improve chip evacuation and tool life. In geometry and kinematics, trochoidal motion can model certain rolling motions and related trajectories, and it generalizes to trochoids generated by rolling circles around other circles (epitrochoids and hypotrochoids being related families when the base path is circular rather than linear). The concept unites several distinct curves under the common mechanism of a point whose path is defined by rolling rotation.