Trochoids

Trochoids are a family of plane curves traced by a fixed point attached to a circle of radius R as the circle rolls without slipping along a straight line. The shape of the trochoid depends on the distance d from the circle’s center to the tracing point. If d equals R, the curve is a cycloid; if d is less than R, it is a curtate trochoid; if d is greater than R, it is a prolate trochoid.

The standard parametric form uses the rotation angle θ of the rolling circle. As the circle rolls, its center moves along the line by a distance Rθ, and the tracing point, located a distance d from the center, has coordinates (d cos θ, d sin θ) relative to the center. This yields the trochoid equations: x(θ) = Rθ − d sin θ, y(θ) = R − d cos θ. The cycloid case (d = R) becomes x(θ) = R(θ − sin θ), y(θ) = R(1 − cos θ).

Shape characteristics vary with d. The cycloid (d = R) has cusps at θ multiples of 2π. Curtate trochoids (d < R) lie above the baseline and have no cusps, while prolate trochoids (d > R) can exhibit loops and self-intersections.

Related curves include epitrochoids and hypotrochoids, which arise when the rolling circle moves around the outside or inside of another circle, respectively. Trochoids provide a unified framework for studying cycloidal motion and its generalizations in kinematics and geometry.