Cycloid & Trochoid

Trace the path of a point on a rolling circle

Cycloid (d = r)

x = r(t - sin t)

y = r(1 - cos t)

Radius (r): 50

Distance (d): 50 (d = r)

Animation Speed: 1.0x

Show Rolling Circle Show Tracing Point Show Grid

What are Cycloid and Trochoid?

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.

Cycloid (d = r): The tracing point is on the circumference of the rolling circle. The curve has cusps where it touches the ground.
Curtate Cycloid (d < r): The tracing point is inside the rolling circle. The curve is wavy and never touches the ground.
Prolate Cycloid / Trochoid (d > r): The tracing point is outside the rolling circle. The curve forms loops and self-intersections.

Gear Teeth Design: Cycloid curves are used in the design of gear teeth for smooth motion transfer.
Architecture: The inverted cycloid (brachistochrone) is the curve of fastest descent.
Spirograph: Toy art based on trochoid patterns created by rolling circles.