Cycloid and Rolling Circle
Roll a circle without slipping and connect its traced point to cycloid coordinates, cusps and arc length.
By Riano · Published · Updated · Report a correction
Rolling into Cycloids
The path traced by a point on the rim of a circular wheel as it rolls along a straight line without slipping is called a cycloid. This curve elegantly connects rotational motion with translational motion. It appears in nature, mechanics, and classic calculus problems such as the brachistochrone and tautochrone curves.
Parametric Equations of a Cycloid
Consider a circle of radius r rolling along the x-axis. The center of the circle, C, translates horizontally. If the circle has rotated through an angle θ, its center has moved a distance equal to the arc length rolled, which is rθ. Therefore, the coordinates of the center are (rθ, r).
A point P on the rim, initially at the origin (0, 0), rotates around the center. The relative position of P with respect to C is given by (−r sin θ, −r cos θ). Adding this to the center's coordinates yields the parametric equations for the cycloid:
- x(θ) = r(θ − sin θ)
- y(θ) = r(1 − cos θ)
The speed of the point P relative to the angle θ is given by the magnitude of its derivative vector. Calculating dx/dθ = r(1 − cos θ) and dy/dθ = r sin θ, we find the speed per radian to be 2r sin(θ / 2). Integrating this speed from 0 to θ yields the arc length formula 4r(1 − cos(θ / 2)).
Worked example: a half-turn with radius 2
Let us calculate the properties of the cycloid when a circle of Radius r = 2 has rotated through an Angle θ = π (which means θ / π = 1), reaching the top of its arch.
- The angle is θ = π radians.
- The center C is at (2π, 2), which is approximately (6.283185, 2).
- The point P is at x = 2(π − sin π) = 2π ≈ 6.283185 and y = 2(1 − cos π) = 2(1 − (−1)) = 4.
- The speed per radian is 2(2)sin(π / 2) = 4(1) = 4.
- The arc length is 4(2)(1 − cos(π / 2)) = 8(1 − 0) = 8.
At this top point, the point is moving horizontally at its maximum speed, and the distance it has traveled along the curved path is exactly 8.
Try the interactive rolling circle
Open the interactive Cycloid and Rolling Circle module to visualize the circle rolling and the cycloid being traced.
- Set Radius r to 1 and change Angle theta / pi to observe the circle rolling. Notice how the traced cyan path follows the point P on the wheel's rim.
- Observe that when Angle theta / pi = 0, the point P is at the origin, the speed is exactly 0, and a cusp forms where the path meets the ground.
- Set Angle theta / pi to 2. The circle has completed one full revolution. The point P is back on the ground, creating another cusp.
- Change the Radius r to 2 or 3. You will see that the path scales proportionally in all dimensions—the height doubles or triples, and so does the width of the arch.
Limitations of the model
This module uses an idealized geometric model of rolling without slipping on a perfectly flat line. In reality, wheels can slip, and surfaces are not perfectly straight. The parameter θ is a rotation angle. Converting it to elapsed time requires a separately specified angular speed; this geometric model does not integrate forces or time.
Furthermore, the numerical readouts are computed using floating-point arithmetic. While highly precise, they may exhibit tiny rounding differences compared to exact mathematical expressions. At the cusps, the speed is analytically zero, meaning the point momentarily comes to rest; however, the curvature is not finite there, and division by speed to find normal vectors would be undefined.
References
For a comprehensive introduction to parametric equations and the cycloid, refer to the OpenStax Calculus Volume 2 textbook, specifically the section covering Cycloids.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Cycloid and Rolling Circle module.
Try Cycloid and Rolling Circle on Riano →