Cycloid and Rolling Circle
Roll a circle without slipping and connect its traced point to cycloid coordinates, cusps and arc length.
Free theory articleRoll a circle without slipping and connect its traced point to cycloid coordinates, cusps and arc length.
Concept & interpretation
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.
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.