rollingcircle

Rolling circle refers to a circle that rolls without slipping along a fixed curve, with the motion generating a family of curves known as roulettes. The most familiar case is a circle rolling on a straight line, where a point fixed to the circle traces a cycloid. If the tracing point is at a distance d from the center, the curve is a trochoid: x = r t − d sin t, y = r − d cos t, with r the rolling circle radius and t the rotation angle. When d = r, this reduces to a cycloid; d < r gives a curtate trochoid and d > r a prolate trochoid.

When the rolling circle moves around another circle, two main families arise. If the circle rolls externally around a fixed circle of radius R, the traced curve is an epicycloid; if it rolls internally, the curve is a hypocycloid. These roulettes are described by trigonometric parametric equations and are characterized by the radius ratio R/r. Special cases include a cardioid, which occurs for certain radius ratios in epicycloids, and various polygons-like curves such as the deltoid in hypocycloids. In the hypocycloid family, for example, the case R = 3r yields a three-cusped deltoid, while R = 2r produces a straight-line segment.

The concept of the rolling circle connects geometry, kinematics, and applications such as trochoidal gear design and the study of planetary motion analogies. Its classical study dates to early work on cycloids and roulettes, and it remains a standard example in the theory of planar motion generated by rolling without slipping.