Rollingcirclelike

Rollingcirclelike is a term used in plane geometry to describe a broad class of curves generated by the motion of a circle rolling without slipping along a reference curve, with a fixed point attached to the circle tracing a roulette-like path. The concept generalizes the familiar idea of a roulette curve, such as those produced when a circle rolls on a line or another circle.

Construction and scope

In a rollingcirclelike construction, a circle of radius r is constrained to roll along a smooth reference curve in the plane. A designated point on or inside/outside the circle traces out a curve as the circle advances. The resulting locus depends on the geometry of the reference curve and on the position of the tracing point relative to the circle. When the reference curve is a straight line, common examples include cycloids and their curtate or prolate variants. If the reference curve is another circle, the generated curves include epicycloids and hypocycloids. For more general base curves, the traced curves are sometimes called roulettes, and the rolling circlelike family encompasses a wide range of shapes from simple to highly intricate.

Examples and relatives

- Straight line base: cycloid (point on the circumference), curtate cycloid (point inside the circle), prolate cycloid (point outside the circle).

- Circular base: epicycloid (rolling on the outside), hypocycloid (rolling on the inside).

- General base curves: trochoid-like curves arising from various offsets of the tracing point and different base trajectories.

Properties

Rollingcirclelike curves can exhibit a spectrum of features, including cusps, loops, and smooth wavy forms, depending on the point’s position and the curvature of the base curve. They are studied in kinematics and geometry for their rich structure and connections to classical roulette curves, differential geometry, and applications in design and mechanical modeling.

See also

Cycloid, trochoid, roulette, epicycloid, hypocycloid, curtate cycloid, prolate cycloid.