trochos

Trochos, from the Greek trochos meaning wheel, is a term used in geometry to denote curves generated by a circle rolling without slipping along a straight line or another circle. The family of curves traced in this way is called trochoids. The position of the tracing point relative to the rolling circle determines the type: if the point lies on the circumference (distance from center equal to the radius), the trochoid is a cycloid; if the point is inside the circle, it is a curtate trochoid; if the point is outside, it is a prolate trochoid, which may have loops when the offset is sufficiently large. A standard parametric form for a trochoid is x = R t − d sin t, y = R − d cos t, where R is the rolling circle radius, d is the distance from the center to the tracing point, and t is the rolling angle.

Trochoids arise as roulette curves, describing the locus of a point on a circle as it rolls. They have appeared in classical studies of plane curves and have practical applications in kinematics, gear and cam design, and computer graphics. In recreational mathematics, trochoids can be drawn with drawing toys like spirographs, illustrating a rich variety of shapes depending on the ratio d/R and the orientation of the motion.

See also: cycloid, roulette, epicycloid, hypotrochoid, epitrochoid.