The Cyclanoid

The cycloid is one of the most important parametrized curves, and it is easily illustrated using the example of a bicycle wheel. The cycloid is the curve traced by a point on the rim of a circle of radius a rolling along a line (see figure).

If k is an integer then the cycloid has k cusps. A cycloid segment from a cusp to a cusp is called an arch of the cycloid. If the cycloid is extended by a straight line then it becomes an evolute of the cycloid. An evolute of a cycloid has the same shape as the original cycloid, but it can be unrolled from either end, which makes it a very simple tautochronic curve.

Another way to extend a cycloid is to choose a point not on the rim but at some distance b from it. Then the cycloid extends to the point P 2 displaystyle P2, and the tangent at the point a coincides with the tangent at P 1.

A cycloid is a differentiable curve if its domain is [] and its slope is a constant displaystyle inf or -inf near a cusp. This means that a function f, which has values at each cusp of the cycloid, is monotonically decreasing or increasing from a to b.

The cycloid was one of the most important mathematical problems of the 17th century and it often provoked quarrels between mathematicians. Mersenne was the first to properly define it and to state its obvious properties. It was also a major problem for Newton, and for Bernoulli, who used it as the starting point for his calculus of variations.