A cycloid is a trajectory formed when a fixed point on a circle rolls along a straight line. Cycloids have very interesting mathematical properties, one of which is curvature. The radius of curvature k of a simple cycloid (roller does not slip) can be calculated by the following formula: k = 4l/t 2.
Where l is the vertical distance from the center of the roller to the contact point, which is twice the radius r of the roller. T is the time for the roller to rotate once, and it is also a period of the cycloid. The derivation of this formula involves the parametric equation of calculus and cycloid. In fact, curvature can be regarded as the speed at which a curve changes in the tangent direction at a certain point, so it usually involves the second derivative. But for this special curve of cycloid, the curvature can be directly calculated by the above formula.