Animation of cycloid and spiral curves in companion with instantaneous center of rotation and radius of curvature

In this paper, the animations for the 2D cycloid and the 3D spiral curves are done. The trajectories of instantaneous rotation center and the corresponding radius of curvature are given. We prove that the trajectory of the instantaneous center of rotation is also a cycloid. For a 3D spiral curve, the two radii and the two instantaneous centers of rotation for the spiral curve are also given. It is interesting to find that the two parameters in the Frenet equation have the same meaning of radius of curvature but in different planes. In a similar way of the 2D experience, we also confirm that the trajectory of the instantaneous center of rotation for a spiral curve is also a spiral curve. An example is also given to discuss the Puyuma express incident, a major accident in 2018. The curve of rail is interpolated and the radius of curvature is determined. Discussions on the radius of rail curve and the speed of train for the failure are done. Finally, the animation is implemented by using the MATLAB and the Mathematica software. Not only theoretical derivation for the curvature of a curve but also its real application to rail engineering is proposed.