Analysis of the cycloid and its evolute including intrinsic equations

The cycloid is the name of the curve traced out by a point on a circle rolling along a straight line. An evolute of a given curve is the curve comprised of the centers of curvature with respect to the points on the original curve. Most first studies of curves analyze the properties of the parametric equations that define the curves. These parametric equations are defined in terms of some coordinate systems in which the curve is analyzed. To analyze the curve free of any such  coordinate system, we have to describe the curve in terms of its so-called intrinsic properties only. In the case of plane curves the intrinsic equation of a curve is one that defines the arc length (one intrinsic property) as a function of the curvature (another intrinsic property). Such studies are the early explorations of differential geometry.

In the paper The Cycloid and Its Evolute a detailed discussion of the cycloid and its evolute is presented in order to make clear the steps needed to proceed from the parametric to the intrinsic equation for the cycloid and its evolute. The paper is more of an informal "classroom discussion" as it includes details and review materials that might not be included in a more formal presentation of the results obtained.

It should serve as a template to study other curves.