On Holditch's theorem and related kinematics

Abstract

Holditch's theorem is a classical result on areas of planar curves generated by moving chords. The construction is closely related to other kinds of curves such as parallel curves, constant width curves or bicycle curves. The basic properties of these curves are compiled and a historical review on Holditch's theorem and related theorems in kinematics is given. First, the Holditch planar setting is rigorously defined and problems such as the existence of that construction or the avoidance of retrograde movements of the moving chord are considered. In the statement of Holditch's theorem, the area of a hidden ellipse appears. A polygonal approach to the theorem is used to show geometrically where this ellipse comes from. Moreover, immediate generalizations of Holditch's theorem and related results to other contexts are possible. So, in the second part, an introduction to non-Euclidean geometry is given and the extension of such results to constant curvature surfaces is presented. In addition, hidden closed curves in the constant curvature manifold related to the generalized statement of Holditch's theorem are found. Finally, a new extension of Holditch's theorem to space curves is derived in a natural way leading to the concept of Holditch surface.