The Holditch sickles for the open homothetic motions

Abstract

A. Tutar and N. Kuruoǧlu [1] had given the following theorem as a generalization of the classical Holditch Theorem [2]: During the closed planar homothetic motions with the period T, if the chord AB of fixed lenght a + b is moved around once on an oval k<inf>0</inf>, then a point X ∈ AB̄ (a = AX̄, b = BX̄) describes a closed path k<inf>0</inf>(X) and the Holditch Ring, which is bounded by k<inf>0</inf> and k <inf>0</inf>(X) has the surface area F = h2(t<inf>0</inf>)πab, for t<inf>0</inf> ∈ [0, T]. In this paper, under the open homothetic motions we expressed the Holditch Sickle such that the closed oval is replaced by the boundary of an bounded convex domain and so, the Holditch Sickles given by H. Pottmann [3] for one-parameter Euclidean motions generalized to the homothetic motions. © 2007 Elsevier B.V., All rights reserved.