The Generalized Holditch Theorem for the Homothetic Motions on the Planar Kinematics

Abstract

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E′ be a 1-parameter closed planar Euclidean motion with the rotation number ν and the period T. Under the motion E/E′, let two points A = (0, 0), B = (a + b, 0) ∈ E trace the curves k <inf>A</inf> , k <inf>B</inf> ⊂ E′ and let F <inf>A</inf> , F <inf>B</inf> be their orbit areas, respectively. If F <inf>X</inf> is the orbit area of the orbit curve k of the point X = (a, 0) which is collinear with points A and B then F <inf>X</inf> = [aF <inf>B</inf> + bF <inf>A</inf> ]/a+b -πνab. In this paper, under the 1-parameter closed planar homothetic motion with the homothetic scale h = h(t), the generalization given above by W. Blaschke and H. R. Müller is expressed and F <inf>X</inf> = [aF <inf>B</inf> + bF <inf>A</inf> ]/a+b - h 2 (t <inf>0</inf> ) πνab, is obtained, where ∃t <inf>0</inf> ∈ [0,T].