The generalized Holditch theorem for the homothetic motions on the planar kinematics

Abstract

W. Blaschke and H. R. Muller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E' be a I-parameter closed planar Euclidean motion with the rotation number v and the period T. Under the motion E/E', let two points A = (0, 0), B = (a + b, 0) is an element of E trace the curves k(A), k(B) subset of E' and let F-A, F-B be their orbit areas, respectively. If F-X 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-X = [aF(B) + bF(A)]/ a+b - pivab. 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. Muller is expressed and F-X = [aF(B) + bF(A)]/a+b - h(2) (t(0)) pivab, is obtained, where There Existst(0) is an element of [0, T].