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 _A, k _B ⊂ 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

$$F_X = \frac{{\left[ {aF_B + bF_A } \right]}}{{a + b}} - \pi vab.$$

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

$$F_X = \frac{{\left[ {aF_B + bF_A } \right]}}{{a + b}} - h^2 \left( {t_0 } \right)\pi vab.$$

is obtained, where $\exists t_0 \in \left[ {0,T} \right]$