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

Kuruoğlu, N. ; Yüce, S.

Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 337-340

Résumé

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let $E/E^{\prime }$ be a 1-parameter closed planar Euclidean motion with the rotation number $\nu $ and the period $T$. Under the motion $E/E^{\prime }$, let two points $A = (0, 0)$, $B = (a + b, 0) \in E$ trace the curves $k_A, k_B \subset E^{\prime }$ 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 \[ F_X = {[aF_B + bF_A] \over a + b} - \pi \nu a b. \] 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 \[ F_X = {[aF_B + bF_A]\over a + b} - h^2 (t_0) \pi \nu a b, \] is obtained, where $\exists t_0 \in [0, T]$.

Détail

Classification : 53A17
Keywords: Holditch Theorem; homothetic motion; Steiner formula