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Keywords:
Holditch Theorem; homothetic motion; Steiner formula
Holditch Theorem; homothetic motion; Steiner formula
Summary:
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 $\exists t_0 \in [0, T]$.
References:
[1] A. Tutar and N. Kuruoğlu: The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics. Mech. Machine Theory 34 (1999), 1–6. DOI 10.1016/S0094-114X(98)00028-7 | MR 1738623
[2] H. Holditch: Geometrical Theorem. Q. J. Pure Appl. Math. 2 (1858), 38–39.
[3] M. Spivak: Calculus on Manifolds. W. A. Benjamin, New York, 1965. MR 0209411 | Zbl 0141.05403
[4] W. Blaschke and H. R. Müller: Ebene Kinematik. Oldenbourg, München, 1956. MR 0078790
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