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

Kuruoğlu, Nuri, and Yüce, Salim. "The generalized Holditch theorem for the homothetic motions on the planar kinematics." Czechoslovak Mathematical Journal 54.2 (2004): 337-340. <http://eudml.org/doc/30863>.

@article{Kuruoğlu2004,
abstract = {W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let $E/E^\{\prime \}$ 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 $\exists t_0 \in [0, T]$.},
author = {Kuruoğlu, Nuri, Yüce, Salim},
journal = {Czechoslovak Mathematical Journal},
keywords = {Holditch Theorem; homothetic motion; Steiner formula; Holditch Theorem; homothetic motion; Steiner formula},
language = {eng},
number = {2},
pages = {337-340},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The generalized Holditch theorem for the homothetic motions on the planar kinematics},
url = {http://eudml.org/doc/30863},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Kuruoğlu, Nuri
AU - Yüce, Salim
TI - The generalized Holditch theorem for the homothetic motions on the planar kinematics
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 2
SP - 337
EP - 340
AB - 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 $\exists t_0 \in [0, T]$.
LA - eng
KW - Holditch Theorem; homothetic motion; Steiner formula; Holditch Theorem; homothetic motion; Steiner formula
UR - http://eudml.org/doc/30863
ER -