| Title:
|
The generalized Holditch theorem for the homothetic motions on the planar kinematics (English) |
| Author:
|
Kuruoğlu, N. |
| Author:
|
Yüce, S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
337-340 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\exists t_0 \in [0, T]$. (English) |
| Keyword:
|
Holditch Theorem |
| Keyword:
|
homothetic motion |
| Keyword:
|
Steiner formula |
| MSC:
|
53A17 |
| idZBL:
|
Zbl 1080.53011 |
| idMR:
|
MR2059254 |
| . |
| Date available:
|
2009-09-24T11:12:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127891 |
| . |
| Reference:
|
[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. MR 1738623, 10.1016/S0094-114X(98)00028-7 |
| Reference:
|
[2] H. Holditch: Geometrical Theorem.Q. J. Pure Appl. Math. 2 (1858), 38–39. |
| Reference:
|
[3] M. Spivak: Calculus on Manifolds.W. A. Benjamin, New York, 1965. Zbl 0141.05403, MR 0209411 |
| Reference:
|
[4] W. Blaschke and H. R. Müller: Ebene Kinematik.Oldenbourg, München, 1956. MR 0078790 |
| . |
