On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes

Öz

In this present paper, we will take three affine Cayley-Klein planes into consideration: A , P ò ò and  P ò . The plane  P ò is a fixed plane relative to two other moving affine Cayley-Klein (CK)-planes. We will describe one-parameter motions A A / , /  ò ò ò ò P P and /  ò ò P P and discuss the relationship between the motions A A / , /  ò ò ò ò P P and /  ò ò P P by evaluating their derivative formulae, velocity vectors and pole points. Also, we will observe moving coordinate system and after that, we will examine the canonical relative system for one-parameter planar motions in the affine CK-planes by using the notions of moving coordinate system. Moreover, Euler-Savary formula, which gives the relationship between the
curvatures of trajectory curves, will be obtained with the help of canonical relative system for oneparameter motions in affine CK-planes planes by using the method given by H. R. Müller in 1956 [1].

Anahtar Kelimeler

• Cayley-Klein planes
• one-parameter planar motion
• moving coordinate system
• kinematics
• Euler-Savary Formula

On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes

Öz

Bu çalışmada A , P ò ò ve  P ò üç afin Cayley-Klein düzlemi gözönüne alınmıştır.  P ò düzlemi diğer iki hareketli afin Cayley-Klein (CK)-düzlemine göre sabittir. Çalışmada bir parametreli
A A / , /  ò ò ò ò P P ve/  ò ò P P hareketleri tarif edilecek; türev formülleri, hız vektörleri ve pol noktaları elde edilerek A A / , /  ò ò ò ò P P ve /  ò ò P P hareketleri arasındaki ilişki tartışılacaktır. Ayrıca afin (CK)-düzlemlerinde hareketli koordinat sistemi araştırılarak bu hareketli koordinat sisteminin kavramları ile kavramları bir parametreli hareketler için kanonik izafe sistemi incelenecektir. Bu ifadelere ek olarak, kanonik izafe sistemi yardımıyla afin (CK)-düzlemlerinde bir parametreli hareketler için yörünge eğrilerinin eğrilikleri arasındaki ilişkiyi veren Euler Savary formülü H. R. Müller tarafında 1956 yılında verilen metodla elde edilecektir [1].

Anahtar Kelimeler

• Cayley-Klein düzlemleri
• bir parametreli düzlemsel hareket
• hareketli koordinat sistemi
• kinematik
• Euler-Savary formülü

Kaynakça

1. Blaschke, W., Müller, H. R. 1956. Ebene Kinematik, R. Oldenbourg, München, 269p.
2. Klein F. 1985. Ueber die sogenannte Nicht-Euklidische Geometrie. In: Gauß und die Anfänge der nicht-euklidischen Geometrie. Teubner-Archiv zur Mathematik, Springer, Vienna, Vol 4., pp. 224-238
3. Klein, F., 1967. Vorlesungen über nicht-Euklidische Geometrie, Springer, Berlin, 330p.
4. Yaglom, I. M. 1979. A simple non-Euclidean geometry and its Physical Basis, Springer-Verlag, New York, 307p.
5. Röschel, O. 1992. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen I, Journal of Geometry, Vol. 44, No. 1-2, pp. 160-170.
6. Röschel, O. 1993. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen II, Journal of Geometry, Vol. 47, No. 1-2, pp. 131-140.
7. Es, H. 2003. Motions and Nine Different Geometry, Ankara University, Graduate School of Natural and Applied Sciences, PhD Thesis, 130p, Ankara, Turkey.
8. Helzer, G. 2000. Special Relativity with Acceleration, The American Mathematical Monthly, Vol. 107, No. 3, pp. 219-237.

9. Herranz, F. J., Santader, M. 1997. Homogeneous Phase Spaces: The Cayley-Klein framework. http://arxiv.org/pdf/physics/9702030v1.pdf (Access Date: 26.03.2013).
10. Salgado, R. 2006. Space-Time Trigonometry. In: AAPT Topical Conference: Teaching General Relativity to Undergraduates,AAPT Summer Meeting, July 20-21, Syrauce University, NY, 22-26.
11. Sanjuan, M. A. F. 1984. Group Contraction and Nine Cayley-Klein Geometries, International Journal of Theoretical Physics, Vol. 23, No.1, pp 1-14.
12. Spirova, M. 2009. Propellers in Affine Cayley-Klein Planes, Journal of Geometry, Vol. 93, pp. 164-167.
13. Urban, H. 1994. Über drei zwangslaufig gegeneinander bewegte Cayley/Klein-Ebenen, Geometriae Dedicata, Vol. 53, pp. 187-199.
14. McRae, A.S. 2009. Clifford Fibrations and Possible Kinematics, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 5, 072, 18p. [15] Kisil, V. V. 2012. Geometry of Möbius Transformations: Eliptic, Parabolic and Hyperbolic Actions of ,Imperial College Press, London, 208p.
15. Kisil, V. V. 2012. Geometry of Möbius Transformations: Eliptic, Parabolic and Hyperbolic Actions of ,Imperial College Press, London, 208p.
16. Dijskman, E. A. 1976. Motion Geometry of Mechanism, Cambridge University Press, Cambridge, 280p.
17. Hall, A. S. Jr. 1986. Kinematics and Linkage Design, Waveland Press, Inc., Prospect Heights, Illinois (Originally published by Prentice-Hall,Inc.,),162p.
18. [18] Ergin, A. A. 1991. On the one-parameter Lorentzian motion, Communications, Faculty of Science, University of Ankara, Series A, Vol. 40, pp. 59-66.
19. Akar, M., Yüce, S., Kuruoğlu, N. 2013. One-Parameter Planar Motion in the Galilean Plane, International Electronic Journal of Geometry (IEJG), Vol. 6, No. 1, pp. 79-88.
20. (Bayrak) Gürses, N., Yüce, S. 2014. One-Parameter Planar Motions in Affine Cayley-Klein Planes, European Journal of Pure and Applied Mathematics, Vol. 7, No. 3, pp. 335-342.
21. Tutar, A., Kuruoğlu, N., Düldül, M. 2001. On the Moving Coordinate System and Pole Points on the Lorentzian Plane, International Journal of Applied Mathematics, Vol. 7, No. 4, pp. 439-445.
22. Akbıyık, M., Yüce, S. 2015. The Moving Coordinate System and Euler-Savary's Formula for the One- Parameter Motions On Galilean (Isotropic) Plane, International Journal of Mathematical Combinatorics, Vol. 2, pp. 88-105.
23. Ergüt, M., Aydın, A. P., Bildik, N. 1988. The Geometry of the Canonical Relative System and the One-Parameter Motions in 2-dimensional Lorentzian Space, The Journal of Fırat University, Vol. 3, No. 1, pp. 113-122.
24. Akbıyık, M.2012. Moving coordinate system and Euler Savary formula on Galilean plane, Yıldız Technical University, Graduate School of Natural and Applied Sciences, Master Thesis, 90p, Istanbul, Turkey.
25. Ergin, A. A. 1992. Three Lorentzian Planes Moving With Respect to One Another And Pole Points, Communications, Faculty of Science, University of Ankara, Series A, Vol. 14, pp. 79-84.
26. Aytun, I. 2002. Euler-Savary formula for one-parameter Lorentzian plane motion and its Lorentzian geometrical interpretation, Celal Bayar University, Graduate School of Natural and Applied Sciences Master Thesis, 54p, Manisa, Turkey.
27. Ikawa, T. 2003. Euler-Savary's Formula on Minkowski Geometry, Balkan Journal of Geometry and Its Applications, Vol. 8, No. 2, pp. 31-36.
28. Dooner, D. B., Griffis, M. W. 2007. On the Spatial Euler-Savary Equations for Envelopes, Journal of Mechanical Design, Vol. 129, No. 8, pp. 865-875.
29. Buckley, R., Whitfield, E. V. 1949. The Euler-Savary Formula, The Mathematical Gazette, Vol. 33, No. 306, p. 297-299.
30. Garnier, R. 1951. Cours de cinématique, géométrie et cinématique cayleyennes, Gauthier-Villars, Paris.
31. Sandor, G. N., Xu, Y., Weng, T-C.1990. A Graphical Method for Solving the Euler-Savary Equation, Mechanism and Machine Theory, Vol. 25, No. 2, pp. 141-147.
32. Sandor, G. N., Arthur, G.E., Raghavacharyulu, E. 1985. Double Valued Solutions of the Euler-Savary Equation and Its Counterpart in Bobillier's Construction, Mechanism and Machine Theory, Vol. 20, No. 2, pp. 145-178.
33. Ergüt, M., Aydın, A.P., Bildik, N. 1989. The Curvature of the Trajectory curves of the pole curves , and the point correspondence in 2-dimensional Lorentzian Plane The Journal of Fırat University, Vol. 4, No. 1, pp. 2-33.
34. Ito, N., Takahashi, K. 1999. Extension of the Euler-Savary Equation to Hypoid Gears, JSME Int. Journal. Ser C. Mech Systems, Vol. 42, No. 1, pp. 218-224.
35. Röschel, O. 1983. Zur Kinematik der isotropen Ebene, Journal of Geometry, Vol. 21, pp. 146-156.
36. Röschel, O. 1985. Zur Kinematik der isotropen Ebene II., Journal of Geometry, Vol. 24, pp. 112-122.