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

Year 2019, Volume: 21 Issue: 61, 137 - 148, 15.01.2019

Abstract

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].

References

  • Blaschke, W., Müller, H. R. 1956. Ebene Kinematik, R. Oldenbourg, München, 269p.
  • 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
  • Klein, F., 1967. Vorlesungen über nicht-Euklidische Geometrie, Springer, Berlin, 330p.
  • Yaglom, I. M. 1979. A simple non-Euclidean geometry and its Physical Basis, Springer-Verlag, New York, 307p.
  • Röschel, O. 1992. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen I, Journal of Geometry, Vol. 44, No. 1-2, pp. 160-170.
  • Röschel, O. 1993. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen II, Journal of Geometry, Vol. 47, No. 1-2, pp. 131-140.
  • Es, H. 2003. Motions and Nine Different Geometry, Ankara University, Graduate School of Natural and Applied Sciences, PhD Thesis, 130p, Ankara, Turkey.
  • Helzer, G. 2000. Special Relativity with Acceleration, The American Mathematical Monthly, Vol. 107, No. 3, pp. 219-237.
  • 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).
  • 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.
  • Sanjuan, M. A. F. 1984. Group Contraction and Nine Cayley-Klein Geometries, International Journal of Theoretical Physics, Vol. 23, No.1, pp 1-14.
  • Spirova, M. 2009. Propellers in Affine Cayley-Klein Planes, Journal of Geometry, Vol. 93, pp. 164-167.
  • Urban, H. 1994. Über drei zwangslaufig gegeneinander bewegte Cayley/Klein-Ebenen, Geometriae Dedicata, Vol. 53, pp. 187-199.
  • 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.
  • Kisil, V. V. 2012. Geometry of Möbius Transformations: Eliptic, Parabolic and Hyperbolic Actions of ,Imperial College Press, London, 208p.
  • Dijskman, E. A. 1976. Motion Geometry of Mechanism, Cambridge University Press, Cambridge, 280p.
  • Hall, A. S. Jr. 1986. Kinematics and Linkage Design, Waveland Press, Inc., Prospect Heights, Illinois (Originally published by Prentice-Hall,Inc.,),162p.
  • [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.
  • 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.
  • (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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Ikawa, T. 2003. Euler-Savary's Formula on Minkowski Geometry, Balkan Journal of Geometry and Its Applications, Vol. 8, No. 2, pp. 31-36.
  • 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.
  • Buckley, R., Whitfield, E. V. 1949. The Euler-Savary Formula, The Mathematical Gazette, Vol. 33, No. 306, p. 297-299.
  • Garnier, R. 1951. Cours de cinématique, géométrie et cinématique cayleyennes, Gauthier-Villars, Paris.
  • 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.
  • 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.
  • 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.
  • 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.
  • Röschel, O. 1983. Zur Kinematik der isotropen Ebene, Journal of Geometry, Vol. 21, pp. 146-156.
  • Röschel, O. 1985. Zur Kinematik der isotropen Ebene II., Journal of Geometry, Vol. 24, pp. 112-122.

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

Year 2019, Volume: 21 Issue: 61, 137 - 148, 15.01.2019

Abstract

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].

References

  • Blaschke, W., Müller, H. R. 1956. Ebene Kinematik, R. Oldenbourg, München, 269p.
  • 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
  • Klein, F., 1967. Vorlesungen über nicht-Euklidische Geometrie, Springer, Berlin, 330p.
  • Yaglom, I. M. 1979. A simple non-Euclidean geometry and its Physical Basis, Springer-Verlag, New York, 307p.
  • Röschel, O. 1992. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen I, Journal of Geometry, Vol. 44, No. 1-2, pp. 160-170.
  • Röschel, O. 1993. Zur Krümmungsverwandtschaft von zwanglaufen in affinen CK-Ebenen II, Journal of Geometry, Vol. 47, No. 1-2, pp. 131-140.
  • Es, H. 2003. Motions and Nine Different Geometry, Ankara University, Graduate School of Natural and Applied Sciences, PhD Thesis, 130p, Ankara, Turkey.
  • Helzer, G. 2000. Special Relativity with Acceleration, The American Mathematical Monthly, Vol. 107, No. 3, pp. 219-237.
  • 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).
  • 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.
  • Sanjuan, M. A. F. 1984. Group Contraction and Nine Cayley-Klein Geometries, International Journal of Theoretical Physics, Vol. 23, No.1, pp 1-14.
  • Spirova, M. 2009. Propellers in Affine Cayley-Klein Planes, Journal of Geometry, Vol. 93, pp. 164-167.
  • Urban, H. 1994. Über drei zwangslaufig gegeneinander bewegte Cayley/Klein-Ebenen, Geometriae Dedicata, Vol. 53, pp. 187-199.
  • 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.
  • Kisil, V. V. 2012. Geometry of Möbius Transformations: Eliptic, Parabolic and Hyperbolic Actions of ,Imperial College Press, London, 208p.
  • Dijskman, E. A. 1976. Motion Geometry of Mechanism, Cambridge University Press, Cambridge, 280p.
  • Hall, A. S. Jr. 1986. Kinematics and Linkage Design, Waveland Press, Inc., Prospect Heights, Illinois (Originally published by Prentice-Hall,Inc.,),162p.
  • [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.
  • 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.
  • (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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Ikawa, T. 2003. Euler-Savary's Formula on Minkowski Geometry, Balkan Journal of Geometry and Its Applications, Vol. 8, No. 2, pp. 31-36.
  • 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.
  • Buckley, R., Whitfield, E. V. 1949. The Euler-Savary Formula, The Mathematical Gazette, Vol. 33, No. 306, p. 297-299.
  • Garnier, R. 1951. Cours de cinématique, géométrie et cinématique cayleyennes, Gauthier-Villars, Paris.
  • 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.
  • 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.
  • 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.
  • 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.
  • Röschel, O. 1983. Zur Kinematik der isotropen Ebene, Journal of Geometry, Vol. 21, pp. 146-156.
  • Röschel, O. 1985. Zur Kinematik der isotropen Ebene II., Journal of Geometry, Vol. 24, pp. 112-122.
There are 36 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Nurten Gürses 0000-0001-8407-854X

Salim Yüce 0000-0002-8296-6495

Publication Date January 15, 2019
Published in Issue Year 2019 Volume: 21 Issue: 61

Cite

APA Gürses, N., & Yüce, S. (2019). On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen Ve Mühendislik Dergisi, 21(61), 137-148.
AMA Gürses N, Yüce S. On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes. DEUFMD. January 2019;21(61):137-148.
Chicago Gürses, Nurten, and Salim Yüce. “On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes”. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen Ve Mühendislik Dergisi 21, no. 61 (January 2019): 137-48.
EndNote Gürses N, Yüce S (January 1, 2019) On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi 21 61 137–148.
IEEE N. Gürses and S. Yüce, “On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes”, DEUFMD, vol. 21, no. 61, pp. 137–148, 2019.
ISNAD Gürses, Nurten - Yüce, Salim. “On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes”. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi 21/61 (January 2019), 137-148.
JAMA Gürses N, Yüce S. On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes. DEUFMD. 2019;21:137–148.
MLA Gürses, Nurten and Salim Yüce. “On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes”. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen Ve Mühendislik Dergisi, vol. 21, no. 61, 2019, pp. 137-48.
Vancouver Gürses N, Yüce S. On the Moving Coordinate System and Euler-Savary Formula in Affine Cayley-Klein Planes. DEUFMD. 2019;21(61):137-48.

Dokuz Eylül Üniversitesi, Mühendislik Fakültesi Dekanlığı Tınaztepe Yerleşkesi, Adatepe Mah. Doğuş Cad. No: 207-I / 35390 Buca-İZMİR.