Hyperbolic Number Forms of the Euler-Savary Equation

Yıl 2022, Cilt: 15 Sayı: 2, 343 - 358, 31.10.2022

Duygu Çağlar Nurten Gürses

https://doi.org/10.36890/iejg.1127959

Öz

This study deals with hyperbolic number forms of the Euler-Savary Equation (ESE) that find one of the four points on a pole ray, provided the other three are known. These hyperbolic number forms are examined under one-parameter planar hyperbolic motions that are examined according to the osculating circles contacting through three infinitesimally close points. The hyperbolic number approach gives more detailed information than the traditional method. Thus, it eliminates sign errors and provides convenience in the application. As a final part, examples are given to show the utility of the practical way in the application.

Anahtar Kelimeler

Euler-Savary Equation, hyperbolic number, one-parameter planar motion, kinematics

Kaynakça

• [1] Akbıyık, M., Yüce, S.: Euler Savary’s formula on complex plane C∗. Appl. Math. E-Notes 16, 65-71 (2016).
• [2] Birman, G. S., Nomizu, K.: Trigonometry in Lorentzian geometry. Am Math Mon 91 (9), 543-549 (1984).
• [3] Birman, G. S.: On L2 and L3. Elem. Math. 43 (2), 46-50 (1988).
• [4] Blaschke, W., Müller, H. R.: Ebene kinematik. Verlag Von R. Oldenbourg. München (1956).
• [5] Bottema, O.: Cardan motion in elliptic geometry . Canadian Journal of Mathematics, 27(1), 37-43 (1975).
• [6] Buckley, R., Whitfield, E. V.: The Euler-Savary formula. Math. Gaz. 33 (306), 297-299 (1949).
• [7] Catoni, F., Zampetti, P.: Two-dimensional space-time symmetry in hyperbolic functions. Nuovo Cimento Soc. Ital. Fis. B 115 (12), 1433-1440 (2000).
• [8] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: Hyperbolic trigonometry in two-dimensional space-time geometry. Nuovo Cimento Soc. Ital. Fis. B 118 (5), 475-492 (2003).
• [9] Cockle, J.: On a new imaginary in algebra. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (226), 37-47 (1849).
• [10] Çağlar, D., Gürses, N.: New hyperbolic-number forms of the Euler Savary equation: The consideration of future pointing timelike pole rays for spacelike pole curves. In: International Conference on Mathematics and Its Applications in Science and Engineering (ICMASE), July 09-10 2020, Ankara, TURKIYE. Abstract Book 39-41 (2020).
• [11] Erdman, A. G., Sandor, G. N.: Mechanism design analysis and synthesis. Prentice-Hall Inc. (1997).
• [12] Eren, K., Ersoy, S.: Cardan positions in the Lorentzian plane. Honam Mathematical Journal, 40(1), 187-198 (2018).
• [13] Ersoy, S., Akyiğit, M.: One-parameter homothetic motion in the hyperbolic plane and Euler-Savary formula. Adv. Appl. Clifford Algebr. 21 (2), 297-313 (2011).
• [14] Fjelstad, P.: Extending special relativity via the perplex numbers. Amer. J. Phys. 54 (5), 416-422 (1986).
• [15] Fjelstad, P., Gal, S. G.: n-dimensional hyperbolic complex numbers. Adv. Appl. Clifford Algebr. 8 (1), 47-68 (1998).
• [16] Fjelstad, P., Gal, S. G.: Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers. Adv. Appl. Clifford Algebr. 11 (1), 81-107 (2001).
• [17] Gürses, N., Akbıyık, M., Yüce, S.: One-parameter homothetic motions and Euler-Savary formula in generalized complex number plane. Adv. Appl. Clifford Algebr. 26 (1), 115-136 (2016).
• [18] Hahn, L. S.: Complex numbers and geometry. American Mathematical Soc. (2019).
• [19] Hall, A. S.: Kinematics and linkage design. Waveland Press Inc. (1986).
• [20] Harkin, A. A., Harkin, J. B.: Geometry of generalized complex numbers. Math. Mag. 77 (2), 118-129 (2004).
• [21] Hartenberg, R., Danavit, J.: Kinematic synthesis of linkages. New York: McGraw-Hill. (1964).
• [22] Hirschhorn, J.: Kinematics and dynamics of plane mechanisms. McGraw-Hill. (1962).
• [23] Masal, M., Tosun, M., Pirdal, A. Z.: Euler Savary formula for the one parameter motions in the complex plane C. Int. J. Phys. Sci. 5 (1), 6-10 (2010).
• [24] Nešović, E., Petrović-Torgašev, M.: Some trigonometric relations in the Lorentzian plane. Kragujevac J. Math. 33 (25), 219-225 (2003).
• [25] Nešović, E.: Hyperbolic angle function in the Lorentzian plane. Kragujevac J. Math. 28 (28), 139-144 (2005).
• [26] Nešović, E., Petrović-Torgašev, M., Verstraelen L.: Curves in Lorentzian spaces. Bolletino dell Unione Mat. Ital. 8 (3), 685-696 (2005).
• [27] Nitta, T., Kuroe, Y.: Hyperbolic gradient operator and hyperbolic back-propagation learning algorithms. IEEE Trans. Neural Netw. Learn. Syst. 29 (5), 1689-1702 (2017).
• [28] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. (1983).
• [29] Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers. Anal. Univ. Oradea, Fasc. Math 11 (71), 110 (2004).
• [30] Rothbart, H., Brown, T. H.: Mechanical design handbook, measurement, analysis, and control of dynamic systems. McGraw-Hill Education. (2006).
• [31] Sandor, G. N., Erdman, A. G., Hunt, L., Raghavacharyulu, E.: New complex-number forms of the Euler-Savary equation in a computer-oriented treatment of planar path-curvature theory for higher-pair rolling contact. J. Mech. Des. 104, 227-232 (1982).
• [32] Sandor, G. N., Erdman, A. G., Raghavacharyulu, E.: Double-valued solutions of the Euler-Savary equation and its counterpart in Bobillier’s construction. Mech. Mach. Theory 20 (2), 145-148 (1985).
• [33] Sandor, G. N., Xu, Y., Weng, T.: A graphical method for solving the Euler-Savary equation. Mech. Mach. Theory 25 (2), 141-147 (1990).
• [34] Sobczyk, G.: The hyperbolic number plane. College Math. J. 26 (4), 268-280 (1995).
• [35] Sobczyk, G.: New foundations in mathematics: The geometric concept of number. Birkhauser. (2013).
• [36] Yaglom, I. M.: Complex numbers in geometry. Academic Press. (1968).
• [37] Yaglom, I. M.: A simple non-Euclidean geometry and its physical basis. Springer-Verlag. (1979).
• [38] Yüce, S., Kuruoğlu, N.: One-parameter plane hyperbolic motions. Adv. Appl. Clifford Algebr. 18 (2), 279-285 (2008).