A New Approach to Homothetic Motions with Tessarines in Semi-Euclidean Space E2-4

Abstract

In this study, by using tessarines in 4-dimension semi-Euclidean space, we describe a variety of algebraic properties and give a matrix that is similar to Hamilton operators and we show that the hypersurfaces are obtained and a new motion is defined in

𝐸⁴₂. Then, this motion is proven to be homothetic motion. For this one parameter homothetic motion, we defined some theorems about velocities, pole points, and pole curves. Finally, It is found that this motion defined by the regular curve of order r on the hypersurface 𝑀𝑖₃, at every 𝑡- instant, has only one acceleration centre of order (𝑟 −1). Due to the way in which the matter is given with tessarines, the study gives some formulas, facts and properties about homothetic motion and variety of algebraic properties which are not generally known.

Keywords

• Tessarines,Homothetic motions,Pole curves,Hypersurface

TESSARİNELER İLE HOMOTETİK HAREKETLERE 𝑬𝟐 𝟒 YARI- ÖKLİD UZAYINDA YENİ BİR YAKLAŞIM

Abstract

Bu çalışmada, 4 boyutlu yarı Öklid uzayında tessarinesleri kullanarak, Hamilton operatörlerine benzer bir matris verdik ve çeşitli cebirsel özelliklerini tanımladık. Daha sonra bu hareketin homotetik hareket olabilmesi ispatlandı. Bir parametreli homotetik hareket için, pol noktaları , pol eğrileri ve hız merkezleri hakkında bazı teoremler tanımladık. Sonunda, her 𝑡 anında, bir 𝑀𝑖₃ hiperyüzeyi üzerinde eğrilerin türevleri ve 𝑟’ inci dereceden regular eğriler tarafından tanımlanan hareketin sadece (𝑟 − 1)’ inci derecen bir hız merkezine sahip olduğu bulundu. Tessarinesler ile verilen konudaki yöntemden dolayı, çalışma homotetik hareket hakkında bilinmeyen cebirsel özellikleri ve bazı formulleri , gerçekleri ve özellikleri veriyor.

Keywords

• Tessarineler,Homotetik hareketler,Pol eğrileri,Hiperyüze

References

1. [1] J. Cockle, On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, Philosophical magazine, LondonDublin-Edinburgh, 1848.
2. [2] J. Cockle, On a New Imaginary in Algebra Philosophical magazine, series3, London-Dublin-Edinburgh, 34, pp. 37--47, 1849.
3. [3] J. Cockle, On the Symbols of Algebra and on the Theory of Tessarines, 34, pp. 406-410, Philosophical magazine, series3, London-Dublin-Edinburgh, 1849.
4. [4] J. Cockle, On Impossible Equations, on Impossible Quantities and on Tessarines, ,Philosophical magazine, London-DublinEdinburgh, 1850.
5. [5] J. Cockle, On the True Amplitude of a Tessarine, Philosophical magazine, London-Dublin-Edinburgh, 1850.
6. [6] Y. Yaylı , Homothetic Motions at E4. Mech. Mach. Theory., 27 (3), 303-305, 1992.
7. [7] F. Babadağ, Homothetic Motions And Bicomplex Numbers, Algebras, Groups And Geometrıes, Vol. 26, Number 4,193-201, 2009.
8. [8] F. Babadağ, Y. Yaylı and N. Ekmekci, Homothetic Motions at (E ⁸ ) with Bicomplex Numbers (C ₃ ), Int. J. Contemp. Math. Sciences, Vol. 4, no. 33, 1619 - 1626, 2009.

9. [9] F. Babadağ, The Real Matrices forms of the Bicomplex Numbers and Homothetic Exponential motions, Journal of Advances in Mathematics, ISSN 2347-1921, Vol 8, No. 1, 1401 - 1406, 2014.
10. [10] B. O'Neill, Semi-Riemannian geometry, Academic Press, New York, 1983.