Reflections with Respect to Line and Hyperplane from Quaternionic Point of View

Abstract

In this study, the reflections in and are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case.

Keywords

• Quaternions,Reflections,Eigenvalues,Eigenvectors,Rigid Motions

Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar

Abstract

Bu çalışmada,  ve  uzayında yansımalar birim kuaterniyonlar ile incelenmiştir. İlk olarak,  uzayında orjinden geçen ve kuaterniyona dik doğrultudaki doğru boyunca yansımayı belirten bir lineer dönüşüm tanımlanmıştır. Ardından, ortaya çıkan sonuçlar örneklendirilmiştir. Benzer şekilde, uzayında orjinden geçen hiperdüzlem ve kuaterniyon doğrultusundaki doğru boyunca yansımalara karşılık gelen dönüşümler tanıtılmıştır. Son olarak bu yansıma dönüşümlerinin matris temsilleri elde edilmiş ve her durum için bu özdeğer ve özvekörlerin hesaplanması ile geometrik yorumlar kuaterniyon katsayıları ile analiz edilmiştir. 

Keywords

• Kuaterniyonlar,Yansımalar,Özdeğerler,Özvektörler,Katı hareketler

References

1. Erdoğdu M, Özdemir M, 2015. Cayley Formula in Minkowski Space-time. International Journal of Geometric Methods in Modern Physics, 12.
2. Erdoğdu M, Özdemir M, 2018. Generating Four Dimensional Rotation Matrices, in progres.
3. Friedberg AJ, Insel LE, Spence LE, 2003. Linear Algebra. Prentice Hall Pearson Education International, New Jersey.
4. Gracia AP, Thomas F, 2017. On Cayley’s Factorization of 4D Rotations and Applications. Advances in Applied Clifford Algebras, 27: 523-538.
5. Gonzalez GA, Aragon JL, Rodriguez-Andrade MA, Verde Star L, 2009. Reflections, Rotations and Pythagorean Numbers. Advances in Applied Clifford Algebras, 19:1-14.
6. Gürlebeck K, Sprössig W, 1997. Quaternionic and Clifford Calculus for Physicists and Engineers, Series: Mathematical Methods in Practice 1. Wiley.
7. Hacısalihoğlu H.H, 1983. Hareket Geometrisi ve Kuaterniyonlar Teorisi.Hacısalihoğlu Yayıncılık, Türkiye.
8. Jadczyk A, Szulga J, 2016. Lorentzian Transformations from Elementary Point of Wiew. The Electronic Journal of Linear Algebra, 31: 794-813.

9. Jadczyk A, Szulga J, 2014. A Comment on "On the Rotation Matrix in Minkowski Space-time" by Özdemir and Erdoğdu. Reports on Mathematical Physics, 74: 39-44.
10. Keçelioğlu O, Özkaldı S, Gündoğan H, 2012. Rotations and Screw Motion with Timelike Vector in 3-Dimensional Lorentzian Space. Advances in Applied Clifford Algebras, 22:1081-1091.
11. Nesovic E, 2016. On Rotation About Lightlike Axis in Three Dimensional Minkowski Space. Advances in Applied Clifford Algebras, 26: 237-251.
12. Özdemir M, 2016. An Altenative Approach to Eliptical Motion. Advances in Applied Clifford Algebras, 26: 279-304.
13. Özdemir M, Ergin AA, 2006. Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics, 56: 322-336.
14. Özdemir M, Erdoğdu M, Şimşek H, 2014. On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions. Advances in Applied Clifford Algebras, 24: 179-192.
15. Özdemir M, Erdoğdu M, 2014. On the Rotation Matrix in Minkowski Space-time. Reports on Mathematical Physics, 74: 27-38.
16. Özkaldı S, Gündoğan H, 2010. Cayley Formula, Euler Paremeters and Rotations in 3-Dimensional Lorentzian Space. Advances in Applied Clifford Algebras, 20: 367-377.
17. Roman S, 2008. Advanced Linear Algebra, Graduade text in mathematics. Springer, USA.
18. Şenyurt S, Çalışkan A, 2018. The Quaternionic Expression of Ruled Surfaces. Filomat, 32: 5753-5766.
19. Şenyurt S, Cevahir C, Altun Y, 2017. On Spatial Quaternionic Involute Curve A New View. Advances in Applied Clifford Algebras, 27: 1815-1824.
20. Şenyurt S, Grill L, 2015. Spherical Indicatrix Curves of Spatial Quaternionic Curvesç Applied Mathematical Sciences, 9: 4469-4477.
21. Şimşek H, Özdemir M, 2017. Rotations on Lightcone in Minkowski 3-Space. Advances in Applied Clifford Algebras, 27: 2841-2853.
22. Şimşek H, Özdemir M, 2016. Generating Hyperbolical Rotation Matrix for a Given Hyperbolid. Linear Algebra and Its Applications, 496: 221-245.
23. Ünal D, Güngör MA, Tosun M, 2016. Homethetic Cayley Formula and Its Applications. Advances in Applied Clifford Algebras, 26: 809-824.
24. Wilkins DR, 1844-1850. On Quaternions or On A New System of Imaginaries in Algebra by William Rowan Hamilton. Philosophical Magazine.
25. Zhang F, 1997. Quaternions and Matrices of Quaternions. Linear Algebra and Its Applications, 251: 21-57.