Simple, Double and Isoclinic Rotations with a Viable Algorithm

Melek Erdoğdu; Mustafa Özdemir

doi:10.36753/mathenot.642208

EN

Simple, Double and Isoclinic Rotations with a Viable Algorithm

Abstract

The main topic of this study is to investigate rotation matrices in four dimensional Euclidean space in two different ways. The first of these ways is Rodrigues formula and the second is Cayley formula.The most important common point of both formulas is the use of skew symmetric matrices. However, depending on the skew symmetric matrix used, it is possible to classify the rotation matrices by both formulas. Therefore, it is also revealed how the rotation matrices obtained by both formulas are classified as simple, double

or isoclinic rotation. Eigenvalues of skew symmetric matrices play the major role in this classification. With the use of all results, it is also seen which skew symmetric matrix is obtained from a given rotation matrix by Rodrigues and Cayley formula, respectively. Finally, an algorithm for classification of rotations is given with the help of the obtained datas and explained with an example.

Keywords

Euclidean Four Space,Rodrigues Rotation Formula,Rotation matrix,Cayley Rotation Formula

References

[1] J.H. Gallier : Geometric Methods and Applications, For Computer Science and Engineering. Texts in Applied Mathematics 38 (2011): 680p.
[2] J. H. Gallier : Notes on Differential Geometry and Lie Groups. University of Pennsylvania (2014): 730p. [3] C. M. Geyer : Catadioptric Projective Geometry: Theory and Applications. Doctorial Dissertation,University of Pennsylvania (2003).
[4] J. Gallier and D. Xu : Computing Exponentials of Skew Symmetric Matrices and Logarithms of Orthogonal Matrices. International Journal of Robotics and Automation 18 (2000): 10-20.
[5] J. E. Mebius : Derivation of Euler-Rodrigues Formula for three-dimensional rotations from the general formula for four dimensional rotations, arxiv: math.GM (2007).
[6] D. Serre : Matrices: Theory and Applications, Graduate text in Mathematics, Springer - Verlag, London (2002).
[7] C.L. Bottasso and M. Barri : Integrating finite rotations. Computer Methods in Applied Mechanics and Engineering 164 (1998): 307-331.
[8] A. Ibrahimbegovic : On the choice of finite rotation parameters. Computer Methods in Applied Mechanics and Engineering 149 (1997): 49-71.
[9] A. Jadczyk, J. Szulga : A Comment on "On the Rotation Matrix in Minkowski Space-time" by Özdemir and Erdoğdu. Reports on Mathematical Physics 74 (2014): 9-44.
[10] B. Bükçü : On the Rotation Matrices in Semi-Euclidean Space. Commun. Fac. Sci. Univ. Ank. Series A1. 55 (2006): 7-13.
[11] M. Özdemir, M. Erdoğdu : On the Rotation Matrix in Minkowski Space-time. Reports on Mathematical Physics 74 (2014): 27-38.

[12] M. Erdoğdu, M. Özdemir : Cayley formula in Minkowski space-time.International Journal of Geometric Methods in Modern Physics 12 (2015): 1-11.
[13] E.B. Dam, M. Koch, M. Lillholm : Quaternions, Interpolation and Animation. DIKU, Technical Report (1998): TR-98/5.
[14] J. Schmidt, H. Nieman : Using Quaternions for Parametrizing 3-D Rotations in Unconstrained Nonlinear Optimization, Vision Modeling and Visualization. Stuttgart, Germany (2001): 399-406.
[15] L. Vicci : Quaternions and Rotations in 3-Space, The Algebra and Geometric Interpretation. Microelectric Systems Laboratory, Department of Computer Sciences, UNC (2001): TR01-014.
[16] E. Celledoni and N. Safström: A Hamiltonian and Multi-Hamiltonian formulation of a rod model using quaternions. Computer Methods in Applied Mechanics and Engineering 199 (2010): 2813-2819.
[17] M. Özdemir, A.A. Ergin : Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics 56 (2006): 322-336.
[18] M. Özdemir, M. Erdoğdu, H. ¸Sim¸sek : On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions. Adv. Appl. Clifford Algebras 24 (2014): 179-192.
[19] J. E. Mebius : A Matrix Based Proof of the Quaternion Representation Theorem for Four Dimensional Rotations (2005) arxiv: math.GM/0501249v1.
[20] J. L. Weiner and G.R. Wilkens : Quaternions and Rotations in E4: The American Mathematical Monthly (2005):69-76.
[21] R.W. Brackett : Robotic Manipulators and the Product of Exponentials Formula. Mathematical Theory and Networks and Systems. Proceeding of International Symposium. Berlin, Springer-Verlag (1984): 120-127.
[22] M.R. Murray, Z. Li, S.S. Sastry : A Mathematical Introduction to Robotic Manipulation. Boca Raton F.L. CRC Press (1994).
[23] L. Kula, M.K. Karacan, Y. Yaylı : Formulas for the Exponential of Semi Symmetric Matrix of order 4: Mathematical and Computational Applications 10 (2005): 99-104.
[24] T. Politi : A Formula for the Exponential of a Real Skew-Symmetric Matrix of Order 4. BIT Numerical Mathematics 41 (2001): 842-845.
[25] J.M. Selig : Cayley Maps for SE(3). 12th IFToMM World Congress, Besancon (2007): 18-21.
[26] A. Cayley : Sur Quelques Proprietes des Determinants Gauches. The Collected Papers of Arthur Cayley SC.D.F.R.S. Cambridge University Press (1889).
[27] A. N. Norris : Euler-Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors. Mathematics and Mechanics of Solids 13 (2008): 465-498.
[28] S. Özkaldı, H. Gündo˘gan : Cayley Formula, Euler Parameters and Rotations in 3- Dimensional Lorentzian Space. Advances in Applied Clifford Algebras 20 (2010): 367-377.
[29] D. Eberly : Constructing Rotation Matrices Using Power Series. Geometric Tools LLC (2007): http://geometrictools.com/.
[30] L. Pertti : Clifford algebras and spinors. Cambridge University Press (2001): ISBN:978-0-521-00551-7.
[31] G. Gallego, A. Yezzi : A compact Formula for Derivative of a 3-D Rotation in Exponential Coordinates (2014): DOI 10.1007/s10851-014-0528-x.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Melek Erdoğdu ^*
0000-0001-9610-6229
Türkiye

Mustafa Özdemir
0000-0002-1359-4181

Publication Date

March 20, 2020

Submission Date

November 4, 2019

Acceptance Date

March 20, 2020

Published in Issue

Year 2020 Volume: 8 Number: 1

DOI

https://doi.org/10.36753/mathenot.642208

IZ

https://izlik.org/JA28PN36WY

APA

Erdoğdu, M., & Özdemir, M. (2020). Simple, Double and Isoclinic Rotations with a Viable Algorithm. Mathematical Sciences and Applications E-Notes, 8(1), 11-24. https://doi.org/10.36753/mathenot.642208

AMA

1.Erdoğdu M, Özdemir M. Simple, Double and Isoclinic Rotations with a Viable Algorithm. Math. Sci. Appl. E-Notes. 2020;8(1):11-24. doi:10.36753/mathenot.642208

Chicago

Erdoğdu, Melek, and Mustafa Özdemir. 2020. “Simple, Double and Isoclinic Rotations With a Viable Algorithm”. Mathematical Sciences and Applications E-Notes 8 (1): 11-24. https://doi.org/10.36753/mathenot.642208.

EndNote

Erdoğdu M, Özdemir M (March 1, 2020) Simple, Double and Isoclinic Rotations with a Viable Algorithm. Mathematical Sciences and Applications E-Notes 8 1 11–24.

IEEE

[1]M. Erdoğdu and M. Özdemir, “Simple, Double and Isoclinic Rotations with a Viable Algorithm”, Math. Sci. Appl. E-Notes, vol. 8, no. 1, pp. 11–24, Mar. 2020, doi: 10.36753/mathenot.642208.

ISNAD

Erdoğdu, Melek - Özdemir, Mustafa. “Simple, Double and Isoclinic Rotations With a Viable Algorithm”. Mathematical Sciences and Applications E-Notes 8/1 (March 1, 2020): 11-24. https://doi.org/10.36753/mathenot.642208.

JAMA

1.Erdoğdu M, Özdemir M. Simple, Double and Isoclinic Rotations with a Viable Algorithm. Math. Sci. Appl. E-Notes. 2020;8:11–24.

MLA

Erdoğdu, Melek, and Mustafa Özdemir. “Simple, Double and Isoclinic Rotations With a Viable Algorithm”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 1, Mar. 2020, pp. 11-24, doi:10.36753/mathenot.642208.

Vancouver

1.Melek Erdoğdu, Mustafa Özdemir. Simple, Double and Isoclinic Rotations with a Viable Algorithm. Math. Sci. Appl. E-Notes. 2020 Mar. 1;8(1):11-24. doi:10.36753/mathenot.642208

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Simple, Double and Isoclinic Rotations with a Viable Algorithm

Abstract

Simple, Double and Isoclinic Rotations with a Viable Algorithm

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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