Elliptic Quaternion and Elliptic linear Interpolation

Year 2024, Volume: 27 Issue: 3, 1189 - 1195, 25.07.2024

Melisa Rahebi , Yusuf Yaylı

https://doi.org/10.2339/politeknik.1444759

Abstract

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have been done elliptic quaternion linear interpolation on ellipsoid using elliptic quaternions. This interpolation curve is called Elerp elliptic linear interpolation. In addition, ESquad (spline elliptic quaternion interpolation) is defined by using the group structure feature of elliptic quaternion on ellipsoid.

Keywords

Quaternions, Elliptic Quaternions, Interpolation, Euclidean Space, Elerp, Spline, Esquad.

Eliptik Kuaterniyon ve Eliptik Lineer İnterpolasyon

Abstract

Öklid uzayında küre üzerinde kuaterniyonlar kullanılarak küresel spline kuaterniyon enterpolasyonu yapılmıştır. Bu yazıda, eliptik kuaterniyonları kullanarak elipsoid üzerinde eliptik kuaterniyon doğrusal enterpolasyonu yaptık. Bu enterpolasyon eğrisine Elerp eliptik doğrusal enterpolasyon adı verilir. Ayrıca ESquad (spline eliptik kuaterniyon enterpolasyonu), eliptik kuaterniyonun elipsoid üzerinde grup yapısı özelliği kullanılarak tanımlanır.

Keywords

Kuaterniyonlar, Eliptik Kuaterniyonlar, İnterpolasyon, Öklid Uzayı, Elerp, Spline, Esquad.

References

• [1] Dam, E. B., Koch M., Lillholm, M., “Quaternions, interpolation and animation”, Technical Report DIKU-TR-98/5 Institute of computer science University of Copenhagen, Denmark, July 17, (1998).
• [2] Edward P., Jon A, W., “Quaternions in computer vision and robotics”, Technical Report Department of Computer Science, Carnegie-Mellon University, (1982).
• [3] Eberly, D., “Quaternion Algebra and Calculus”, http://www.geometrictools.com/Documentation/Documentation.html.
• [4] Ghadami, R., Rahebi, J., Yaylı, Y., “Linear interpolation in Minkowski space”, International Journal of Pure and Applied Mathematics, 77(4): 469-484, (2012).
• [5] Hamilton, W. R., “Researches respecting quaternions”, Transactions of the Royal Irish Academy 21: 199-296, (1848).
• [6] Kincaid, D., Cheney, W., “Numerical Analysis”, Brooks/Cole Publishing Company, Pacific Grove, California, (1991).
• [7] Noakes, L., “A Note on Spherical Splines”, Journal of the Royal Statistical Society. Series B, 47(3): 482-488, (1985).
• [8] O'Neill, B., “Semi Riemannian Geometry with applications Storelativity”, Academic Press Inc., London, (1983).
• [9] Pletinckx, D., “Quaternion calculus as a basic tool in Computer graphics”, The Visual Computer, 5(2): 2-13, (1989).
• [10] Shoemake, K., “Animating rotation with quaternion curves”, ACM siggraph, 19(3): 245-254, (1985).
• [11] Ghadami, R., Rahebi, J., Yaylı, Y., “Spline Split Quaternion Interpolation in Minkowski Space”. Advances in Applied Clifford Algebras 23(4): 849–862, (2013).
• [12] László Szirmay-K, Magdics M., “Adapting Game Engines to Curved”. The Visual Computer (2021). https://dx.doi.org/10.1007/s00371-021-02303-2
• [13] Özdemir, M., “Elliptic Quaternions and Generating Elliptical Rotation Matrices” (2016). https://www.researchgate.net/publication/291975543).
• [14] Ghadami, R., Rahebi, J., Yaylı, Y., “Fast methods for spherical linear interpolation in minkowski space”. Advances in Applied Clifford Algebras, 25: 863-873, (2015).