Trajectory curves and surfaces: A new perspective via projective geometric algebra

Year 2024, Volume: 73 Issue: 1, 64 - 75, 16.03.2024

Ferhat Taş

https://doi.org/10.31801/cfsuasmas.1170867

Abstract

The aim of this work is to define quaternion curves and surfaces and their conjugates via operators in Euclidean projective geometric algebra (EPGA). In this space, quaternions were obtained by the geometric product of vector fields. New vector fields, which we call trajectory curves and surfaces, were obtained by using this new quaternion operator. Moreover, dual quaternion curves are determined by a similar method and then their generated motion is studied. Illustrative examples are given.

Keywords

Quaternions, geometric algebra, rotation, projective geometric algebra

References

• Vince, J., Imaginary Mathematics for Computer Science, Springer, ISBN-10: 3319946366, 2018. https://doi.org/10.1007/978-3-319-94637-5
• Argand, J. R., Essai Sur Une Maniere de Representer des Quantites Imaginaires Dans les Constructions Geometriques, 2nd edn. Gauthier-Villars, Paris, 1874.
• Hamilton, W. R., On quaternions: or a new system of imaginaries in algebra, Phil. Mag.3rd ser., 25(163) (1844), 10-13. doi.org/10.1080/14786444408644923
• Clifford, W. K., Preliminary sketch of bi-quaternions, Proceedings of the London Mathematical Society, s1–4(1) (1873), 381–395. doi.org/10.1112/plms/s1-4.1.381
• Study, E., Geometrie der Dynamen, Teubner, Leipzig, 1901.
• Bottema, O., Roth, B., Theoretical Kinematics (Vol. 24), Courier Corporation, ISBN 10:0486663469, ISBN 13: 9780486663463, 1990.
• Hestenes, D., New Foundations for Classical Mechanics (Vol. 15), Springer Science & Business Media, 2012. doi.org/10.1007/0-306-47122-1
• Hestenes, D., Sobczyk, G., Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Vol. 5), Springer Science & Business Media, 2012. doi.org/10.1007/978-94-009-6292-7
• Hestenes, D., Space-Time Algebra (Vol. 67), Birkhauser, Basel, 2015. doi.org/10.1007/978-3-319-18413-5
• Selig, J. M., Geometric Fundamentals of Robotics, Springer Science & Business Media, 2004. doi.org/10.1007/b138859
• Hildenbrand, D., Geometric computing in computer graphics using conformal geometric algebra Computers & Graphics, 29(5) (2005), 795-803. doi.org/10.1016/j.cag.2005.08.028
• Dorst, L., Fontijne, D., Mann, S., Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, Elsevier, 2019. ISBN: 9780123749420
• Gunn, C. G., Doing Euclidean plane geometry using projective geometric algebra, Adv. Appl. Clifford Algebras, 27, (2017), 1203–1232. https://doi.org/10.1007/s00006-016-0731-5
• Gunn, C., Geometric algebras for Euclidean geometry, Adv. Appl. Clifford Algebras 27, 185–208 (2017). https://doi.org/10.1007/s00006-016-0647-0
• Vince, J. A., Geometric Algebra for Computer Graphics, Springer, 2008. doi.org/10.1007/978-1-84628-997-2
• Perwass, C., Edelsbrunner, H., Kobbelt, L., Polthier, K., Geometric Algebra with Applications in Engineering (Vol. 4), Springer, Berlin, 2009. doi.org/10.1007/978-3-540-89068-3
• Bayro-Corrochano, E., Daniilidis, K., Sommer, G., Motor algebra for 3D kinematics: the case of the hand-eye calibration. Journal of Mathematical Imaging and Vision 13, (2000), 79–100. https://doi.org/10.1023/A:1026567812984
• Kanatani, K., Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, CRC Press. 2015. https://doi.org/10.1201/b18273
• Vaz, J., Da Rocha, J., An Introduction to Clifford Algebras and Spinors, Oxford University Press, Oxford. 2016. https://doi.org/10.1093/acprof:oso/9780198782926.001.0001
• Josipovic, M., Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics, Birkhauser, Basel, 2019. https://doi.org/10.1007/978-3-030-01756-9
• Lasenby, J. (Ed.)., Guide to Geometric Algebra in Practice (pp. 371-389), Springer, New York, 2011. https://doi.org/10.1007/978-0-85729-811-9
• Aslan, S., Yaylı, Y., Motions on curves and surfaces using geometric algebra, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), (2022), 39-50. doi.org/10.31801/cfsuasmas.878766
• Shoemake, K., Animating rotation with quaternion curves, In Proceedings of the 12th annual conference on Computer graphics and interactive techniques, (1985), 245-254. https://doi.org/10.1145/325334.325242
• https://enkimute.github.io/ganja.js/