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.
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
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/
Year 2024,
Volume: 73 Issue: 1, 64 - 75, 16.03.2024
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
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
Taş, F. (2024). Trajectory curves and surfaces: A new perspective via projective geometric algebra. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(1), 64-75. https://doi.org/10.31801/cfsuasmas.1170867
AMA
Taş F. Trajectory curves and surfaces: A new perspective via projective geometric algebra. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2024;73(1):64-75. doi:10.31801/cfsuasmas.1170867
Chicago
Taş, Ferhat. “Trajectory Curves and Surfaces: A New Perspective via Projective Geometric Algebra”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 1 (March 2024): 64-75. https://doi.org/10.31801/cfsuasmas.1170867.
EndNote
Taş F (March 1, 2024) Trajectory curves and surfaces: A new perspective via projective geometric algebra. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 1 64–75.
IEEE
F. Taş, “Trajectory curves and surfaces: A new perspective via projective geometric algebra”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 1, pp. 64–75, 2024, doi: 10.31801/cfsuasmas.1170867.
ISNAD
Taş, Ferhat. “Trajectory Curves and Surfaces: A New Perspective via Projective Geometric Algebra”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/1 (March 2024), 64-75. https://doi.org/10.31801/cfsuasmas.1170867.
JAMA
Taş F. Trajectory curves and surfaces: A new perspective via projective geometric algebra. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:64–75.
MLA
Taş, Ferhat. “Trajectory Curves and Surfaces: A New Perspective via Projective Geometric Algebra”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 1, 2024, pp. 64-75, doi:10.31801/cfsuasmas.1170867.
Vancouver
Taş F. Trajectory curves and surfaces: A new perspective via projective geometric algebra. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(1):64-75.