×

Computing perspective projections in 3-dimensions using rotors in the homogeneous and conformal models of Clifford algebra. (English) Zbl 1335.65030

This paper is focused on the usage of rotations and spherical inversions in the homogeneous and conformal models of Clifford algebras to compute affine and projective transformations applied in computer graphics. It is shown how to model translation, rotation, reflection and perspective projections as versors and rotors in a single Clifford algebra. An explanation of modelling a perspective projection in three-dimensional space as rotations in four-dimensional space is first given. Then, this procedure is applied on modelling of perspective projections in three-dimensional spaces by rotors in both elliptical and parabolic homogeneous models of Clifford algebras. Finally, computations of perspective projection using rotor representations expressed as composites of two spherical inversions in the conformal model of Clifford algebras is presented. Additionally, several interesting ideas about open problems are mentioned. The results presented can be applied in many scientific and engineering areas such as physics, robotics, computer graphics, etc.

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry

Software:

GAviewer
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Bayro-Corrochano, Motor algebra approach for visually guided robotics. Pattern Recognition, 35 (1), (2002). · Zbl 0988.68194
[2] L. Dorst, D. Fontijne and S. Mann, Geometric Algebra for Computer Science. Morgan-Kaufmann, 2007.
[3] J. Foley, A. van Dam, S. Feiner and J. Hughes, Computer Graphics: Principles and Practice. Second Edition, Addison Wesley, 1990. · Zbl 0875.68891
[4] D. Fontijne and L. Dorst, GAviewer, interactive visualization software for geometric algebra, 2002-2010. Downloadable at www.geometricalgebra.net/downloads.
[5] R. Goldman, Understanding Quaternions. In: GraphicalModels, Vol. 73 (2011), pp. 21-49.
[6] R. Goldman, A homogeneous model for 3-dimensional computer graphics based on Clifford algebra for R3In L. Dorst and J. Lasenby, editors, Guide to Geometric Algebra in Practice, pages 329-352. Springer-Verlag, 2011. · Zbl 1290.68124
[7] R. Goldman, Modeling Perspective Projections in 3-Dimensions by Rotations in 4-Dimensions, In: Graphical Models, Vol. 75 (2013) pp. 41-55.
[8] R. Goldman and S. Mann, Using R(4, 4) As a Model for Computer Graphics. In preparation. · Zbl 1312.65020
[9] C. Gunn, On the homogeneous model of Euclidean geometry. In L. Dorst and J. Lasenby, editors, Guide to Geometric Algebra in Practice, pages 297-328. Springer, 2011. · Zbl 1291.15066
[10] Perwass C.: Geometric Algebra with Applications in Engineering. Springer- Verlag, Berlin (2009) · Zbl 1179.15025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.