We show how to compute perspective projections in 3-dimensions using rotations and spherical inversions in the homogeneous and conformal models of Clifford Algebra. One achievement of our paper is to show that although perspective is a purely projective operation, while a Clifford algebra by its very definition is a metric tool, nevertheless and surely somewhat surprisingly we show that perspective projection can also be modeled by rotors in the homogeneous and conformal models of Clifford Algebra.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
E. Bayro-Corrochano, Motor algebra approach for visually guided robotics. Pattern Recognition, 35 (1), (2002).
L. Dorst, D. Fontijne and S. Mann, Geometric Algebra for Computer Science. Morgan-Kaufmann, 2007.
J. Foley, A. van Dam, S. Feiner and J. Hughes, Computer Graphics: Principles and Practice. Second Edition, Addison Wesley, 1990.
D. Fontijne and L. Dorst, GAviewer, interactive visualization software for geometric algebra, 2002-2010. Downloadable at www.geometricalgebra.net/downloads.
R. Goldman, Understanding Quaternions. In: GraphicalModels, Vol. 73 (2011), pp. 21-49.
R. Goldman, A homogeneous model for 3-dimensional computer graphics based on Clifford algebra for R 3In L. Dorst and J. Lasenby, editors, Guide to Geometric Algebra in Practice, pages 329–352. Springer-Verlag, 2011.
R. Goldman, Modeling Perspective Projections in 3-Dimensions by Rotations in 4-Dimensions, In: Graphical Models, Vol. 75 (2013) pp. 41–55.
R. Goldman and S. Mann, Using R(4, 4) As a Model for Computer Graphics. In preparation.
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.
Perwass C.: Geometric Algebra with Applications in Engineering. Springer- Verlag, Berlin (2009)
About this article
Cite this article
Goldman, R., Mann, S. & Jia, X. Computing Perspective Projections in 3-Dimensions Using Rotors in the Homogeneous and Conformal Models of Clifford Algebra. Adv. Appl. Clifford Algebras 24, 465–491 (2014). https://doi.org/10.1007/s00006-014-0439-3
- Clifford Algebra
- Conformal Model
- Homogeneous Model
- Perspective Projection