Abstract
We construct a homogeneous model for Computer Graphics using the Clifford Algebra for ℝ3. To incorporate points as well as vectors within this model, we employ the odd-dimensional elements of this graded eight-dimensional algebra to represent mass-points by exploiting the pseudoscalars to represent mass. The even-dimensional elements of this Clifford Algebra are isomorphic to the quaternions, which operate on the odd-dimensional elements by sandwiching. Along with the standard sandwiching formulas for rotations and reflections, this paradigm allows us to use sandwiching to compute perspective projections.
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Notes
- 1.
Editorial note: The reader may find the geometrical view of Gunn (in Chap. 15, this volume) enlightening: the basis vectors represent normal vectors of coordinate planes, and the point at the origin is then the trivector representing the intersection of those three coordinate planes.
- 2.
Editorial note: Since this chapter uses the algebra ℝ3, bivectors from its 3-D space ℝ3 are always also 2-blades, but the author prefers to use the term ‘bivector’. In contrast, he describes the planes in the 4-D representational space consistently as ‘planes’, giving their spanning 4-D vectors but not representing them algebraically.
- 3.
Editorial note: Note that this chapter gives a geometric algebra description of a perspective projection onto a plane. For a geometric algebra representation of a general projective transformation in 3-D, the reader is referred to Chap. 13 in this volume, which uses ℝ3,3.
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Acknowledgements
I would like to thank Leo Dorst and Steve Mann for reading a preliminary draft of this manuscript and providing valuable comments, criticisms, and suggestions. I would also like to thank the anonymous referees for their constructive criticisms. This work is much improved as a result of the observations of these people. Any mistakes that still remain are, of course, entirely my own.
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© 2011 Springer-Verlag London Limited
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Goldman, R. (2011). A Homogeneous Model for Three-Dimensional Computer Graphics Based on the Clifford Algebra for ℝ3 . In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_16
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DOI: https://doi.org/10.1007/978-0-85729-811-9_16
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