Abstract
We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for Euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blades. We identify sandwich operators in the algebra that provide all Euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct Euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of group elements. We conclude with an elementary account of Euclidean kinematics and rigid body motion within this framework.
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Notes
- 1.
We use superscripts for W and subscripts for W ∗ since W ∗ will be the more important algebra for our purposes.
- 2.
Note that the orientation of e 31 is reversed; this is traditional since Plücker introduced these line coordinates.
- 3.
Editorial note: The reader may find the alternative coordinate-free construction in Sect. 18.3 enlightening.
- 4.
Blurring the distinction between these two spaces may have led some authors to incorrect conclusions about the homogeneous model of Euclidean geometry, see [21, p. 11].
- 5.
Orientation is an interesting topic which lies outside the scope of this article.
- 6.
In fact, the validity of most of the above calculations requires that I 2=0.
- 7.
The presence of a minus sign (or “the factor (−1)nk”) in some literature arises from the fact that the desired reflection is in the hyperplane orthogonal to the 1-vector appearing in the sandwich. Since 1-vectors represent hyperplanes here, in the dual algebra, no such correction factor is required.
- 8.
A convention apparently introduced by Klein, see [18].
- 9.
A polarity is an involutive projectivity that swaps points and planes.
- 10.
- 11.
We restrict ourselves to the case of a finite set of mass points, since extending this treatment to a continuous mass distribution presents no significant technical problems; summations have to be replaced by integrals.
- 12.
It remains to be seen if this approach represents an improvement over the linear algebra approach, which could also be maintained in this setting.
- 13.
From corpus, Latin for body.
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Gunn, C. (2011). On the Homogeneous Model of Euclidean Geometry. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_15
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