Abstract
Geometric algebra was initiated by W.K. Clifford over 140 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This introduction explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, the popular conformal model, and projective geometric algebra. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc.
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Notes
- 1.
The inner product defines the measurement of length and angle.
- 2.
This setting amounts to an algebra generating relationship.
- 3.
No product sign will be introduced, simple juxtaposition implies the geometric product just like \(2x = 2 \times x\).
- 4.
A \(\mathbb {K}\)-isometry between two inner-product spaces is a \(\mathbb {K}\)-linear mapping preserving the inner products.
- 5.
A \(\mathbb {K}\)-algebra is a \(\mathbb {K}\)-vector space equipped with an associative and multilinear product. An inner-product \(\mathbb {K}\)-algebra is a \(\mathbb {K}\)-algebra equipped with an inner product structure when taken as \(\mathbb {K}\)-vector space.
- 6.
Important fields are real \(\mathbb {R}\) and complex numbers \(\mathbb {C}\), etc.
- 7.
That is a \(\mathbb {K}\)-linear homomorphism preserving the inner products, i.e., a \(\mathbb {K}\)-linear mapping preserving both the products of the algebras when taken as rings, and the inner products of the algebras when taken as inner-product vector spaces.
- 8.
The setting \(e_{\emptyset }=1\) is frequently used in geometric algebra, but not necessary.
- 9.
See also Remark 1.
- 10.
- 11.
Note that reflections at hyperplanes are nothing but the Householder transformations [44] of matrix analysis.
- 12.
A hyperplane of a nD space is a \((n-1)\)D subspace, thus a hyperplane of \(\mathbb {R}^2\), \(n=2\), is a 1D (\(2-1=1\)) subspace, i.e. a line. Every hyperplane is characterized by a vector normal to the hyperplane.
- 13.
Reversion is an anti-automorphism. Often a dagger \(A^{\dagger }\) is used instead of the tilde, as well as the term transpose.
- 14.
Note, that in general for Clifford algebras Cl(n, 0) of Euclidean spaces \(\mathbb {R}^{n,0}\) we have the identity \(\textrm{Spin}(n) = \textrm{Spin}_+(n)\), where \(\textrm{Spin}(n) = \textrm{Spin}(n,0)\). The reason is that \(A\widetilde{A} < 0\) is only possible for non-Euclidean spaces \(\mathbb {R}^{p,q}\), with \(q>0\).
- 15.
Two-fold covering means, that there are always two elements \(\pm A\) in \(\textrm{Pin}(2,0)\), \(\textrm{Spin}(2,0)\), and \(\textrm{Spin}_+(2,0)\), representing one element in \(\textrm{O}(2,0)\), \(\textrm{SO}(2,0)\) and \(\textrm{SO}_+(2,0)\), respectively.
- 16.
Also called grade projection.
- 17.
Strictly speaking, Cl(3, 0) is the algebra of directions in three dimensions, respectively the algebra of hyperplanes (and subspaces) passing through the origin. It is closely related to the linear algebra of \(\mathbb {R}^3\).
- 18.
The minus signs are only chosen, to make the product of two bivectors identical to the third, and not minus the third.
- 19.
Obviously, we can alternatively select any unit bivector \({{\textbf {u}}}\) and obtain the tessarines \(\{1, {{\textbf {u}}}i_3, {{\textbf {u}}}, i_3 \}\). The two specifications are related by \({{\textbf {u}}}= u i_3\) or \(u = -{{\textbf {u}}}i_3 = {{\textbf {u}}}^*\).
- 20.
In the context of blade subspaces, whenever a blade B contains another blade A as factor, then the geometric product is reduced to left or right contraction: \(A B = A\rfloor B, B A = B \lfloor A\).
- 21.
- 22.
To include \(e_{23}\) one can simply compute \(A-P_B(A) = e_3+e_{23}\).
- 23.
The symbol \(\vee \) stems from Grassmann-Cayley algebra.
- 24.
For definition and computation of the meet see also Sect. 4 of [5].
- 25.
Theorem 4 of [61] shows in full generality how to compute the characteristic polynomial of any \(M \in Cl(p,q)\), including the determinant and adjugate.
- 26.
Note that some authors prefer opposite signature Cl(3, 1), e.g., [37]. Furthermore, various names are in use, like Clifford’s geometric algebra of spacetime, geometric algebra of spacetime, Clifford algebra of spacetime, or geometric algebra of Minkowski spacetime, etc.
- 27.
Unique up to a nonzero scalar factor.
- 28.
Note the alternative notations: e instead of \({{{\boldsymbol{e}}}}_0\) in [49], or \(\bar{n}\) for \(e_0\) and n for \({{{\boldsymbol{e}}}}_{\infty }\) with \(n\cdot \bar{n} = 2\) in [46], or \( o \) for \({{{\boldsymbol{e}}}}_0\) and \(\infty \) for \({{{\boldsymbol{e}}}}_{\infty }\) with \( o \cdot \infty =-2\) in [12], etc.
- 29.
We use for null vectors the notation \(\textbf{e}_{o}\), and \(\textbf{e}_{\infty }\) with added indexes 1,2,3, because this intuitive notation for CGA null vectors became widespread with [10], replacing the earlier notation \(\overline{n}\) and n. The notation \(\overline{n}\) and n with added indexes 1,2,3 was used in [57], but [6] consistently combined instead \(\textbf{e}_{o}\), and \(\textbf{e}_{\infty }\) with added indexes 1,2,3, etc. [36] adopts the notation for basis vectors following [6].
- 30.
The parameters \(\lambda _i\), \(i=1,2,3\), parameterize a continuous set of horospheres [15].
- 31.
- 32.
In the case of Cl(p, q) with \(q>0\) one can use the principal reverse \(g(\boldsymbol{x})^T\) of page xx instead of the reverse \(\widetilde{g(\boldsymbol{x})}\).
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Acknowledgements
E.H.: Now to the King eternal, immortal, invisible, the only God, be honor and glory for ever and ever. Amen. [Bible, 1 Tim. 1:17] E.H. thanks his family for their patient support. We warmly thank L. Dorst for advice on Sect. 4, and for the permission to use Fig. 1. We thank S. Breuils for permission to use Fig. 2.
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Hitzer, E., Hildenbrand, D. (2024). Introduction to Geometric Algebra. In: Araujo Da Silva, D.W.H., Hildenbrand, D., Hitzer, E. (eds) Advanced Computational Applications of Geometric Algebra. ICACGA 2022. Springer Proceedings in Mathematics & Statistics, vol 445. Springer, Cham. https://doi.org/10.1007/978-3-031-55985-3_1
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