Abstract
The formal language of Clifford’s algebras is attracting an increasingly large community of mathematicians, physicists and software developers seduced by the conciseness and the efficiency of this compelling system of mathematics. This contribution will suggest how these concepts can be used to serve the purpose of scientific visualization and more specifically to reveal the general structure of complex vector fields. We will emphasize the elegance and the ubiquitous nature of the geometric algebra approach, as well as point out the computational issues at stake.
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Notes
- 1.
The geometric interpretation of the decomposition of the geometric product in outer and inner products will be explained again for \(V = \mathbb{R}^{3}\) at the beginning of Sect. 2.2.
- 2.
A general demonstration (also valid for a degenerate Q) is given for example in [9, p. 88]. In Euclidean spaces, the well-known Gram-Schmidt orthogonalization can be used.
- 3.
The dualization introduced in Sect. 2.2 makes more general equations for M and J possible.
- 4.
For two blades A and B of grades a and b, the left contraction A⌋B is \(\langle A\,B\rangle _{b-a}\) when a ≤ b, it is zero otherwise. When blade A is contained in blade B, it equals the geometric product A B [7].
- 5.
Quite naturally, the exponential of a blade A is defined with the usual power series \(\sum _{k=0}^{\infty }\frac{A^{k}} {k!}\). The additivity exp(A + B) = exp(A) exp(B) is not true in general. The circular and hyperbolic functions of blades are also defined with power series.
- 6.
This explains the notation F r = (r ∗∇)F for the directional differentiation.
- 7.
In 2D, the index corresponds the number of turns the field makes around a critical point.
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Acknowledgements
This work undertaken (partially) in the framework of CALSIMLAB is supported by the public grant ANR-11-LABX-0037-01 overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-11-IDEX-0004-02).
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Ausoni, C.O., Frey, P. (2015). Geometric Algebra for Vector Field Analysis and Visualization: Mathematical Settings, Overview and Applications. In: Bennett, J., Vivodtzev, F., Pascucci, V. (eds) Topological and Statistical Methods for Complex Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44900-4_11
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