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Clifford Algebra and Flows Hans Hagen and Gerik Scheuermann

Abstract.

Clifford algebra is an extension of the usual vector space description for geometric spaces. It combines Grassmann’s inner and outer products into a single associative product. The resulting algebra allows a description of projections and rotations without matrices and has other nice properties. With this new model for geometric spaces, an analysis can be introduced that uses the additional properties compared to conventional vector spaces. After an introduction to the subject, some advantages of this model with respect to planar vector fields are discussed. For threedimensional vector fields, we derive flow surfaces in piecewise linear vector fields. It is assumed that the flow surfaces are defined by linear splines. It can be shown that these stream surfaces can be calculated with the same precision as stream lines. The computation of stream surfaces enables a better understanding of tridimensional fluid flows than stream lines, since they contain information about rotation and divergence.

§1. Introduction Vector fields and their associated flows have enjoyed increasing interest among different scientific communities in recent years. Mechanical and aeronautical engineers study fluid flows, usually air or water, around car, ship or aircraft bodies and inside turbines. Chemists try to understand the changing geometry of the border between reacting fluids, for example in combustion. Physicists have a strong interest in the field lines of magnetic, electric and gravitational fields. The mathematical description of these phanomena uses vector fields for the local tangential movement. The integral curves of the vector field lead to the actual movement. Most engineers, scientists and applied mathematicians take the usual vector algebra to express their considerations. Nevertheless, some people have rediscovered Clifford algebra and calculus for this purpose over the last decades [1,2,3,4]. Clifford algebra extends the usual vector space description for geometric spaces by combining Grassmann’s inner and outer products into a single associative product. This results in an algebra allowing a description of projections and rotations by elements of the algebra along Mathematical Methods in CAGD: Oslo 2000 Tom Lyche and Larry L. Schumaker (eds.), pp. 1–9. Copyright c 2001 by Vanderbilt University Press, Nashville, TN. ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.

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H. Hagen and G. Scheuermann

with other nice properties. The algebra can be modeled by matrices, but this is only useful for general considerations, and is rather distracting in practical calculations. With a new language for linear algebra at hand, a new language for analysis, Clifford analysis, can be developed. The additional operation of Clifford algebra allows here a unification of different differential operators. In the case of vector fields, we obtain a union of curl and divergence. Also, the formulation of curl by a plane segment instead of a normal vector generalizes nicely to higher dimensions. For some time now, we have experienced an increasing interest in topological and related properties of vector fields. We recall a simplification of topological questions about planar vector fields by considerations based on Clifford analysis. A planar vector can be seen as a unit vector multiplied by a spinor or complex number. Therefore, a vector field, can be seen as an unit vector multiplied by a complex function. For polynomial vector fields, we derive a strong relation between algebraic properties of the polynomial, and the topological properties of the vector field. The roots of the polynomial are the critical points of the vector field. If the polynomial splits into factors, the critical points of the “factor fields” are the critical points of the whole field and their Poincar´e-indices sum up. In three dimensions, we discuss some ideas on the calculation of stream surfaces in piecewise linear vector fields. §2. Clifford Algebra Clifford algebra is a way to extend the usual description of geometry by a multiplication of vectors. We give a basic introduction in the two-dimensional case but it can be done in any dimension. We start with a vector v ∈ IR2 . Together with the Euclidean standard basis {e1 , e2 } it can be written as v = v 1 e1 + v 2 e2 . The standard description as a column vector gives 0 v1 1 . = + v2 v = v1 v2 1 0 If we use square matrices instead, we could take v = v1

0 1 1 0

+ v2

1 0

0 −1

=

v2 v1

v1 −v2

.

This looks a little bit strange, but it allows a matrix multiplication of vectors: vw = =

v2 v1

v1 −v2

v 1 w1 + v 2 w2 v 1 w2 − v 2 w1

w2 w1

w1 −w2

v 2 w1 − v 1 w2 v 1 w1 + v 2 w2

.

Clifford Algebra and Flows

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With a suitable choice of the remaining two basis vectors of the square matrices, we get vw = (v1 w1 + v2 w2 ) With the terms 1 :=

1 0

1 0 0 1

0 1

,

+ (v1 w2 − v2 w1 )

i :=

0 1

−1 0

0 −1 1 0

.

,

we end up with a four-dimensional algebra G with the following rules for the multiplication: 1ej = ej , ej 1 = e j ,

j = 1, 2, j = 1, 2,

12 = 1, e2j = 1,

j = 1, 2,

i2 = −1,

e1 e2 = −e2 e1 = i. The following projections are useful for computations: < · >0 : G2 → IR ⊂ G2 a1 + be1 + ce2 + di 7→ a1

< · >1 : G2 → IR2 ⊂ G2 a1 + be1 + ce2 + di 7→ be1 + ce2 < · >0 : G2 → IRi ⊂ G2

a1 + be1 + ce2 + di 7→ di. We get

vw = (v • w) + (v ∧ w) for two vectors v, w ∈ IR2 ⊂ G2 , where • is the usual scalar product and ∧ the outer product of Grassmann. Now, we have a unification of these two products into an associative multiplication. We can find constructions like this one for every dimension n by using 2 n dimensional subalgebras of a complex matrix algebra M at(m, C) [4]. These algebras are models for a Clifford algebra describing n-dimensional Euclidean space. More details can be found in the literature [3,4,5]. In our 2D-case we have another important fact, because we can interpret the elements a1 + bi ∈ G2 of our algebra as complex numbers.

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H. Hagen and G. Scheuermann §3. Clifford Analysis

After extending the structure of the linear algebra, one may also change the analysis. This leads to a differential operator that does not depend on the coordinates as we demonstrate below. Our maps will be multivector fields A : IR2 → G2 r 7→ A(r) . A Clifford vector field is just a multivector field with values in IR2 ⊂ G2 : v : IR2 → IR2 ⊂ G2

xe1 + ye2 7→ v1 (x, y)e1 + v2 (x, y)e2 . The directional derivative of A in direction b ∈ IR2 is defined by 1 Ab (r) = lim [A(r + b) − A(r)]. →0 This allows the definition of the vector derivative of A at r ∈ IR2 , ∂A(r) : IR2 → G2 r 7→ ∂A(r) =

2 X

g k Agk (r).

k=1

This is independent of the basis {g1 , g2 } of IR2 . The vectors g1 =

i g2 γ

i g 2 = − g1 γ

with g1 ∧ g2 = γi are called reciprocal vectors. For a vector field v : IR2 → IR2 , we get in Euclidean coordinates, ∂v =

2 X j=1

ej vej =

2 X j=1

ej (

∂v1 ∂v2 e1 + e2 ) ∂ej ∂ej

∂v1 ∂v2 ∂v2 ∂v1 + )1 + ( − )i ∂e1 ∂e2 ∂e1 ∂e2 ∂v = (divv)1 + (curlv)i. =(

This differential operator integrates divergence and rotation.


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§4. Planar Vector Fields and Topology

For our analysis of vector fields, it is necessary to look at v : IR2 → IR2 ⊂ G2 in suitable coordinates. Let z = x + iy, z¯ = x − iy be complex numbers in the algebra. This means x=

1 (z + z¯) 2

y=

1 (z − z¯). 2i

We get v(r) = v1 (x, y)e1 + v2 (x, y)e2 1 1 1 1 = [v1 ( (z + z¯), (z − z¯)) − iv2 ( (z + z¯), (z − z¯))]e1 2 2i 2 2i = E(z)e1 , where E : C → C ⊂ G2 1 1 1 1 z 7→ v1 ( (z + z¯), (z − z¯)) − iv2 ( (z + z¯), (z − z¯)) 2 2i 2 2i is a complex-valued function of a complex variable. The idea is to analyze E instead of v and get topological results directly from the formulas in some interesting cases. Let us first assume that E and v are linear. Theorem 1. Let v(r) = (az + b¯ z + c)e1 be a linear vector field. For |a| 6= |b|, it has a unique zero at z0 e1 ∈ IR2 . For |a| > |b|, v has one saddle point with index −1. For |a| < |b|, it has one critical point with index +1. The special types in this case can be obtained from the following list : [ (1)] Re(b) = 0 ⇔ circle at z0 . [ (2)] Re(b) 6= 0, |a| > |Im(b)| ⇔ node at z0 . [ (3)] Re(b) 6= 0, |a| < |Im(b)| ⇔ spiral at z0 . [ (4)] Re(b) 6= 0, |a| = |Im(b)| ⇔ focus at z0 . In cases (2) − (4) one has a sink for Re(b) < 0 and a source for Re(b) > 0. For |a| = |b|, one gets a whole line of zeros. We included this simple theorem to show that this description gives topological information directly. Let us look now at the general polynomial case. Theorem 2. Let v : IR2 → IR2 ⊂ G2 be an arbitrary polynomial vector field with isolated critical points. Let E : C → C be the polynomial so that v(r) = E(z)e1 . Let Fk Q : C → C, k = 1, . . . , n be the irreducible components n of E, so that E(z) = k=1 Fk (z). Then the vector fields wk : IR2 → IR2 , wk (r) = Fk (r)e1 , have only isolated zeros z1 , . . . , zm . These are the zeros of v, and for the Poincar´e-indices we have indzj v =

n X

k=1

indzj wk .

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Fig. 1. Two critical points in an analytic vector field..

For illustration, we include two small examples. In Figure 1, we show the vector field v : IR2 → IR2 ze1 → (z − 2)¯ z e1 .

In Figure 2, we show the vector field v : IR2 → IR2 ze1 → (¯ z+

1 1 1 1 − i)(z − i)(z − )e1 . 4 2 2 4

More examples can be found in [6]. §5. Piecewise Linear Vector Fields in 3-Space and Flow Surfaces The definitions and constructions of the previous two sections can be extended to three dimensions. Since the notations do not change significantly, we omit this here. We study piecewise linear vector fields over a tetrahedrization T of a domain D ⊂ IR3 , v : D → IR3 v|T : T → IR3

x=

3 X i=0

βi pi 7→ v =

3 X i=0

βi vi ,


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Fig. 2. Three critical points in an analytic vector field..

P3 with i=0 βi = 1. pi are the vertices of the tetrahedron T , and vi given vector values. A stream line through a point a ∈ D is defined by ca : IR ⊃ Ia → IR3 τ 7→ ca (τ )

∂ca (τ ) = v(ca (τ )) ca (0) = a.

On a tetrahedron T , we have an exact solution for the stream lines, ca : IR ⊃ Ia → T τ 7→ exp(Aτ )a, where A is a matrix describing our linear vector field v : T → IR3 , v(x) = Ax + b. Detailed formulas for the calculation of exp(Aτ ) have been published by Nielson [7]. A stream surface through a curve b : [0, 1] → D σ 7→ b(σ) is defined by Sb : [0, 1] × IR → D

(σ, τ ) 7→ S(σ, τ ) ∂τ S(σ, τ ) = v(S(σ, τ )) S(σ, 0) = b(σ).

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Fig. 3. A stream surface inside a tetrahedron defined by a line segment..

(We assume that each streamline on the surface stays at the boundary of D forever to allow the simple notation [0, 1] × IR for the domain of the stream surface.) On a tetrahedron T , there is an exact solution for a stream surface Sb : [0, 1] × IR → T

(σ, τ ) 7→ (1 − σ) exp(Aτ )b(0) + σ exp(Aτ )b(1).

This allows a stream surface calculation with arbitrary precision > 0 in each tetrahedron, and finally a quite precise stream surface calculation in the whole domain. The calculation in one tetrahedron is illustrated in Figure 3. In Figure 4, a complete stream surface is shown.

Fig. 4. A stream surface in a domain divided into tetrahedra..


9 §6. Conclusion

We have introduced Clifford algebra and analysis as an alternative to conventional vector spaces. The application to planar vector fields has given relations between algebraic description and topological properties. In three dimensions, we have discussed some ideas for the precise calculation of stream surfaces which look promising. So far, we are not using the full power of Clifford analysis here, so there is still a lot of work to do. References 1. Brackx, F., R. Delanghe, and F. Sommen, Clifford Analysis, Research Notes in Mathematics 76. Pitman, London, 1982. 2. Hestenes, D., and G. Sobczyk, Clifford Algebra to Geometric Calculus, D. Reidel Publishing Company, Dordrecht, 1984. 3. Hestenes, D., New Foundations for Classical Mechanics, Kluwer Academic Publishers, Dordrecht, 1986. 4. Snygg, J., Clifford Algebra : A Computational Tool for Physicists, Oxford University Press, Oxford, 1997. 5. Gilbert, J. E., and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991. 6. Scheuermann, G., Topological Vector Field Visualization with Clifford Algebra, PhD thesis, University of Kaiserslautern, Kaiserslautern, Germany, 1999. 7. Nielson, G. M., Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains, IEEE Transactions on Visualization and Computer Graphics 5(4)(1999):360–372.. Hans Hagen Computer Science Department University of Kaiserslautern PO Box 3049 D-67653 Kaiserslautern Germany [email protected] Gerik Scheuermann Center for Image Processing and Integrated Computing University of California 2343 Academic Surge Bld. Davis, CA 95616 USA [email protected]