An Interesting Class of Polynomial Vector Fields Gerik Scheuermann, Hans Hagen and Heinz Kr¨ uger The visualization of vector fields is one of the most important topics in visualization. Of special interest over the last years have been topology-based methods. We present a method to construct polynomial vector fields after determine the critical points and their index to generate test data and help analysing vector field topology. The idea is based on Clifford algebra and analysis. We start with a formulation of a plane real vector field in Clifford algebra giving more topological information directly from the formulas than the usual description in cartesian coordinates. The theorem presented here is an essential improvement to our previous result in [9], and has implications to general polynomial vector fields.
Abstract.
§1. Introduction Scientific Visualization often deals with a large amount of data. Especially in the visualization of vector fields one usually has the problem that it is nearly impossible to show the whole data. One has to reduce the information in a useful way. One very successful approach is to look for the topology of the field and visualize its essential parts [4]. The testing of new algorithms in this direction was done up to now by comparing with handdrawings based on experiments. The problem is that in the literature ([1], [3], [7]) one only finds an analysis of linear fields and some special equations, where the position, number and type of critical points is not arbitrary. The reason is probably that it is difficult to analyse nonlinear fields in the classical way with computing the integral curves. We have found and published a theorem in [9] which allows a description of the topology of some polynomial vector fields. This was used in [10] to derive a new algorithm for vector field visualization which can detect and visualize higher order singularities. In this paper we give an essential generalization of this theorem with a similar result for a larger class of polynomial vector fields. We also can show how the analysis of a arbitrary polynomial vector field can be simplified. This result is obtained by using Clifford algebra and analysis which are described in Section 2 and 3. The result is derived and proved in Section 4. Mathematical Methods for Curves and Surfaces II Morten Dæhlen, Tom Lyche, Larry L. Schumaker (eds.), pp. 1–10. Copyright c 1998 by Vanderbilt University Press, Nashville, TN. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.
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G. Scheuermann, H. Hagen and H. Kr¨ uger
The last section gives some examples showing how one can actually construct non-trivial polynomial vector fields by this result which should be useful for testing algorithms. §2. Clifford Algebra We need Clifford algebra only in two dimensions, so for simplicity we should stay there. Let {e1 , e2 } be an orthogonal basis of IR2 with standard scalar product. The Clifford algebra G2 is then the IR-algebra of maximal dimension containing IR and IR2 such that for each vector x ∈ IR2 holds x2 = kxk2 . This implies the following rules e2j = 1
j = 1, 2
e1 e2 + e 2 e1 = 0 We get a 4-dimensional algebra with the vector basis {1, e1 , e2 , e1 e2 }. We now have the real vectors xe1 + ye2 ∈ IR2 ⊂ G2 and the real numbers x1 ∈ IR ⊂ G2 both in the algebra. We may also define the projections h·ik , k = 0, 1, 2 by h·i0 : G2 → IR ⊂ G2 a1 + be1 + ce2 + de1 e2 7→ a1
h·i1 : G2 → IR2 ⊂ G2 a1 + be1 + ce2 + de1 e2 7→ be1 + ce2
h·i2 : G2 → IRe1 e2 ⊂ G2
a1 + be1 + ce2 + de1 e2 7→ de1 e2
For two vectors one can then describe the new product by already known products. Let v = v1 e1 +v2 e2 , w = w1 e1 +w2 e2 be two vectors, v1 , v2 , w1 , w2 ∈ IR. Then vw = (v1 w1 + v2 w2 )1 + (v1 w2 − w1 v2 )e1 e2 = hvwi0 + hvwi2
=v·w+v∧w
where · denotes the scalar (inner) product and ∧ denotes the outer product of Grassmann. This unification of inner and outer product is the starting point of the geometric interpretation. We will not need it here, so see [5] for a good introduction. More important for us is that the complex numbers can also be canonically embedded by recognizing (e1 e2 )2 = −1, so set i := e1 e2 .
An Interesting Class of Polynomial Vector Fields
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Then a1 + bi ∈ C ⊂ G2 is a subalgebra. The next section introduces a bit of Clifford analysis. §3. Clifford Analysis Here we need again only the two-dimensional case, so we limit our definitions to that case to avoid technical overload. Our basic 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
r = xe1 + ye2 7→ v(r) = 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 ∈ IR. This allows the definition of the vector derivative of A at r ∈ IR2 by ∂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 integral in Clifford analysis is defined as follows : Let M ⊂ IR2 be an oriented r-manifold and A, B : M → G 2 be two piecewise continous multivector fields. Then one defines Z
AdXB = lim M
n→∞
n X
A(xi )∆X(xi )B(xi ),
i=0
where ∆X(xi ) is a r-volume in the usual Riemannian sense. This allows the definition of the Poincar´e-index of a vector field v at a ∈ IR2 as Z 1 v ∧ dv inda v = lim , →0 2πi S 1 v2 where S1 is a circle of radius around a.
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G. Scheuermann, H. Hagen and H. Kr¨ uger §4. Analysis of Polynomial Vector Fields
For our result it is necessary to look at v : IR2 → IR2 ⊂ G2 in a different way. Let z = x + iy, z¯ = x − iy. 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, z¯)e1 , where E : C 2 → C ⊂ G2 1 1 1 1 (z, z¯) 7→ v1 ( (z + z¯), (z − z¯)) − iv2 ( (z + z¯), (z − z¯)) 2 2i 2 2i is a complex-valued function of two complex variables. The idea is now to analyse E instead of v and get topological results directly from the formulas in some interesting cases. Let us first assume that E and also v is 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| has v 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 got 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.
Proof: A computation of the derivatives of the components v1 , v2 and a comparison with the classic classification gives this result. We included this easy theorem to show that this description gives topological information more directly. Let us look now at the general polynomial case :
An Interesting Class of Polynomial Vector Fields
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Theorem 2. Let v : IR2 → IR2 ⊂ G2 be an arbitrary polynomial vector field with isolated critical points. Let E : C 2 → C be the polynomial so that v(r) = E(z, z¯)e1 . Let Fk : C 2 → QnC, k = 1, . . . , n, be the irreducible components of E, so that E(z, z¯) = k=1 Fk . Then have the vector fields wk : IR2 → IR2 , wk (r) = Fk (r)e1 also only isolated zeros z1 , . . . , zm . These are then the zeros of v and for the Poincar´e-indices we have indzj v =
n X
indzj wk ,
k=1
Proof: The wk have only isolated zeros because otherwise v would have also not isolated zeros. It is also obvious that a zero of a wk is a zero of v and a zero of v must be a zero of one of the wk . For the derivatives we get n
X ∂Fk ∂E =a ∂z ∂z ∂E =a ∂ z¯
k=1 n X
k=1
∂Fk ∂ z¯
n Y
l=1,l6=k n Y
Fl Fl .
l=1,l6=k
For the computation of the Poincar´e-index, we assume zj = 0 after a change of the coordinate system and that is so small that there are no other zeros inside S1 . We get Z v ∧ dv 1 indzj v = 2πi S1 v 2 Z n n n X Y 1 ∂Fk Y 1 Fk e1 [dza ha Fl + = 2πi S1 v 2 ∂z d¯ za
n X
k=1
Z
∂Fk ∂ z¯
k=1 n Y
l=1,l6=k n X
k=1
l=1,l6=k
Fl ]e1 i2
n Y 1 ∂Fk ∂Fk Fk (dz Fl F¯l i2 ha¯ a + d¯ z ) 2 ∂z ∂ z¯ S1 v k=1 l=1,l6=k Z n X 1 ∂Fk 1 ∂Fk hFk e1 (dz = + d¯ z )e1 i2 ¯ 2πi S1 Fk Fk ∂z ∂ z¯ k=1
1 = 2πi
= =
n X
k=1 n X
indzj Fk e1 indzj wk .
k=1
For experiments it is nice to use linear factors because one has good insights in their behavior from Theorem 1.
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G. Scheuermann, H. Hagen and H. Kr¨ uger
Theorem 3. Let v : IR2 → IR2 ⊂ G2 be the vector field v(r) = E(z, z¯)e1 with E(z, z¯) =
n Y
(ak z + bk z¯ + ck )
k=1
|ak | 6= |bk |,
and let zk be the unique zero of ak z + bk z¯ + ck . Then has v zeros at zj , j = 1, . . . , n and the Poincar´e-index of v at zj is the sum of the indices of the (ak z + bk z¯ + ck )e1 at zj . Proof: Special case of Theorem 2 §5. Examples The following examples show the topological structure of vector fields with the algebraic structure of Theorem 3. They shall illustrate its usefulness for the testing of topological vector field visualization algorithms by showing that one has full control over position and index of the critical points. Our first example is the vector field v(r) =(¯ z − (−0.51 − 0.51i))(¯ z − (0.38 − 0.92i))(z − (0.63 + 0.46i)) (z − (0.74 + 0.35i))(¯ z − (−0.33 + 0.52i))e1
in the square [−1, 1]×[−1, 1] in cartesian coordinates. Figure 1 shows that the field has saddles at (0.63, 0.46) and (0.74, 0.35) produced by the factors with z. There are further singularities of index 1 at (−0.51, −0.51), (0.38, −0.92) and (−0.33, 0.52) produced by the factors with z¯. The sampled arrows indicate the direction of the unit vector field at a 20 × 20 quadratic grid.
Fig. 1. Two saddles and three index 1 singularities.
An Interesting Class of Polynomial Vector Fields
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The second example is z − (0.51 + 0.27i))(z − (0.68 − 0.59i)) v(r) =(¯ z − (−0.58 − 0.64i))(¯
(¯ z − (−0.12 − 0.84i))2 (z − (−0.11 − 0.72i))(z − (0.74 + 0.35i))e1
again in the square [−1, 1] × [−1, 1] in cartesian coordinates. Figure 2 shows that the field has saddles at (0.68, −0.59) ,(−0.11, −0.72) and (0.74, 0.35 − 0.18, 0.46) produced by the factors with z. There are further singularities of index 1 at (−0.58, −0.64) and (0.51, 0.27) produced by the factors with z¯. There is also an index 2 singularity at (−0.12, −0.84) stemming from the squared factor. Again the sampled arrows indicate the direction of the unit vector field.
Fig. 2. Three saddles, two index 1 and one index 2 singularity.
Acknowledgments. This work was partly made possible by financial support by the Deutscher Akademischer Auslandsdienst (DAAD). The first author got a “DAAD-Doktorandenstipendium aus Mitteln des zweiten Hochschulsonderprogramms” for his stay at the Arizona State University from Oct. 96 to Jan. 97. We also want to thank Alyn Rockwood, Greg Nielson and David Hestenes from Arizona State University for many comments, suggestions and inspiration. Special thanks go to Shoeb Bhinderwala who wrote the excellent user interface to create the pictures in the examples [2]. References 1. Arnold, V. I., Gew¨ ohnliche Differentialgleichungen, Deutscher Verlag der Wissenschaften, Berlin, 1991.
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G. Scheuermann, H. Hagen and H. Kr¨ uger 2. Bhinderwala, S. A., Design and visualization of vector fields, master thesis, Arizona State University, 1997. 3. Guckenheimer, J., and P. Holmes, Nonlinear Oszillations, Dynamical Systems and Linear Algebra, Springer, New York, 1983. 4. Helman, J. L., and L. Hesselink, Visualizing vector field topology in fluid flows, IEEE Computer Graphics and Applications 11:3 (1991), 36–46. 5. Hestenes, D., New Foundations for Classical Mechanics, Kluwer Academic Publishers, Dordrecht, 1986. 6. Kr¨ uger, H., and M. Menzel, Clifford-analytic vector fields as models for plane electric currents, in Analytical and Numerical Methods in Quaternionic and Clifford Analysis, W. Spr¨ ossig, K. G¨ urlebeck (eds.), Seiffen, 1996. 7. Hirsch, M. W., and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York 1974 8. Milnor, J. W., Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, 1965. 9. Scheuermann, G., H. Hagen, and H. Kr¨ uger, Clifford algebra in vector field visualization, accepted for Vismath ’97.
10. Scheuermann, G., H. Hagen, H. Kr¨ uger, M. Menzel, and A. P. Rockwood, Visualization of higher order singularities in vector fields, IEEE Visualization Proc., ACM Press, New York, 1997, 67–74. Gerik Scheuermann and Hans Hagen Dept. of Computer Science University of Kaiserslautern Postfach 3049 D-67653 Kaiserslautern Germany scheuer,
[email protected] [email protected] [email protected] Heinz Kr¨ uger Dept. of Physics University of Kaiserslautern Postfach 3049 D-67653 Kaiserslautern Germany
[email protected]
