An Interesting Class of Polynomial Vector Fields

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1 A Iterestig Class of Polyomial Vector Fields Gerik Scheuerma, Has Hage ad Heiz Krüger Abstract. The visualizatio of vector fields is oe of the most importat topics i visualizatio. Of special iterest over the last years have bee topology-based methods. We preset a method to costruct polyomial vector fields after determie the critical poits ad their idex to geerate test data ad help aalysig vector field topology. The idea is based o Clifford algebra ad aalysis. We start with a formulatio of a plae real vector field i Clifford algebra givig more topological iformatio directly from the formulas tha the usual descriptio i cartesia coordiates. The theorem preseted here is a essetial improvemet to our previous result i [9], ad has implicatios to geeral polyomial vector fields.. Itroductio Scietific Visualizatio ofte deals with a large amout of data. Especially i the visualizatio of vector fields oe usually has the problem that it is early impossible to show the whole data. Oe has to reduce the iformatio i a useful way. Oe very successful approach is to look for the topology of the field ad visualize its essetial parts [4]. The testig of ew algorithms i this directio was doe up to ow by comparig with haddrawigs based o experimets. The problem is that i the literature ([], [3], [7]) oe oly fids a aalysis of liear fields ad some special equatios, where the positio, umber ad type of critical poits is ot arbitrary. The reaso is probably that it is difficult to aalyse oliear fields i the classical way with computig the itegral curves. We have foud ad published a theorem i [9] which allows a descriptio of the topology of some polyomial vector fields. This was used i [0] to derive a ew algorithm for vector field visualizatio which ca detect ad visualize higher order sigularities. I this paper we give a essetial geeralizatio of this theorem with a similar result for a larger class of polyomial vector fields. We also ca show how the aalysis of a arbitrary polyomial vector field ca be simplified. This result is obtaied by usig Clifford algebra ad aalysis which are described i Sectio 2 ad 3. The result is derived ad proved i Sectio 4. Mathematical Methods for Curves ad Surfaces II Morte Dæhle, Tom Lyche, Larry L. Schumaker (eds.), pp. 0. Copyright oc 998 by Vaderbilt Uiversity Press, Nashville, TN. ISBN -xxxxx-xxx-x. All rights of reproductio i ay form reserved.

2 2 G. Scheuerma, H. Hage ad H. Krüger The last sectio gives some examples showig how oe ca actually costruct o-trivial polyomial vector fields by this result which should be useful for testig algorithms. 2. Clifford Algebra We eed Clifford algebra oly i two dimesios, so for simplicity we should stay there. Let {e, e 2 } be a orthogoal basis of IR 2 with stadard scalar product. The Clifford algebra G 2 is the the IR-algebra of maximal dimesio cotaiig IR ad IR 2 such that for each vector x IR 2 holds x 2 = x 2. This implies the followig rules e e 2 + e 2 e = 0 e 2 j = j =, 2 We get a 4-dimesioal algebra with the vector basis {, e, e 2, e e 2 }. We ow have the real vectors xe + ye 2 IR 2 G 2 ad the real umbers x IR G 2 both i the algebra. We may also defie the projectios k, k = 0,, 2 by 0 : G 2 IR G 2 a + be + ce 2 + de e 2 a : G 2 IR 2 G 2 a + be + ce 2 + de e 2 be + ce 2 2 : G 2 IRe e 2 G 2 a + be + ce 2 + de e 2 de e 2 For two vectors oe ca the describe the ew product by already kow products. Let v = v e +v 2 e 2, w = w e +w 2 e 2 be two vectors, v, v 2, w, w 2 IR. The vw = (v w + v 2 w 2 ) + (v w 2 w v 2 )e e 2 = vw 0 + vw 2 = v w + v w where deotes the scalar (ier) product ad deotes the outer product of Grassma. This uificatio of ier ad outer product is the startig poit of the geometric iterpretatio. We will ot eed it here, so see [5] for a good itroductio. More importat for us is that the complex umbers ca also be caoically embedded by recogizig (e e 2 ) 2 =, so set i := e e 2.

3 A Iterestig Class of Polyomial Vector Fields 3 The a + bi C G 2 is a subalgebra. The ext sectio itroduces a bit of Clifford aalysis. 3. Clifford Aalysis Here we eed agai oly the two-dimesioal case, so we limit our defiitios to that case to avoid techical overload. Our basic maps will be multivector fields A : IR 2 G 2 r A(r). A Clifford vector field is just a multivector field with values i IR 2 G 2 v : IR 2 IR 2 G 2 r = xe + ye 2 v(r) = v (x, y)e + v 2 (x, y)e 2 The directioal derivative of A i directio b IR 2 is defied by A b (r) = lim ɛ 0 [A(r + ɛb) A(r)], ɛ ɛ IR. This allows the defiitio of the vector derivative of A at r IR 2 by A(r) : IR 2 G 2 r A(r) = 2 g k A gk (r). This is idepedet of the basis {g, g 2 } of IR 2. The itegral i Clifford aalysis is defied as follows : Let M IR 2 be a orieted r-maifold ad A, B : M G 2 be two piecewise cotious multivector fields. The oe defies M AdXB = lim A(x i ) X(x i )B(x i ), i=0 where X(x i ) is a r-volume i the usual Riemaia sese. This allows the defiitio of the Poicaré-idex of a vector field v at a IR 2 as v dv id a v = lim ɛ 0 2πi v 2, where is a circle of radius ɛ aroud a.

4 4 G. Scheuerma, H. Hage ad H. Krüger 4. Aalysis of Polyomial Vector Fields For our result it is ecessary to look at v : IR 2 IR 2 G 2 i a differet way. Let z = x + iy, z = x iy. This meas x = (z + z) 2 y = (z z). 2i We get v(r) = v (x, y)e + v 2 (x, y)e 2 = [v ( 2 (z + z), 2i (z z)) iv 2( 2 (z + z), 2i (z z))]e = E(z, z)e, where E : C 2 C G 2 (z, z) v ( 2 (z + z), 2i (z z)) iv 2( 2 (z + z), (z z)) 2i is a complex-valued fuctio of two complex variables. The idea is ow to aalyse E istead of v ad get topological results directly from the formulas i some iterestig cases. Let us first assume that E ad also v is liear. Theorem. Let v(r) = (az + b z + c)e be a liear vector field. For a b it has a uique zero at z 0 e IR 2. For a > b has v oe saddle poit with idex. For a < b it has oe critical poit with idex. The special types i this case ca be got from the followig list : () Re(b) = 0 circle at z 0. 2) Re(b) 0, a > Im(b) ode at z 0. 3) Re(b) 0, a < Im(b) spiral at z 0. 4) Re(b) 0, a = Im(b) focus at z 0. I cases 2) 4) oe has a sik for Re(b) < 0 ad a source for Re(b) > 0. For a = b oe gets a whole lie of zeros. Proof: A computatio of the derivatives of the compoets v, v 2 ad a compariso with the classic classificatio gives this result. We icluded this easy theorem to show that this descriptio gives topological iformatio more directly. Let us look ow at the geeral polyomial case :

5 A Iterestig Class of Polyomial Vector Fields 5 Theorem 2. Let v : IR 2 IR 2 G 2 be a arbitrary polyomial vector field with isolated critical poits. Let E : C 2 C be the polyomial so that v(r) = E(z, z)e. Let F k : C 2 C, k =,...,, be the irreducible compoets of E, so that E(z, z) = F k. The have the vector fields w k : IR 2 IR 2, w k (r) = F k (r)e also oly isolated zeros z,..., z m. These are the the zeros of v ad for the Poicaré-idices we have id zj v = id zj w k, Proof: The w k have oly isolated zeros because otherwise v would have also ot isolated zeros. It is also obvious that a zero of a w k is a zero of v ad a zero of v must be a zero of oe of the w k. For the derivatives we get E z = a E z = a F k z F k z l=,l k l=,l k For the computatio of the Poicaré-idex, we assume z j = 0 after a chage of the coordiate system ad that ɛ is so small that there are o other zeros iside Sɛ. We get id zj v = 2πi = 2πi d za = 2πi = = = 2πi v dv v 2 v 2 a F k z l=,l k v 2 aā id zj F k e id zj w k. F k e [dza F l ]e 2 F l F l. F k z l=,l k F k (dz F k z + d z F k z ) F l + l=,l k F k Fk F k e (dz F k z + d z F k z )e 2 F l Fl 2 For experimets it is ice to use liear factors because oe has good isights i their behavior from Theorem.

6 6 G. Scheuerma, H. Hage ad H. Krüger Theorem 3. Let v : IR 2 IR 2 G 2 be the vector field v(r) = E(z, z)e with E(z, z) = (a k z + b k z + c k ) a k b k, ad let z k be the uique zero of a k z + b k z + c k. The has v zeros at z j, j =,..., ad the Poicaré-idex of v at z j is the sum of the idices of the (a k z + b k z + c k )e at z j. Proof: Special case of Theorem 2 5. Examples The followig examples show the topological structure of vector fields with the algebraic structure of Theorem 3. They shall illustrate its usefuless for the testig of topological vector field visualizatio algorithms by showig that oe has full cotrol over positio ad idex of the critical poits. Our first example is the vector field v(r) =( z ( i))( z ( i))(z ( i)) (z ( i))( z ( i))e i the square [, ] [, ] i cartesia coordiates. Figure shows that the field has saddles at (0.63, 0.46) ad (0.74, 0.35) produced by the factors with z. There are further sigularities of idex at ( 0.5, 0.5), (0.38, 0.92) ad ( 0.33, 0.52) produced by the factors with z. The sampled arrows idicate the directio of the uit vector field at a quadratic grid. Fig.. Two saddles ad three idex sigularities.

A Iterestig Class of Polyomial Vector Fields 7 The secod example is v(r) =( z ( 0.58 0.64i))( z (0.5 + 0.27i))(z (0.68 0.59i)) ( z ( 0.2 0.84i)) 2 (z ( 0. 0.72i))(z (0.74 + 0.

7 A Iterestig Class of Polyomial Vector Fields 7 The secod example is v(r) =( z ( i))( z ( i))(z ( i)) ( z ( i)) 2 (z ( i))(z ( i))e agai i the square [, ] [, ] i cartesia coordiates. Figure 2 shows that the field has saddles at (0.68, 0.59),( 0., 0.72) ad (0.74, , 0.46) produced by the factors with z. There are further sigularities of idex at ( 0.58, 0.64) ad (0.5, 0.27) produced by the factors with z. There is also a idex 2 sigularity at ( 0.2, 0.84) stemmig from the squared factor. Agai the sampled arrows idicate the directio of the uit vector field. Fig. 2. Three saddles, two idex ad oe idex 2 sigularity. Ackowledgmets. This work was partly made possible by fiacial support by the Deutscher Akademischer Ausladsdiest (DAAD). The first author got a DAAD-Doktoradestipedium aus Mittel des zweite Hochschulsoderprogramms for his stay at the Arizoa State Uiversity from Oct. 96 to Ja. 97. We also wat to thak Aly Rockwood, Greg Nielso ad David Hestees from Arizoa State Uiversity for may commets, suggestios ad ispiratio. Special thaks go to Shoeb Bhiderwala who wrote the excellet user iterface to create the pictures i the examples [2]. Refereces. Arold, V. I., Gewöhliche Differetialgleichuge, Deutscher Verlag der Wisseschafte, Berli, 99.

8 8 G. Scheuerma, H. Hage ad H. Krüger 2. Bhiderwala, S. A., Desig ad visualizatio of vector fields, master thesis, Arizoa State Uiversity, Guckeheimer, J., ad P. Holmes, Noliear Oszillatios, Dyamical Systems ad Liear Algebra, Spriger, New York, Helma, J. L., ad L. Hesselik, Visualizig vector field topology i fluid flows, IEEE Computer Graphics ad Applicatios :3 (99), Hestees, D., New Foudatios for Classical Mechaics, Kluwer Academic Publishers, Dordrecht, Krüger, H., ad M. Mezel, Clifford-aalytic vector fields as models for plae electric currets, i Aalytical ad Numerical Methods i Quaterioic ad Clifford Aalysis, W. Sprössig, K. Gürlebeck (eds.), Seiffe, Hirsch, M. W., ad S. Smale, Differetial Equatios, Dyamical Systems ad Liear Algebra, Academic Press, New York Milor, J. W., Topology from the Differetiable Viewpoit, The Uiversity Press of Virgiia, Charlottesville, Scheuerma, G., H. Hage, ad H. Krüger, Clifford algebra i vector field visualizatio, accepted for Vismath Scheuerma, G., H. Hage, H. Krüger, M. Mezel, ad A. P. Rockwood, Visualizatio of higher order sigularities i vector fields, IEEE Visualizatio Proc., ACM Press, New York, 997, Gerik Scheuerma ad Has Hage Dept. of Computer Sciece Uiversity of Kaiserslauter Postfach 3049 D Kaiserslauter Germay scheuer,hage@iformatik.ui-kl.de scheuer@iformatik.ui-kl.de hage@iformatik.ui-kl.de Heiz Krüger Dept. of Physics Uiversity of Kaiserslauter Postfach 3049 D Kaiserslauter Germay krueger@physik.ui-kl.de

CHAPTER I: Vector Spaces

CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig