Hexagonal 3-web by two pencils of straight lines and a pencil of circles

HEXAGONAL 3-WEB BY TWO PENCILS OF STRAIGHT LINES AND A PENCIL OF CIRCLES SUMMARY In this work, all of the hexagonal 3-webs formed by two pencils of straight lines and a pencil of circles are determined. The work consists of three parts. In the first part, some fundamental concepts of the Web Geometry are given. Let Ci (i = 1,2,3) be three families of lines which cover the region R simply and suppose that throught each point (re, y) of R there passes one and only one line of each family, three families of lines be given by the equations Ui(x,y) = Ui = sbt (» = 1,2,3). (0.1.1) Assume that the partial derivatives dUi dUi dx ' dy ' (0.1.2) are not both zero and that the functions Ui(x,y) are analytic in R. The functional determinants If' are supposed to be non-zero and any two lines each of which belongs to different families have no more than one common point. Three families of lines satisfying the above properties are said to form a 3-web and is denoted by M. A web M is said to be hexagonal in region R, if it is formed by three families of parallel straight lines [1]. An expression such as oj=p(x,y)dx + q(x,y)dy, (0.1.4) is named as a Pfaffian in two variables. The poler product of two Pfaffians «i = Pi(x,y)dx + qi(x,y) (i = 1,2), (0.1.5) is defined by where [wi, wa] = (PİÇ2 - P2qi)[dx, dy], (0.1.6) [dx,dy] = - [dy, dx].The poler product has the follawing properties: [wı,W2İ + [w2,Wı] = 0, [wi,W2 + a*] = [wi,wj] + [wi,ws]. (0.1.7) [/wı,wa] = /[wi,wa], where f is a scalar. Exterior differential of a Pfaffiau, denoted by dw, is defined as dw = [dp, dx] + [dq, dy] = (qx - py)[dx, dy]. (0.1.8) In the second part, using the concept of inversion, two pencils of straight lines and a pencil of circles are obtained from three appropriate pencils of circles. Consider a circle with center O and of radius R. The point A' is called the inversion of a point A with respect to the given circle [2], if the following conditions are saisfied: i) The points A and A' are on the line through O. ii) OA'.OA = R2. From this defmation it follows that _ fc2(X - a) _ k2(Y - b) x- (X-a)2 + (Y-by+a ' y~ (x-ay + (Y-by+b- (0'L9) Let a; be a complex number and let y be its conjugate. Then, the equation of an elliptic pencil with base points t\, e2 is xy- -xy2-x - sx^2-y + ' 2 2 + (0.1.10) A.[(ei - e2)ar - (ex - e2)y + txe'2 - e[e2].i = 0, while the equation of an hyperbolic pencil with null-circles hi and hi is xy+ (-h[)x + (h^y + hh[ + (0.1.11) X.[(h'2 - h\)x + {hi - hx)y + hxh'x - h2h'2] = 0. On the other hand, if a is the base point and u is the straight line carrying the centres of the circles of the pencil, then the equation of a parabolic pencil is of the form xy - a'x - ay + aa' + X.(u'x + uy - a'u - au') = 0 (0.1.12) By taking inversion with respect to one of the base points of an elliptic pencil, this pencil transforms to a pencil of straight lines. In the same way, by taking inversion with respect to the base point of a parabolic pencil, it transforms to a pencil of parallel straight lines. To obtain pencil of straight lines, the inversion elliptic or parabolic pencils must be taken [3], VIIn the third part, all the hexagonal 3- webs formed by two pencils of straight lines and a pencil of circles are obtained. In order to simplify the calculations, equation of pencils of straight lines are taken as ai(x + y) + 2&i + A.(-t'd(a: - y) + 2dx) = 0, (0.1.13) and (d - ibx)x + (ai + ibi)y = 2A. (0.1.14) For example, let the pencils be _ oi (s+3/)+26i 1 - ic\ (x- y)+2di ' W2 - -.^(«-yj+adi ' (U.I.lOJ xy- h\x- hıy+hıh[ U3 ~ {h'2-h>1)x+(h2-k1)y+hih'1-h2h'2. By differentiation we get, from (0.1.15) A'dx + Ady = 0, B'dx + Bdy = Q, (0.1.16) C'dx + Cdy = 0, where we have put A = %a\C\y + 1ib\C\ - 2a\d\, B = 2ia2c2y + 2ib2c2 - 2a2d2, (0.1.17) C = (h'2 - h[)x2 - (h'2 - h[)(h2 + hi)x + (h'2 - h\)hxh2, The differantial form A)[dx,dy], (0.1.24) dKeyword: Doku geometrisi = Web geometry ; Geometri = Geometry