UNIT - 2 BILINEAR TRANSFORMATION

1 BILINEAR TRANSFORMATION UNIT - 2 BILINEAR TRANSFORMATION . LESSON STRUCTURE. Objective Introduction BILINEAR TRANSFORMATION Resultant or Product of two BILINEAR Transformations Cross-ratio Preservation of Cross Ratio Fixed point and normal form of a BILINEAR Transforma- tions Some special cases of BILINEAR Transformations conformal mapping Necessary and Sufficient Condition for a mapping to be conformal The mappings, W z n ,W z 2 and the inverse mapping W z1/ 2. Summary Solved Examples Model Questions References Objective In this unit in addition to BILINEAR Transformations, Cross ratio, normal form of BILINEAR Transformations will be discussed.

2 Special emphasis will be given on conformal mapping , special BILINEAR TRANSFORMATION , the mapping , W z n ,W z 2 and the inverse mapping W z1/ 2 . At the end extended complex plane will be discussed. Introduction Before going to discuss BILINEAR TRANSFORMATION , we shall define ( 41 ). BILINEAR TRANSFORMATION TRANSFORMATION or mapping , conformal TRANSFORMATION and linear TRANSFORMATION . u u x, y . The equation .. (1) define a TRANSFORMATION or mapping by means v v x, y . of which a correspondence between uv plane and xy -plane can be established.

3 If corresponding to each point of xy -plane, there is a unique point of uv -plane and conversely, then such a TRANSFORMATION is called one-one TRANSFORMATION . The corresponding points in the two planes are called images of each other. Hence, by means of equations (1), a region or curve of xy - plane is said to transform or mapped on or represented by the corresponding curve of the uv -plane. The linear TRANSFORMATION : A TRANSFORMATION of the form w az b , is called a linear TRANSFORMATION , where a and b are complex constants. BILINEAR TRANSFORMATION or Mobius TRANSFORMATION : az b A TRANSFORMATION of the form w.

4 (1) is called a BILINEAR TRANSFORMATION cz d of linear fractional TRANSFORMATION , where a,b,c,d are complex constants. Such type of TRANSFORMATION was first studied by Mobius and hence it is sometimes called mobius TRANSFORMATION . Equation (1) may be derived as CWz Wd az b 0 which is clearly a linear both in W and Z. That is why, it is called BILINEAR . Here we assume that ad bc 0 which is called the determinant of the TRANSFORMATION . The TRANSFORMATION (1) is called normalized if ad bc 1 / if the determinant vanishes, then w is merely a constant as shown below.

5 Let W1 and W2 be two values of W corrosponding to Z1 and Z 2 in (1), then az 2 b az1 b W2 W1 . cz 2 d Cz1 d acz1z 2 bcz1 az 2d bd acz1z 2 bcz 2 az1d bd = (Cz1 d )(Cz 2 d ). z 2 (ad bc ) z1 (ad bc ). = (Cz1 d )(Cz 2 d ). ( 42 ). BILINEAR TRANSFORMATION (z 2 z1 )(ad bc ). = (Cz d )(Cz d ) 0 if ad - bc = 0. 1 2. W2 W1 0 if ad - bc = 0 This shows that W is a constant Critical Points : Let az b w .. (1). Cz d b wd z .. (2). wc a dw ad bc From (1) dz cz d 2.. dw This means if z d / c dz dw and 0 if z . dz The points z d c ' z are called critical points where the conformal property does not hold good.

6 Also from (2) it is clear that for each z d c we have a value of W and for each W a c there corresponds a value of Z and the correspondence between W. and Z is one-one. The execptional points Z d c and W a c are mapped into the points W abd Z respectively. Extended complex plane : These points will not remain exceptional if we adjoin a new point called point at infinity denoted by to the complex plane and the complex plane in this case is called extended complex plane. Thus the critical point Z corresponds to the point W a c . Hence we can now say that the BILINEAR sets up a one-one correspendence between the points of entire extended complex z-plane upon the entire extended w-plane.

7 ( 43 ). BILINEAR TRANSFORMATION Resultants or Product of two BILINEAR transformations Consider two BILINEAR TRANSFORMATION az b W such that ad bc 0 .. (1). cz d a1w d1. and c w d such that a1d1 b1c1 0 .. (2). 1 1. Putting the value of W from (1) in equation (2), we get az b . a1 b1 aa b c z b d a b . cz d 1 1 1 1.. az b . c1 c1a d1c z d1d c1b . d1. cz d . Writing A aa1 b1c, B b1d a1b, C c1a d1c, D d1d c1b AZ B. We get . CZ D. Here AD-BC aa1 b1c d1d c1b b1d a1b c1a d1c . aa1d1d bb1cc1 b1c1ad a1d1bc = (a1d1 b1c1) (ad bc) 0 by (1) & (2). AZ B.

8 Thus such that AD BC 0. CZ D. This is a BILINEAR TRANSFORMATION . This BILINEAR TRANSFORMATION is called the resultant or product of the TRANSFORMATION resultant (1) and (2). Thus we may say BILINEAR TRANSFORMATION forms a group. Geometrical Inversion : Every BILINEAR TRANSFORMATION is the resultant of BILINEAR TRANSFORMATION with simple geometrical imports. Consider the BILINEAR TRANSFORMATION az b w .. (1). cz d Where ad bc 0, c 0. ( 44 ). BILINEAR TRANSFORMATION w . b a z a .. a . 1 . b d . a c . from (1) c z d c . c .. z d . c . a bc ad 1.

9 W 2.. c c z d c l bc ad Putting Z 1 z d , Z 2 ,Z3 .Z 2. c z1 c2. a We obtain w Z 3 which is similar to Z1 z d . c c The above three auxiliary transformations namely Z1 , Z 2 , Z 3 are of the form 1. W Z , W , W z z This proves that every general BILINEAR TRANSFORMATION is the resultant of the BILINEAR transformations 1. W Z , W , W z z Where (i) W Z represents translation 1. (ii) W represents inversion z and (iii) W z represents dilation. Cross Ratio Z 4 Z1 Z 2 Z3 . If Z1 , Z 2 , Z 3 , Z 4 are distinct points, then the ratio is called Z 2 Z1 Z 4 Z3.

10 The cross ratio of Z1 , Z 2 , Z 3 , Z 4 . This ratio is invariant under the BILINEAR TRANSFORMATION . Number of four distinct cross-ratio : From four points Z1 , Z 2 , Z 3 and Z 4 lying in the Z-plane, we can obtain different cross ratios according to the order in which the points are taken. Since the four points can permute themselves in be only six-distinct cross ratios. This is so ( 45 ). BILINEAR TRANSFORMATION because if we inter change any points and then interchange remaining two, the cross ratios of the points in this new order will be the same.