Bilinear transformation

BILINEAR TRANSFORMATION

Introduction :

The transformation w = az + b/ cz + d’ – bc ≠ 0 where a, b, c, d are complex numbers, is called a bilinear transformation.

This transformation was first introduced by A.F. Mobius. So it is also called Mobius transformation.

A bilinear transformation is also called a linear fractional transformation because az + b / cz + d is a fraction formed by the linear functions az + b and cz + d.

Theorem 3.5.1.

Under a bilinear transformation no two points in z plane go to the same point in w plane.

Proof :

Suppose z₁ and z₂ go to the same point in the w plane under the transformation w = az + b/ cz + d'

az₁ + b/cz₁ + d = az₂ + b / cz₂ + d

i.e.,(az₁ + b) (cz₂ + d) - (az₂ +b) (cz₁ + d) = 0

acz₁z₂ + adz₁ + bcz₂+ bd - acz₁z₂ - adz₂ - bcz₁ - bd = 0

 (ad - bc) (z₁ - z₂) = 0

or z₁ = z₂  [ ad - bc #0]

This implies that no two distinct points in the z plane go to the same point in the w plane. So, each point in the z plane goes to a unique point in the w plane.

Definition 3.5. (a) Fixed points (or) Invariant points (OR) Prove that a bilinear transformation has atmost two fixed points. [A.U. M/J 2012]

The fixed points of the transformation

w = az + b/cz + d is obtained from

z = az + b/cz + d (or) cz₂ + (d - a) z - b = 0

These points are two in number unless the discriminant is zero in which case the number of points is one.

Remark: Though a bilinear transformation involves a, b, c, d effectively it involves only three constants because az + b and cz + d can be divided by one of a, b, c, d which is non zero so that one of them is 1.

This shows that there is just one bilinear transformation that maps three given distinct points z₁, z₂, z₃ into three specified points w₁, w₂, w₃.

If a ≠ 0 and c ≠ 0, then the transformation can be assumed to be w = A z + B/z + C  (where A = a/c, B = b/a, C = d/c) otherwise, it can not be done so.

Theorem 3.5.2. The bilinear transformation which transforms z₁, z₂,z₃ into W₁, W₂, W₃ is (w – w₁) (w₂ – w₃)/ (w – w₃) (w₂ – w₁) = (z – z₁) (z₂ – z₃)/ (z – z₃) (z₂ – z₁)

Proof: If the required transformation is w = az + b/cz + d

= az + b/cz + d ,  Here a = Aw₁ – Bw₃ b = Bw₃z₁ – Aw₁z₃ c = A – B , d = B z₁ – Az₃

Definition : 3.5.(b) Cross ratio

Given four points z₁, z₂, z₃, z₄ in this order, the ratio (z₁ – z₂) (z₃ – z₄) /(z₂ – z₃) (z₄ – z₁) is called the cross ratio of the points.

Note (1): w = az + b / cz + d can be expressed as cwz + dw - (az + b) = 0

It is linear both in w and z that is why, it is called bilinear.

Note (2): This transformation is conformal only when dw/dz # 0

i.e., ad – bc /(cz+d)² ≠ 0

i.e., ad – bc ≠ 0

If ad - bc = 0, every point in the z plane is a critical point.

Note (3) : Now, the inverse of the transformation w = az + b/cz + d z = -dw + b / cw – a which is also a bilinear transformation except w = a/c.

Note (4) Each point in the plane except z = -d/c corresponds to a unique point in the w-plane.

The point z = -d/c corresponds to the point at infinity in the w plane.

Note (5): The cross ratio of four points

(w₁ - w₂) (w₃ - w₄)/ (w₂ – w₃) (w₄ – w₁) =  (z₁ - z₂) (z₃ - z₄)/ (z₂ – z₃) (z₄ – z₁) is invariant under bilinear transformation.

Note (6) : If one of the points is the point at infinity the quotient of those difference which involve this point is replaced by 1.

Suppose z₁ = ∞, then we replace z – z₁ / z₂ – z₁ by 1 (or ) Omit the factors involving ∞