Hello Saeed M. Ahmadian

       The bilinear transformation w = (az+b)/(z+c)  will transform a straight line y = mx + n in the z plane into another straight line in the w plane if and only if  c = n/m and the resulting image in the w-plane is given by

    v(b-ac) = u[2na - m(b+ac)] + mab

The constants a and b can be determined from the equations w1 = (az1+b)/(z1+c) and  w2 = (az2 + b)/(z2+c)

 w = u +iv and  z = x+iy

Hello Amaechi J. Anyaegbunam

Thank you for your comment. How about circle with infinite radius? is it the same?

Actually I am working with Schwarz-Christoffel transformation to transform W plane ( Complex flow plane) to an auxiliary half plane for example (t ' plane) . After this transformation, now I want to transform this (t ' plane)  to another auxiliary half plane (t).

As we know Bi-linear transformation could be used to transform a circle to circle and line to line conformally, so I want to know how to transform this circle with infinite radius by this transformation.

thank you again.

Hello Saeed M. Ahmadian

 A circle of infinite radius is simply a straight line. Generally, a bilinear transformation transforms a circle and a straight line into either a circle or a straight line.  It is essential that b - ac is not equal to zero to achieve conformality.

Hello Hello Saeed M. Ahmadian

Am I to understand you wish to find out how to use bilinear transformation to transform  the points on W plane  ( W1=-a, W2=b, W3=d and W4=-c) onto the points on t plane.[ t1= -La, t2=lb, t3=(1/m) and t4=(-1/m)] ?

Hello Amaechi J. Anyaegbunam

Yes, I want  those points to be transformed. 

Thank you 

Hello Saeed

 you said you are solving a potential problem using Schwarz-Christoffel tramsformation

The answer to your question is  that the transformation equation is 

w  = (et + f)/(t + g)  where e =  {d - mL[ab+d(b-a)]} /(1-mLd) 

f = Lab and g = L[d+a-b- mLab] /(1-mLd]   where m is to determined from the quadratic equation

m2L2ab(c+d) - mL [ 2(ab+cd)+ (b-a)(d-c) ]  +  c +d = 0

I hope this satisfies the demands of your problem

Hello Amaechi J. Anyaegbunam,

Hope you are doing well,  please accept my deepest thanks. I will use it in my calculations and will inform you if there was any issue. 

Hello Saeed

What is the thanks all about.? You have not said whether you have verified that  my answer is correct..   

 Hello Amaechi J. Anyaegbunam,

Hope been well sir,Sorry that I am replying late, because now I am working on this problem. I think some misunderstandings were happened because the parameters (La, Lb or lb ) are unique parameters, not a product of L and a or L and b. Maybe it was my mistake that I did not mention that those parameters are unique.  I am waiting for your answer

thanks a lot

Hello Saeed

 Sorry that I have not replied you. I have been sick and is just recovering. I will get back to you in a few hours time.

Hello Amaechi J. Anyaegbunam,

Wish you got well, and healthy. Thank you for replying.

Hello Saeed

The bilinear equation that transforms w(-a, b, -c, d) into t (-p, q, -1/m, 1/m)

respectively is

w = (et + f )/(t + g)

where

den = (a+b)(1+mp)- m(a+d)(p+q)

e = [m(a+d)(pa-qb) + (a+b)(d - mpa) ]/den

f = [ pb(a+d) + qa(b-d) - mpqd(a+b)]/ den

g = [(p+q)(d - mpa) - (1+mp)(qb - pa)]/den

where m is the solution of the quadratic equation

pq(a+b)(c+d)m2 - m(p+q) [ (a+d)(b+c) + (a-c)(b-d)] + (a+b)(c+d) = 0

where I have replaced La by p and Lb by q. den is a single symbol where as mpa = m times p times a etc

I hope that this solves your problem.  Make sure you verify these expressions with numerical values and inform me if there is any problem.