conformal mapping

Question

let w = (az+b)/(cz+d) ,ad-bc≠0 be the required mobius transformation. the point w=1 and w=∞ correspond the points 

z= -(b-d)/(a-c) and z = - d/c respectively. since, w=1 and w= ∞ are inverse points with respect to the circle |w-1|=1 .

 

i am very much confused about how come w=1 and w=∞ are inverse point wrt the circle |w-1|=1 ..

please help me, i request you very very much sir, please help me please

Answer 1

The mirror image of a point P about a circle C is defined as follows.

 

Take any point P is outside the circle.

 

Draw two lines through P that are tangent to the circle.

 

Draw a chord connecting the tangency points.

 

Draw a line L through P and perpendicular to the chord.

 

The intersection of L and the chord is P' inside the circle

 

P' is the mirror image of P.

 

We also consider P the mirror image of P'.

 

If the circle is in the complex plane and P and P' are complex numbers then we say that P and P' are inverses with respect to the circle.

 

Now imagine P going out to ∞. Then it's mirror image (or inverse) will approach the circle's center.

 

Therefore the center of a circle is considered to be the inverse of ∞ with respect to that circle.

 

The circle |w-1| = 1 is a unit circle centered at w=1, so 1 is the inverse of inverse of ∞ in this case.