Bilinear map of -i, 0, i to -1, i, 1

Definition

A bilinear (linear fractional, Möbius) transformation is a mapping

w = (az + b)/(cz + d), where a, b, c, d are complex constants and ad − bc ≠ 0.

It maps the extended z-plane one-to-one onto the extended w-plane, and it is determined uniquely by three points and their images.

Finding the transformation

Use the cross-ratio form with (z₁, z₂, z₃) = (−i, 0, i) and (w₁, w₂, w₃) = (−1, i, 1):

(w − w₁)(w₂ − w₃) / [(w − w₃)(w₂ − w₁)] = (z − z₁)(z₂ − z₃) / [(z − z₃)(z₂ − z₁)]

Right side: (z + i)(0 − i) / [(z − i)(0 + i)] = −(z + i)/(z − i).

Left side: (w + 1)(i − 1) / [(w − 1)(i + 1)]. Since (i − 1)/(i + 1) = i, this equals i(w + 1)/(w − 1).

i (w + 1)/(w − 1) = −(z + i)/(z − i)
(w + 1)/(w − 1)   = i (z + i)/(z − i) = (iz − 1)/(z − i)

Write T = (iz − 1)/(z − i). Then w + 1 = T(w − 1), so w = (T + 1)/(T − 1):

w = [(iz − 1) + (z − i)] / [(iz − 1) − (z − i)]
  = (1 + i)(z − 1) / [(i − 1)(z + 1)]

Because (1 + i)/(i − 1) = −i,

Answer: w = −i (z − 1)/(z + 1) = (i − iz)/(z + 1).

Check

• z = −i: w = −i(−i − 1)/(1 − i) = i(1 + i)/(1 − i) = i·i = −1
• z = 0: w = −i(−1)/1 = i
• z = i: w = −i(i − 1)/(i + 1) = −i·i = 1

All three images agree with the given points.