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

Short answer

Use the cross-ratio form equating the w and z cross-ratios. Simplifying gives w=iz, which maps 1 to i, i to -1, and -1 to -i.

Find the bilinear transformation which maps points z₁ = 1, z₂ = i, z₃ = -1 into the points w₁ = i, w₂ = -1, w₃ = -i respectively.

Given z₁ = 1, z₂ = i, z₃ = −1 and w₁ = i, w₂ = −1, w₃ = −i.

Cross-ratio form of the bilinear transformation:

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

Right side

(z − 1)(i + 1) / [(z + 1)(i − 1)]. Since (i + 1)/(i − 1) = −i, this equals −i (z − 1)/(z + 1).

Left side

(w − i)(−1 + i) / [(w + i)(−1 − i)]. Now (−1 + i)/(−1 − i) = (−1 + i)²/[(−1 − i)(−1 + i)] = (−2i)/2 = −i.

So the left side is −i (w − i)/(w + i).

Equate

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

Cross-multiply: (w − i)(z + 1) = (w + i)(z − 1)

wz + w − iz − i = wz − w + iz − i → 2w = 2iz → w = iz.

Answer: w = iz (a rotation of the plane by 90° about the origin).

Check: z = 1 gives i ✓. z = i gives i² = −1 ✓. z = −1 gives −i ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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