---
title: "Bilinear map of 1, i, -1 to i, -1, -i"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:bilinear-1-i-minus1-to-i-minus1-minus-i"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-1-i-minus1-to-i-minus1-minus-i
site: BCT Engineering
---

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

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2072 Asoj, 5 marks
- Video: [Bilinear Transformation Cross Ratio Property - Problem 1 - Engineering Mathematics 3](https://www.youtube.com/watch?v=7eOd2KpEkRM) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-1-i-minus1-to-i-minus1-minus-i
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

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

## 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.

## Answer

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
```text
−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 ✓.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The method uses:
   A. Equal derivatives
   B. Taylor series
   C. Equal moduli
   D. Equal cross ratios
   Answer: D. Invariance. (Key point: Method)

2. (i+1)/(i−1) equals:
   A. −1
   B. 1
   C. i
   D. −i
   Answer: D. Rationalise. (Key point: RHS)

3. The equation reduces to:
   A. (w−i)/(w+i) = (z−1)/(z+1)
   B. (w+i)/(w−i) = z
   C. w = z
   D. w = 1/z
   Answer: A. The −i factors cancel. (Key point: Simplification)

4. w =
   A. −iz
   B. iz
   C. z/i²
   D. 1/z
   Answer: B. 2w = 2iz. (Key point: Result)

## Close variants set in other exams

- [Conformal mapping; bilinear map to -3,-1,1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconformal-bilinear-0-1-inf) — Applied Mathematics, chapter 1, asked 2×

## Related in Applied Mathematics

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- [Bilinear map of -i, 0, i to -1, i, 1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Abilinear-minus-i-0-i) — Applied Mathematics, chapter 1, asked 4×
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- [Conformal mapping; map 1, i, -1 to 0, 1, ∞](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconformal-map-1-i-minus1) — Applied Mathematics, chapter 1
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