---
title: "Linear fractional map 2i, -2, -2i to -2, -2i, 2"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:linear-transformation-2i-minus2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/linear-transformation-2i-minus2
site: BCT Engineering
---

# Linear fractional map 2i, -2, -2i to -2, -2i, 2

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2068 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/linear-transformation-2i-minus2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define conformal mapping. Find the linear fractional transformation that maps three points 2i, -2, -2i onto three given points -2, -2i, 2.

## Short answer

Use the cross-ratio form; simplifying the given three-point mapping yields w=iz. Check: 2i maps to -2, -2 maps to -2i, -2i maps to 2.

## Answer

A **conformal mapping** w = f(z) is a mapping that preserves angles between any two curves, both in magnitude and in sense (direction), at every point where f′(z) ≠ 0. Every analytic function with f′(z) ≠ 0 is conformal.

#### Finding the transformation
z₁ = 2i, z₂ = −2, z₃ = −2i map to w₁ = −2, w₂ = −2i, w₃ = 2.

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 − 2i)(−2 + 2i) / [(z + 2i)(−2 − 2i)]. Now (−2 + 2i)/(−2 − 2i) = (−1 + i)/(−1 − i) = −i (multiply top and bottom by −1 + i: (−1 + i)²/2 = −2i/2 = −i).

So the right side = −i (z − 2i)/(z + 2i).

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

So the left side = −i (w + 2)/(w − 2).

#### Equate
(w + 2)/(w − 2) = (z − 2i)/(z + 2i)

Cross-multiply: (w + 2)(z + 2i) = (w − 2)(z − 2i)

wz + 2iw + 2z + 4i = wz − 2iw − 2z + 4i  →  4iw = −4z  →  w = −z/i = iz.

**Answer: w = iz**

Check: 2i → i·2i = −2 ✓. −2 → −2i ✓. −2i → i(−2i) = 2 ✓.

## Check yourself

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

1. A conformal mapping preserves:
   A. Angles
   B. Distances
   C. Areas
   D. Orientation only of lines
   Answer: A. Standard. (Key point: Conformal)

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

3. The equation becomes:
   A. w = z
   B. (w+2)/(w−2) = (z−2i)/(z+2i)
   C. (w−2)/(w+2) = z
   D. w = 1/z
   Answer: B. −i cancels. (Key point: Reduction)

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

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