---
title: "Bilinear map preserves cross ratio"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:bilinear-invariant-cross-ratio"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-invariant-cross-ratio
site: BCT Engineering
---

# Bilinear map preserves cross ratio

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Bhadra
- Video: [Cross ratio remains invariant under a Bilinear transformation](https://www.youtube.com/watch?v=rnmzobON23g) — RAZA MATHEMATICS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-invariant-cross-ratio
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define conformal mapping. Show that w = (az + b)/(cz + d) is invariant to [(w - w₁)/(w - w₃)] × [(w₂ - w₃)/(w₂ - w₁)] = [(z - z₁)/(z - z₃)] × [(z₂ - z₃)/(z₂ - z₁)].

## Short answer

For w=(az+b)/(cz+d), the difference formula gives w_i-w_j=(ad-bc)(z_i-z_j)/((cz_i+d)(cz_j+d)). Applying it to each cross-ratio factor cancels all denominators and (ad-bc)^2, proving the cross-ratio is invariant.

## 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.

#### Claim
Let w = (az + b)/(cz + d) with ad − bc ≠ 0. Then the cross-ratio is unchanged: the expression in the w-plane equals the same expression in the z-plane.

#### Proof
For any two points zᵢ, zⱼ with images wᵢ, wⱼ:

wᵢ − wⱼ = (azᵢ + b)/(czᵢ + d) − (azⱼ + b)/(czⱼ + d)
 = [(azᵢ + b)(czⱼ + d) − (azⱼ + b)(czᵢ + d)] / [(czᵢ + d)(czⱼ + d)]
 = (ad − bc)(zᵢ − zⱼ) / [(czᵢ + d)(czⱼ + d)].  ... (1)

(The same formula holds for any variable point z with image w.) Now apply (1) to each factor, writing Dᵢ = czᵢ + d and D = cz + d:

```text
(w − w₁)(w₂ − w₃)
 = (ad−bc)² (z − z₁)(z₂ − z₃) / [ D·D₁ · D₂·D₃ ]

(w − w₃)(w₂ − w₁)
 = (ad−bc)² (z − z₃)(z₂ − z₁) / [ D·D₃ · D₂·D₁ ]
```

Both products have the same denominator D·D₁·D₂·D₃ and the same factor (ad − bc)², so they cancel on division:

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

So the cross-ratio is invariant under a bilinear transformation. This is why the relation can be used to find the transformation that maps three given points z₁, z₂, z₃ to w₁, w₂, w₃.

## Check yourself

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

1. The cross ratio under bilinear map is:
   A. Doubled
   B. Zero
   C. Invariant
   D. Inverted
   Answer: C. Claim. (Key point: Claim)

2. wi − wj =
   A. (a)(zi−zj)
   B. (ad+bc)(zi−zj)
   C. (zi−zj)/(czi+d)
   D. (ad−bc)(zi−zj)/[(czi+d)(czj+d)]
   Answer: D. Formula (1). (Key point: Key formula)

3. What cancels?
   A. only d
   B. z values
   C. a and b
   D. (ad−bc)² and the denominators
   Answer: D. Same in numerator and denominator. (Key point: Cancellation)

4. This lets us:
   A. Differentiate
   B. Integrate
   C. Find residues
   D. Find the map from three point pairs
   Answer: D. Standard. (Key point: Use)

## Related in Applied Mathematics

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