---
title: "Homogeneous coordinates"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:homogeneous-coordinates"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/homogeneous-coordinates
site: BCT Engineering
---

# Homogeneous coordinates

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2076 Asoj, 2 marks
- Video: [Homogeneous Coordinates | CG | Computer Graphics | Lec-25 | Bhanu Priya](https://www.youtube.com/watch?v=p4NG28KeruY) — Education 4u
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/homogeneous-coordinates
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

What do you mean by homogeneous coordinates?

## Short answer

Homogeneous coordinates add a third coordinate h to (x,y) as (xh, yh, h), with x = xh/h, y = yh/h (typically h=1). This allows all 2D transformations to be expressed as 3×3 matrix multiplications, enabling concatenation and representing points at infinity (h=0).

## Answer

**Homogeneous coordinates** represent an n-dimensional point by n+1 coordinates. A 2D point (x, y) is written as (x_h, y_h, h) with h ≠ 0, where x = x_h/h and y = y_h/h. Usually h = 1, so (x, y) becomes (x, y, 1).

Why it is used:
- Translation, rotation and scaling can all be written as one matrix multiplication (3×3 in 2D, 4×4 in 3D). Without it, translation needs matrix addition while the others need multiplication.
- A sequence of transformations can be combined (concatenated) into one single matrix, which is faster.
- Points at infinity can be represented with h = 0, which is needed in perspective projection.

Example: translation by (tx, ty) in homogeneous form

```text
| x' |   | 1 0 tx | | x |
| y' | = | 0 1 ty | | y |
| 1  |   | 0 0 1  | | 1 |
```

## Check yourself

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

1. How many coordinates represent a 2D point in homogeneous form?
   A. 3
   B. 2
   C. 4
   D. 1
   Answer: A. n+1 coordinates: (x_h, y_h, h). (Key point: Definition)

2. If a point is (x_h, y_h, h) with h != 0, x equals?
   A. x_h · h
   B. x_h / h
   C. x_h + h
   D. h / x_h
   Answer: B. x = x_h/h. (Key point: Recovering x)

3. Why use homogeneous coordinates?
   A. To make rotation faster than scaling
   B. To remove the need for matrices
   C. Translation can be a matrix multiplication and transformations can be combined
   D. To store color
   Answer: C. All transformations become matrix products and can be concatenated. (Key point: Purpose)

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