---
title: "Affine transformation"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:affine-transformation"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/affine-transformation
site: BCT Engineering
---

# Affine transformation

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2079 Baisakh, 2 marks
- Video: [Affine transformations in 5 minutes](https://www.youtube.com/watch?v=AheaTd_l5Is) — Graphics in 5 Minutes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/affine-transformation
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

What do you understand by affine transformation?

## Short answer

Affine transformation is a linear mapping combined with translation. It preserves straight lines, parallelism, and ratios of distances along a line; lengths and angles may change. Examples include translation, rotation, scaling, reflection, and shearing.

## Answer

An **affine transformation** is a linear transformation followed by a translation. In 2D it has the form

x' = a·x + b·y + tx
y' = c·x + d·y + ty

```text
| x' |   | a b tx | | x |
| y' | = | c d ty | | y |
| 1  |   | 0 0 1  | | 1 |
```

Properties:
- Straight lines stay straight, and parallel lines stay parallel.
- Ratios of distances along a line are preserved, but lengths and angles may change.
- Translation, rotation, scaling, reflection and shearing are all affine, and so is any combination of them.
- The last row is (0 0 1), so it is not a perspective transformation.

## Check yourself

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

1. What is an affine transformation?
   A. Only a translation
   B. A perspective mapping
   C. A linear transformation followed by a translation
   D. Only a rotation
   Answer: C. x' = ax + by + tx etc. (Key point: Definition)

2. What is preserved?
   A. Lengths
   B. Angles
   C. Vanishing points
   D. Parallel lines stay parallel
   Answer: D. Lines stay straight and parallel; lengths and angles may change. (Key point: Preserved)

3. Why is an affine matrix not perspective?
   A. Its last row is (0 0 1)
   B. It has a 4th column
   C. It has no translation
   D. It has zero determinant
   Answer: A. A perspective transformation has a non-trivial last row. (Key point: Last row)

## Related in Computer Graphics

- [Raster display and frame buffer load time](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Araster-system-load-time) — Computer Graphics, chapter 1

## Related in other subjects

- [Low pass to high pass transformation](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alowpass-to-highpass-transformation) — Filter Design, chapter 3, asked 2×
- [RC-CR transformation of Sallen-Key LPF to HPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-cr-transformation-sallen-key) — Filter Design, chapter 6, asked 2×
- [Bruton transformation and FDNR](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abruton-fdnr-grounded-inductor) — Filter Design, chapter 8
- [Bruton transformation; FDNR using GIC](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abruton-importance-fdnr-gic) — Filter Design, chapter 8