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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.

What do you understand by affine transformation?

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

| 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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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