---
tag: P1MKTF7MWQ
kind: concept
name: Projective transformation
msc: [51N15, 51A10]
sources:
  - url: https://en.wikipedia.org/wiki/Homography
same_as:
  wikidata: Q2112539
  wikipedia: Homography
---

## Summary

A projective transformation, or homography, is a map between [[projective spaces|ZT9A6JK8QF]] induced by an invertible linear map of the underlying vector spaces.
In [[homogeneous coordinates|FS9FRJ17D2]] it is $[x] \mapsto [Ax]$ for an invertible matrix $A$, defined up to a nonzero factor.
The projective transformations of $\mathbb{P}^n(k)$ form the group $\mathrm{PGL}(n + 1, k)$.
They are the symmetries of [[projective geometry|N32Z68F0D3]], as Euclidean motions are those of Euclidean geometry.

## Explanations

### L1 human

Take a photograph of a flat floor from an angle: a square tile becomes a general quadrilateral.
The map from the floor to the photo is a projective transformation.
It keeps straight lines straight, but not lengths, angles or parallelism.
On the projective line over $\C$, the Möbius maps $z \mapsto (az + b)/(cz + d)$ with $ad - bc \ne 0$ are exactly the projective transformations.