---
tag: QTZWZR1KWQ
kind: concept
name: Perspectivity
msc: [51A05, 51N15]
sources:
  - url: https://en.wikipedia.org/wiki/Perspectivity
same_as:
  wikidata: Q3924142
  wikipedia: Perspectivity
---

## Summary

A perspectivity maps the points of one line to the points of another line by central projection from a fixed point, the centre.
It is the basic map of [[projective geometry|N32Z68F0D3]] and models what an eye at the centre sees.
A composition of perspectivities is called a projectivity, and on the real projective line the projectivities are exactly the [[projective transformations|P1MKTF7MWQ]].
Perspectivities preserve the [[cross-ratio|09ASMRMD67]].

## Explanations

### L1 human

Stand at a point $O$ and look at points on a line $\ell$.
Each line of sight from $O$ through a point $X$ of $\ell$ hits a second line $m$ at a point $X'$.
The map $X \mapsto X'$ is the perspectivity from $\ell$ to $m$ with centre $O$.
Equal steps on $\ell$ usually become unequal steps on $m$, as in a perspective drawing of a fence.