---
tag: N32Z68F0D3
kind: concept
name: Projective geometry
msc: [51A05, 51N15]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_geometry
same_as:
  wikidata: Q177409
  wikipedia: Projective geometry
---

## Summary

Projective geometry studies the properties of figures that survive projection, such as incidence and the [[cross-ratio|09ASMRMD67]].
Length and angle are not among them.
Its spaces are [[projective spaces|ZT9A6JK8QF]], where any two lines of a plane meet.
It can be built either from linear algebra or from incidence axioms.
In it, points and hyperplanes play symmetric roles, which gives the principle of [[duality|DCMP1A0138]].
It underlies perspective drawing, algebraic geometry and computer vision.

## Explanations

### L1 human

When a lamp casts the shadow of a shape on a wall, lengths and angles change, but straight lines stay straight.
Projective geometry studies exactly what such shadows and perspective views keep.
In this geometry parallel lines are said to meet at a point at infinity, so any two lines in a plane meet.
For example, all ellipses, parabolas and hyperbolas become the same kind of curve, a conic seen from different viewpoints.