---
tag: H5EWB180S6
kind: definition
name: Incidence geometry
msc: [51A05, "51E30"]
defines: CSXT0BG2CM
uses: [FGKNAFNHA3]
sources:
  - url: https://en.wikipedia.org/wiki/Incidence_geometry
---

## Definition

*Incidence geometry* is the study of [[incidence structures|FGKNAFNHA3]] $(P, L, I)$ and of their properties that are preserved by isomorphisms.
An isomorphism of incidence structures is a pair of bijections, on points and on lines, that preserves incidence in both directions.
The structures studied are usually required to satisfy incidence axioms, such as:

1. any two distinct points lie on at most one common line (a *partial linear space*);
2. any two distinct points lie on exactly one common line (a *linear space*).

## Remarks

Example: the statement "two distinct lines of a linear space meet in at most one point" belongs to incidence geometry.
Check: if two distinct lines had two common points, these points would lie on two lines, against clause 2.

Non-example: the statement "the angles of a triangle add up to $\pi$" is not a statement of incidence geometry, since angles are not defined from points, lines and incidence alone.

Main classes are [[affine planes|EC7XT2ZM5S]], [[projective planes|DANFNN8C4Y]], projective spaces, polar spaces and generalized polygons.