Incidence geometry

Definition

Incidence geometry is the study of incidence structures $(P, L, I)$ 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:

any two distinct points lie on at most one common line (a partial linear space);
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, projective planes, projective spaces, polar spaces and generalized polygons.

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Incidence structure concept
Incidence geometry definition

Tag	`H5EWB180S6`
URI	`m2://H5EWB180S6`
Kind	definition
Writing	W0 stub
MSC	51A05, 51E30
Defines	Incidence geometry
Uses	Incidence structure

Incidence geometry

Definition

Remarks

Reading path

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