51Axx Linear incidence geometry
This subtopic introduces the core ideas in linear incidence geometry, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
51A05 General theory and projective geometries
Overview
Projective geometry studies incidence and perspectivity properties that remain invariant under projective transformations.
Related Wikipedia Page
Wikipedia: Projective geometry
Useful Links
Key Ideas
- projective spaces
- cross-ratios
- perspectivities
Typical Uses
Fundamental in computer vision, architecture, and classical geometry.
Applications
- Computer graphics
- Camera calibration
- Synthetic geometry
References
Recommended Textbooks
51A10 Homomorphism, automorphism and dualities in linear incidence geometry
Overview
This topic studies maps between incidence structures that preserve geometric operations and the duality relations between them.
Related Wikipedia Page
Wikipedia: Incidence geometry
Useful Links
Key Ideas
- automorphisms
- duality
- incidence-preserving maps
Typical Uses
Important for classifying geometric structures and understanding symmetry.
Applications
- Finite geometries
- Coding theory
- Synthetic geometry
References
Recommended Textbooks
51A15 Structures with parallelism
Overview
Parallelism gives incidence structures an additional notion of direction and leads to affine and related geometries.
Related Wikipedia Page
Wikipedia: Affine geometry
Useful Links
Key Ideas
- parallel classes
- affine spaces
- direction sets
Typical Uses
Central in elementary geometry and modern incidence theory.
Applications
- Computer graphics
- Affine spaces
- Design theory
References
Recommended Textbooks
51A20 Configuration theorems in linear incidence geometry
Overview
Configuration theorems describe finite arrangements of points and lines whose incidence properties are constrained by geometric laws.
Related Wikipedia Page
Wikipedia: Configuration (geometry)
Useful Links
Key Ideas
- Pappus
- Desargues
- incidence configurations
Typical Uses
Useful for proving structural properties of geometric systems.
Applications
- Synthetic geometry
- Combinatorics
- Finite geometry
References
Recommended Textbooks
51A25 Algebraization in linear incidence geometry
Overview
Algebraization translates incidence geometry into algebraic structures such as vector spaces, fields, and division rings.
Related Wikipedia Page
Wikipedia: Projective space
Useful Links
Key Ideas
- vector spaces
- coordinate systems
- algebraic models
Typical Uses
A bridge between synthetic geometry and algebra.
Applications
- Coordinate geometry
- Finite projective spaces
- Coding theory
References
Recommended Textbooks
51A30 Desarguesian and Pappian geometries
Overview
These geometries satisfy the Desargues and Pappus theorems, giving especially rich algebraic structure and a direct connection to projective spaces over division rings or fields.
Related Wikipedia Page
Wikipedia: Projective geometry
Useful Links
Key Ideas
- Desargues theorem
- Pappus theorem
- coordinate fields
Typical Uses
Central to the classification of projective spaces and incidence structures.
Applications
- Foundations of geometry
- Algebraic geometry
- Finite geometry
References
Recommended Textbooks
51A35 Non-Desarguesian affine and projective planes
Overview
Non-Desarguesian planes are geometric structures that fail to satisfy the Desargues theorem and therefore cannot be coordinatized by a field in the usual way.
Related Wikipedia Page
Wikipedia: Projective plane
Useful Links
Key Ideas
- translation planes
- projective planes
- exceptional geometries
Typical Uses
Important in finite geometry and combinatorial design.
Applications
- Finite projective planes
- Design theory
- Coding theory
References
Recommended Textbooks
51A40 Translation planes and spreads in linear incidence geometry
Overview
Translation planes and spreads are geometric structures built from families of subspaces and are central to the study of finite and infinite projective planes.
Related Wikipedia Page
Wikipedia: Spread (finite geometry)
Useful Links
Key Ideas
- spreads
- translation planes
- subspace partitions
Typical Uses
Used to construct and analyze special incidence geometries.
Applications
- Finite geometry
- Combinatorial design
- Coding theory
References
Recommended Textbooks
51A45 Incidence structures imbeddable into projective geometries
Overview
This area studies incidence structures that can be represented as subsets or substructures of projective spaces.
Related Wikipedia Page
Wikipedia: Projective space
Useful Links
Key Ideas
- embeddings
- substructures
- realizability
Typical Uses
Useful for determining when abstract incidence data is geometrically realizable.
Applications
- Finite geometry
- Combinatorics
- Graph drawing
References
Recommended Textbooks
51A50 Polar geometry, symplectic spaces, orthogonal geometry
Overview
Polar geometry studies incidence relations defined by forms such as symplectic and orthogonal bilinear forms.
Related Wikipedia Page
Wikipedia: Polar space
Useful Links
Key Ideas
- polar spaces
- quadrics
- symplectic forms
Typical Uses
Fundamental in finite geometry, Lie theory, and classical groups.
Applications
- Classical groups
- Finite geometry
- Coding theory
References
Recommended Textbooks