Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

51Nxx Analytic and projective geometry

This subtopic introduces the core ideas in analytic and projective 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

51N05 Descriptive geometry

Overview

Descriptive geometry represents three-dimensional objects through planar projections and constructions.

Related Wikipedia Page

Wikipedia: Descriptive geometry

Useful Links

Key Ideas

  • projections
  • sectioning
  • spatial reconstruction

Typical Uses

Foundational in technical drawing and geometric visualization.

Applications

  • Architecture
  • Engineering drawing
  • CAD education

References

Recommended Textbooks

51N10 Affine analytic geometry

Overview

Affine analytic geometry studies coordinate geometry with affine transformations and invariants as the primary symmetry class.

Related Wikipedia Page

Wikipedia: Affine space

Useful Links

Key Ideas

  • affine coordinates
  • parallelism invariants
  • linear transformations

Typical Uses

Useful for modeling geometric problems where ratios on parallel lines are meaningful but distances are not fixed.

Applications

  • Computer vision
  • Graphics transformations
  • Linear modeling

References

Recommended Textbooks

51N15 Projective analytic geometry

Overview

Projective analytic geometry combines coordinate methods with projective invariants and homogeneous coordinates.

Related Wikipedia Page

Wikipedia: Projective space

Useful Links

Key Ideas

  • homogeneous coordinates
  • projective transformations
  • cross-ratio invariants

Typical Uses

Core for modern geometric vision and many algebraic geometry preliminaries.

Applications

  • Computer vision
  • Camera models
  • Algebraic geometry preparation

References

Recommended Textbooks

51N20 Euclidean analytic geometry

Overview

Euclidean analytic geometry studies geometric figures via Cartesian coordinates, equations, and metric invariants.

Related Wikipedia Page

Wikipedia: Analytic geometry

Useful Links

Key Ideas

  • Cartesian methods
  • distance formulas
  • equations of curves and surfaces

Typical Uses

Widely used as an entry point to modern geometry and modeling.

Applications

  • Engineering mathematics
  • Computer graphics
  • Data modeling

References

Recommended Textbooks

51N25 Analytic geometry with other transformation groups

Overview

This area extends analytic geometry by replacing Euclidean symmetries with alternative transformation groups and corresponding invariants.

Related Wikipedia Page

Wikipedia: Klein geometry

Useful Links

Key Ideas

  • group actions
  • invariant theory
  • geometry via transformations

Typical Uses

Useful for unifying many geometries through symmetry principles.

Applications

  • Relativity models
  • Theoretical physics
  • Computer vision invariants

References

Recommended Textbooks

51N30 Geometry of classical groups in analytic geometry

Overview

Classical matrix groups such as orthogonal, unitary, and symplectic groups are studied through their geometric actions and invariants.

Related Wikipedia Page

Wikipedia: Classical group

Useful Links

Key Ideas

  • group actions on spaces
  • invariants
  • linear and bilinear forms

Typical Uses

Connects analytic geometry with representation theory and mathematical physics.

Applications

  • Quantum mechanics
  • Symmetry analysis
  • Differential equations

References

Recommended Textbooks

51N35 Questions of classical algebraic geometry in analytic geometry

Overview

This subtopic addresses classical algebraic-geometric questions using analytic-geometric techniques and coordinate methods.

Related Wikipedia Page

Wikipedia: Algebraic geometry

Useful Links

Key Ideas

  • analytic methods for algebraic objects
  • classical curves and surfaces
  • projective-analytic links

Typical Uses

Used to bridge computational and conceptual viewpoints between algebraic and analytic geometry.

Applications

  • Computer algebra
  • Vision geometry
  • Complex geometry

References

Recommended Textbooks