This volume studies geometric structures, transformations, and invariants. It covers classical geometry, synthetic methods, and modern structural viewpoints.
Part I. Foundations of Geometry
Chapter 1. Geometric Objects
1.1 Points, lines, planes 1.2 Incidence and betweenness 1.3 Distance and angle 1.4 Congruence and similarity 1.5 Examples
Chapter 2. Axiomatic Geometry
2.1 Euclidean axioms 2.2 Hilbert-style axioms 2.3 Models of geometry 2.4 Independence of axioms 2.5 Examples
Chapter 3. Transformations
3.1 Isometries 3.2 Similarities 3.3 Affine transformations 3.4 Projective transformations 3.5 Invariants
Part II. Euclidean and Metric Geometry
Chapter 4. Euclidean Geometry
4.1 Triangles and circles 4.2 Parallel lines 4.3 Area and volume 4.4 Classical theorems 4.5 Constructions
Chapter 5. Metric Geometry
5.1 Metric spaces 5.2 Geodesics 5.3 Completeness 5.4 Curvature intuition 5.5 Examples
Chapter 6. Convex Geometry
6.1 Convex sets 6.2 Supporting hyperplanes 6.3 Separation theorems 6.4 Polytopes 6.5 Applications
Part III. Projective and Affine Geometry
Chapter 7. Affine Geometry
7.1 Affine spaces 7.2 Affine combinations 7.3 Parallelism 7.4 Affine transformations 7.5 Examples
Chapter 8. Projective Geometry
8.1 Projective spaces 8.2 Homogeneous coordinates 8.3 Duality 8.4 Cross-ratio 8.5 Examples
Chapter 9. Classical Theorems
9.1 Desargues theorem 9.2 Pappus theorem 9.3 Pascal theorem 9.4 Brianchon theorem 9.5 Applications
Part IV. Non-Euclidean Geometry
Chapter 10. Hyperbolic Geometry
10.1 Models of hyperbolic plane 10.2 Geodesics 10.3 Triangles and area 10.4 Isometries 10.5 Applications
Chapter 11. Spherical Geometry
11.1 Great circles 11.2 Spherical triangles 11.3 Area and angle excess 11.4 Navigation examples 11.5 Applications
Chapter 12. Other Geometries
12.1 Elliptic geometry 12.2 Incidence geometries 12.3 Finite geometries 12.4 Synthetic geometries 12.5 Examples
Part V. Geometric Structures
Chapter 13. Transformation Geometry
13.1 Klein’s Erlangen program 13.2 Groups acting on spaces 13.3 Invariants under transformations 13.4 Classification by symmetry 13.5 Examples
Chapter 14. Geometric Algebra
14.1 Vectors and blades 14.2 Products and rotations 14.3 Clifford algebra overview 14.4 Applications 14.5 Examples
Chapter 15. Discrete Geometry
15.1 Arrangements 15.2 Tilings 15.3 Packings and coverings 15.4 Lattices 15.5 Applications
Part VI. Geometry and Algebra
Chapter 16. Coordinate Geometry
16.1 Coordinates and equations 16.2 Lines and conics 16.3 Quadrics 16.4 Transformations 16.5 Examples
Chapter 17. Algebraic Methods
17.1 Polynomial equations 17.2 Varieties overview 17.3 Incidence via algebra 17.4 Computational geometry links 17.5 Applications
Chapter 18. Symmetry and Groups
18.1 Symmetry groups 18.2 Reflection groups 18.3 Crystallographic groups 18.4 Classification problems 18.5 Applications
Part VII. Applications and Computation
Chapter 19. Computational Geometry
19.1 Geometric algorithms 19.2 Convex hulls 19.3 Voronoi diagrams 19.4 Triangulations 19.5 Applications
Chapter 20. Geometry in Physics
20.1 Space and motion 20.2 Coordinate systems 20.3 Symmetry principles 20.4 Relativity overview 20.5 Applications
Chapter 21. Geometry in Graphics and Vision
21.1 Transformations in graphics 21.2 Camera geometry 21.3 Meshes and surfaces 21.4 Shape analysis 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Metric geometry 22.2 Incidence geometry 22.3 Rigidity theory 22.4 Geometric group theory links 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Packing and covering problems 23.2 Rigidity questions 23.3 Incidence bounds 23.4 Computational complexity 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of geometry 24.2 Key contributors 24.3 Euclidean to modern geometry 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Classical theorem reference B. Standard geometric models C. Proof techniques checklist D. Construction methods E. Cross-reference to other MSC branches