Euclidean plane and its relatives

Study overview of the reviewed opening

Petrunin’s introduction describes a course designed to be elementary in its tools while rigorous in its reasoning. It assumes that students already know enough about real numbers and continuity to work with foundational geometry efficiently. Elementary set language, limits and familiar functions are part of that preparation; vector algebra and complex numbers become useful for selected later chapters.

The organising choice is the metric approach. Rather than beginning with an extensive classical axiom system, the book describes the Euclidean plane as a metric space satisfying specified properties. This introduces foundational ideas about distance while supporting the familiar study of lines, angles, triangles and circles.

The introduction also explains the dependencies across the verified chapters. Euclidean results lead to circle geometry and inversion, which support a model of the hyperbolic plane. Neutral geometry removes the parallel postulate from consideration, making comparison between systems possible. Later chapters extend the discussion to affine, projective and spherical geometry, complex coordinates, constructions and area. The author qualifies the rigour of some supplementary chapters. This guide reflects the reviewed introduction and its roadmap, rather than separately summarising the proofs in all twenty chapters.

Book contents

1. Introduction
2. 1 Preliminaries
3. 2 Axioms
4. 3 Half-planes
5. 4 Congruent triangles
6. 5 Perpendicular lines
7. 6 Similar triangles
8. 7 Parallel lines
9. 8 Triangle geometry
10. 9 Inscribed angles
11. 10 Inversion
12. 11 Neutral plane
13. 12 Hyperbolic plane
14. 13 Geometry of the h-plane
15. 14 Affine geometry
16. 15 Projective geometry
17. 16 Spherical geometry
18. 17 Projective model
19. 18 Complex coordinates
20. 19 Geometric constructions
21. 20 Area

Questions for study

The following prompts are Academic Master’s editorial material.

• What changes when a geometric system is defined using distance?
• Why are continuity and the parallel postulate significant in the reading route?
• How does the introduction distinguish the core development from additional topics?

Sources and coverage

Introduction read in full (PDF pages 6–8); arXiv version, licence and contents checked.

Source edition · Source licence