BASIC FACTS OF PROJECTIVE GEOMETRY

Аннотация

This article provides a comprehensive survey of the foundational structures and theorems in real projective geometry. Beginning with the construction of projective space 𝑃𝑛 via homogeneous coordinates and affine charts, we develop the language of incidence, duality, and subspace intersections. We then explore the cross–ratio and its invariance under 𝑃𝐺𝐿-actions, followed by a precise statement and proof sketch of the Fundamental Theorem of Projective Geometry. Finally, we present the classical incidence theorems of Desargues and Pascal-complete with coordinate proofs and geometric interpretations-and discuss extensions to finite fields, complex varieties, and modern applications in computer vision. Keywords: projective space, homogeneous coordinates, affine charts, duality, cross– ratio, projective linear group, fundamental theorem, desargues’ theorem, pascal’s theorem, finite projective planes, homographies.