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BASIC FACTS OF PROJECTIVE GEOMETRY
Mahamadaliyeva O’g’iloy
NamSU, Faculty of Physics and Mathematics
1st student of Mathematics
Ismonjanova Sumbula Sherzod qizi
3rd -cource student of Applied mathematics.
https://doi.org/10.5281/zenodo.15640438
Abstract:
This article provides a comprehensive survey of the foundational structures
and theorems in real projective geometry. Beginning with the construction of projective space
𝑃
𝑛
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.
INTRODUCTION
Projective geometry emerged in the Renaissance as artists sought mathematically
consistent methods for perspective drawing. Early work by Pappus and later by Girard
Desargues (1639) and Blaise Pascal (1639) uncovered striking incidence properties invariant
under “projection.” In the nineteenth century, Möbius, Plücker, and Steiner formalized the
subject, introducing homogeneous coordinates and emphasizing duality. Today, projective
ideas underpin algebraic geometry, coding theory, and computer vision: any two photographs
of a rigid scene are related by a projective transformation (a “homography”), enabling 3D
reconstruction and image stitching.
In this article, we systematize the core algebraic methods and the principal theorems. We
assume familiarity with basic linear algebra and Euclidean geometry but develop all projective
concepts from first principles.
METHODS
Define
𝑃
𝑛
= (𝑅
{𝑛+1}
∖ {0}) ∼, (𝑥
0
, … , 𝑥
𝑛
) ∼ 𝜆(𝑥
0
, … , 𝑥
𝑛
), 𝜆 ≠ 0.
A point is denoted
[𝑥
0
, … , 𝑥
𝑛
]
. The standard affine chart
𝑈
𝑖
is given by
𝑥
𝑖
≠ 0,
with
coordinates
These charts cover
𝑃
𝑛
and exhibit it as an
𝑛
-dimensional manifold.
A
𝑘
-plane in
𝑃
𝑛
is the image of a
(𝑘 + 1) −
dimensional linear subspace of
𝑅
{𝑛+1}
.
Incidence is encoded algebraically: given a point
[𝑥]
and a hyperplane
𝐻 = [𝑎]
defined by
𝑎
0
𝑥
0
+ ⋯ . . +𝑎
𝑛
𝑥
𝑛
= 0,
we have
[𝑥] ∈ 𝐻 ⟺ ⟨𝑎, 𝑥⟩ = 0.
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More generally, the Grassmannian
𝐺𝑟(𝑘, 𝑛)
parametrizes all
𝑘
-planes in
𝑃
𝑛
, with Plücker
coordinates arising from the
(𝑘 + 1) × (𝑘 + 1)
minors of a representative matrix.
Any invertible linear map
𝑀 ∈ 𝐺𝐿
𝑛+1
(𝑅)
induces
𝜙
𝑀
: [𝑥] ↦ [𝑀𝑥].
Since scalar multiples act trivially, the full group of projective automorphisms is
RESULTS
Theorem.
In
𝑃
𝑛
, the correspondence
[𝑥] ⟷ (𝑎
0
𝑥
0
+ ⋯ + 𝑎
𝑛
𝑥
𝑛
= 0)
yields an isomorphism between the lattice of subspaces and its opposite
: 𝑝𝑜𝑖𝑛𝑡𝑠
↔
ℎ𝑦𝑝𝑒𝑟𝑝𝑙𝑎𝑛𝑒𝑠, 𝑙𝑖𝑛𝑒𝑠 ↔ (𝑛 − 2) − 𝑝𝑙𝑎𝑛𝑒𝑠,
etc. Every valid incidence statement has a dual.
Example.
In
𝑃
2
, “any two distinct points determine exactly one line” is dual to “any two
distinct lines meet at exactly one point.”
Let four distinct collinear points
𝐴, 𝐵, 𝐶, 𝐷 ⊂ 𝑃
1
be represented in an affine coordinate
𝑥 ∈ 𝑅 ∪ {∞}.
Their cross–ratio is
(𝐴, 𝐵; 𝐶, 𝐷) =
(𝑐 − 𝑎)(𝑑 − 𝑏)
(𝑐 − 𝑏)(𝑑 − 𝑎)
.
Proposition.
If
𝑇 ∈ 𝑃𝐺𝐿
2
(𝑅) 𝑡ℎ𝑒𝑛
(𝑇(𝐴), 𝑇(𝐵); 𝑇(𝐶), 𝑇(𝐷)) = (𝐴, 𝐵; 𝐶, 𝐷).
Proof sketch.
Any
𝑇
is of the form
𝑥 ↦
𝑝𝑥+𝑞
𝑟𝑥+𝑠
𝐴
direct computation shows that substituting
into the cross–ratio formula cancels all dependence on
𝑝, 𝑞, 𝑟, 𝑠.
The cross–ratio also characterizes harmonic division:
(𝐴, 𝐵; 𝐶, 𝐷) = −1.
Fundamental theorem.
Let
𝜙 : 𝑃
𝑛
→ 𝑃
𝑛
be a bijection sending lines to lines (a
collineation
). Then
𝜙
is induced by a semilinear map of
𝑅
𝑛+1
.
Over
𝑅
, semilinear = linear, so
𝜙 ∈ 𝑃𝐺𝐿
𝑛+1
(𝑅).
Proof sketch.
One shows first that
𝜙
sends triples of collinear points to collinear triples
and preserves cross–ratios in 1D subspaces; then extends to all of
𝑃
𝑛
by choosing bases of points
in general position.
Desargues’ theorem.
Given triangles
△ 𝐴𝐵𝐶
and
△ 𝐴′𝐵′𝐶′
in
𝑃
2
such that lines
𝐴𝐴
′
, 𝐵𝐵
′
, 𝐶𝐶
′
concur at a point
𝑂,
the intersection points
𝑃 = 𝐵𝐶 ∩ 𝐵
′
𝐶
′
, 𝑄 = 𝐶𝐴 ∩ 𝐶
′
𝐴
′
, 𝑅 = 𝐴𝐵 ∩ 𝐴
′
𝐵
′
lie collinearly.
Coordinate proof.
Embed in
𝑃
3
, view the two triangles as planar sections of a 3D pyramid,
then project back to
𝑃
2
.
Pascal’s theorem.
For any hexagon
𝐴
1
𝐴
2
𝐴
3
𝐴
4
𝐴
5
𝐴
6
inscribed in a nondegenerate conic in
𝑃
2
, the three points
𝐴
1
𝐴
2
∩ 𝐴
4
𝐴
5
, 𝐴
2
𝐴
3
∩ 𝐴
5
𝐴
6
, 𝐴
3
𝐴
4
∩ 𝐴
6
𝐴
1
are collinear (the “Pascal line”).
Algebraic proof.
Parameterize the conic as the image of
𝑃
1
under a degree-2 Veronese
embedding, reduce to a statement about four cross–ratios, and verify collinearity via a
determinant vanishing.
DISCUSSION
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Projective geometry’s algebraic formalism renders many proofs linear-algebraic,
replacing case-by-case synthetic arguments. Duality reveals a deep symmetry: statements
about points and hyperplanes are interchangeable. The cross–ratio is the unique projective
invariant on four points and becomes central in complex analysis and Möbius geometry.
Beyond
𝑅
, one studies
𝑃
𝑛
(𝐹
𝑞
),
yielding finite projective planes (e.g.\ the Fano plane
𝑃
2
(𝐹
2
)
with applications in coding theory and combinatorics. Over
𝐶, 𝑃
𝑛
(𝐶)
is the setting for
algebraic varieties, moduli spaces, and enumerative geometry (e.g.\ the count of lines on a cubic
surface).
In computer vision, images are related by planar homographies
𝐻 ∈ 𝑃𝐺𝐿
3
(𝑅).
Given four
known correspondences, one solves a linear system for
𝐻
and uses it to rectify or stitch images.
Reconstruction of 3D scenes from multiple views relies on projective invariants and the
Fundamental theorem to recover camera matrices up to projective ambiguity.
CONCLUSION
Projective geometry reframes classical Euclidean notions in a unified, incidence-based
language, treating “points at infinity” on the same footing as finite points and rendering
parallelism a special case of intersection. By introducing homogeneous coordinates and affine
charts, one gains a powerful linear-algebraic toolkit: subspaces become coordinate subspaces,
incidence reduces to bilinear forms, and transformations become matrix actions in
𝑃𝐺𝐿
𝑛+1
.
The principle of duality reveals an elegant symmetry between points and hyperplanes,
while the cross-ratio emerges as the unique invariant of four collinear points under
homographies. The Fundamental Theorem of Projective Geometry then shows that any
bijection preserving lines must itself be a projective transformation, anchoring the subject’s
rigidity and its deep connection to linear algebra. Finally, the classical incidence theorems of
Desargues and Pascal both illustrate the reach of projective methods-offering coordinate proofs
via lifts to higher dimensions or via determinant identities-and serve as gateways to broader
algebraic structures such as finite fields and division rings.
Beyond its foundational theorems, projective geometry continues to influence modern
mathematics and applications: from coding theory in finite projective planes to the global study
of complex varieties, and from tropical analogues to practical algorithms in computer vision for
image rectification and 3D reconstruction. In this way, the subject stands as both a historical
cornerstone and an ever‐evolving landscape, where deep theoretical insights and real-world
problems meet in the rich interplay of algebra, combinatorics, and geometry.
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