Projective Geometry

Projective geometry is a branch of geometry concerned with properties preserved by projective transformations. Its central objects are points, lines, planes, and their incidence relations: which points lie on which lines or planes, and how these objects intersect. Unlike Euclidean geometry, it does not treat lengths, angles, or parallelism as fundamental invariants. Motivated by perspective, it introduces points at infinity and provides a framework in which many apparently different geometric configurations have the same underlying structure. (vision.stanford.edu)

Projective space and homogeneous coordinates

The standard algebraic construction begins with a vector space $V$ over a field $F$. Its associated projective space, written $\mathbb P(V)$, is the set of all one-dimensional linear subspaces of $V$. If $V$ has dimension $n+1$, then $\mathbb P(V)$ has projective dimension $n$, and is commonly denoted $\mathbb P^n(F)$. Thus, projective points correspond to entire lines through the origin of the underlying vector space, not to individual vectors. (people.maths.ox.ac.uk)

After choosing a basis, a point has homogeneous coordinates

where the coordinates are not all zero and

The brackets therefore represent an equivalence class of nonzero vectors under multiplication by a nonzero scalar. A projective line is the projectivization of a two-dimensional linear subspace; a projective plane comes from a three-dimensional subspace. Incidence becomes inclusion of vector subspaces, connecting projective geometry directly with linear algebra. (people.maths.ox.ac.uk)

Affine charts and points at infinity

In the projective plane, use coordinates $[X:Y:W]$. The points with $W\ne0$ can be written uniquely as

They form an affine plane. The remaining points, with $W=0$, form the line at infinity. Each represents a direction of affine parallel lines. For example, the lines $y=mx+b_1$ and $y=mx+b_2$, with $b_1\ne b_2$, intersect projectively at $[1:m:0]$. (courses.maths.ox.ac.uk)

“Infinity” depends on the chosen chart. Another projective coordinate system can make a formerly infinite point finite. Projective space itself does not distinguish such points. Moreover, the statement that any two distinct lines intersect applies to a projective plane: lines in projective three-space can still be skew. (courses.maths.ox.ac.uk)

Transformations and invariants

An invertible linear map $A:V\to V$ induces a projective transformation

Multiplying $A$ by a nonzero scalar gives the same transformation. In the real projective plane, an invertible $3\times3$ matrix therefore represents a transformation with eight independent parameters. Such a plane transformation is commonly called a homography. Four ordered point correspondences determine it uniquely when each set has no three collinear points. (ai.stanford.edu)

Projective transformations preserve incidence, collinearity, and concurrence, but generally change lengths, angles, midpoints, and parallelism. A line represented by the equation $\ell^{\mathsf T}x=0$ transforms according to

while points transform as $x'=Ax$. The inverse-transpose rule ensures that transformed points remain on their transformed lines. (ai.stanford.edu)

The projective transformation group contains affine and Euclidean transformation groups as more restrictive subgroups. Affine geometry is recovered by requiring preservation of a selected hyperplane at infinity; Euclidean geometry requires further metric structure. This hierarchy exemplifies the Erlangen programme associated with Felix Klein, which characterizes geometries through transformation groups and their invariants. (people.maths.ox.ac.uk)

Cross-ratio

The principal numerical invariant of four distinct points on a projective line is their cross-ratio. With affine coordinates $a,b,c,d$, one convention is

The differences are signed quantities, not unsigned distances. The order of the points matters, and different conventions can produce reciprocal or otherwise related expressions. A point at infinity is accommodated by taking the corresponding limit, or by using homogeneous coordinates directly. (people.maths.ox.ac.uk)

On a projective line, transformations take the fractional-linear form

Substituting this expression into the cross-ratio formula makes the extra factors cancel. Thus ordinary separation ratios may change under perspective while the cross-ratio remains invariant. It also supplies an invariant for four concurrent lines: intersect them with any transverse line and take the cross-ratio of the four intersection points. (vision.stanford.edu)

Duality

Projective duality exchanges points and lines in a projective plane while reversing incidence. The assertion that two distinct points determine a unique line is dual to the assertion that two distinct lines meet in a unique point. Collinearity of points corresponds to concurrence of lines. Applying a theorem to the dual plane therefore produces a corresponding dual theorem. (courses.maths.ox.ac.uk)

Algebraically, hyperplanes in $\mathbb P(V)$ correspond to points of $\mathbb P(V^*)$, where $V^*$ is the dual vector space. A hyperplane is the kernel of a nonzero linear functional, considered up to scalar multiplication. In projective dimension $n$, a $k$-dimensional subspace has a dual subspace of dimension $n-k-1$. Thus, in projective three-space, points and planes are dual, while lines are dual to lines. (people.maths.ox.ac.uk)

Conics and classical theorems

A projective conic is defined by a homogeneous quadratic equation. Over the real or complex numbers, it can be written

with $C$ symmetric; it is nonsingular when $C$ is invertible. All nonsingular real conics with real points are projectively equivalent. Their affine appearances depend on the chosen line at infinity: an ellipse has no real intersection with it, a parabola is tangent to it, and a hyperbola meets it at two distinct real points. Thus these familiar types are not separate projective classes. (people.maths.ox.ac.uk)

A nonsingular conic also defines a polarity, relating points to lines. For a point represented by $p$, its polar has equation $p^{\mathsf T}Cx=0$; when $p$ lies on the conic, this is its tangent line. This correspondence connects duality with quadratic forms. (people.maths.ox.ac.uk)

Several classical results express projective geometry's emphasis on incidence:

• Desargues’ theorem: if the lines joining corresponding vertices of two triangles are concurrent, the intersections of their corresponding sides are collinear, provided the configuration is nondegenerate.
• Pappus’ theorem: for triples $A,B,C$ and $A',B',C'$ on two lines, the intersections $AB'\cap A'B$, $AC'\cap A'C$, and $BC'\cap B'C$ are collinear, under the usual distinctness assumptions.
• Pascal’s theorem: for six points on a nonsingular conic, the three intersections of opposite sides of the resulting hexagon are collinear. Its dual, Brianchon’s theorem, concerns a hexagon circumscribed about a conic, whose three diagonals joining opposite vertices are concurrent. (courses.maths.ox.ac.uk)

These statements include intersections at infinity, avoiding separate parallel-line exceptions that arise in affine formulations. (math.ucr.edu)

Axiomatic and finite projective geometry

Projective geometry can also be developed synthetically from axioms, without beginning with coordinates. A projective plane consists of points, lines, and an incidence relation satisfying:

1. Any two distinct points lie on exactly one line.
2. Any two distinct lines meet at exactly one point.
3. There exist four points, no three of which are collinear. (math.ucr.edu)

These axioms alone do not force the plane to come from a vector space over a field. Desarguesian planes are coordinatizable over division rings, whose multiplication need not be commutative; non-Desarguesian planes also exist. Pappus’ theorem characterizes planes coordinatizable over commutative fields. Consequently, a result proved using field coordinates need not hold in every abstract projective plane. (math.ucr.edu)

Over a finite field with $q$ elements, the plane $\mathrm{PG}(2,q)$ has

points and the same number of lines, with $q+1$ points on each line and $q+1$ lines through each point. For $q=2$, this is the Fano plane: seven points and seven lines, each incident with three points or lines respectively. Such structures connect projective geometry with combinatorial designs. (isibang.ac.in)

Historical development

The development of linear perspective during the Renaissance supplied an important motivation for projective methods. Girard Desargues developed a unified treatment of conics and points at infinity in his 1639 Brouillon project. His work influenced Blaise Pascal, although its wider reception was limited and much of its significance was recognized later. (mathshistory.st-andrews.ac.uk)

Jean-Victor Poncelet's Traité des propriétés projectives des figures, published in 1822, helped establish projective geometry as an independent subject. Nineteenth-century work developed both synthetic methods and coordinate approaches, while Klein's transformation-group viewpoint placed projective geometry within a broader classification of geometries. (mathshistory.st-andrews.ac.uk)

Applications and limits

In computer vision, an ideal perspective camera is represented in homogeneous coordinates by

where $X$ is a three-dimensional projective point, $x$ is its image, and $P$ is a rank-three $3\times4$ camera matrix. Planar scenes viewed by different cameras are related by homographies under suitable nondegeneracy conditions. Projective methods also describe vanishing points and geometric constraints between multiple images. (vision.stanford.edu)

A camera projection must be distinguished from an invertible projective transformation: it reduces dimension and generally maps all points along a viewing ray to one image point. Projective image geometry alone does not recover absolute lengths or angles; additional calibration or metric information is required. (ai.stanford.edu)

In algebraic geometry, projective spaces are ambient spaces for varieties defined by homogeneous polynomials. Projective closure adds the points missing from an affine description, and constructions such as the Segre, Veronese, and Plücker embeddings use projective coordinates to represent more elaborate spaces. Here projective geometry extends beyond perspective and incidence into the study of algebraic equations and their solution spaces. (people.maths.ox.ac.uk)

References

1. ASO: Projective Geometry (2024–25) — University of Oxfordcourses.maths.ox.ac.uk
2. Projective Geometry — Nigel Hitchinpeople.maths.ox.ac.uk
3. Projective Geometry: Printed Lecture Notes — University of Oxfordcourses.maths.ox.ac.uk
4. An Introduction to Projective Geometry (for computer vision) — Stan Birchfieldai.stanford.edu
5. The Klein Programme — Nigel Hitchinpeople.maths.ox.ac.uk
6. Quadrics — Nigel Hitchinpeople.maths.ox.ac.uk
7. Incidence Geometrymath.ucr.edu
8. This Week's Finds in Mathematical Physics, Week 145 — John Baezmath.ucr.edu
9. Projective Planes — Discrete Mathematics Lecture Notesisibang.ac.in