Projective Geometry - Core Definitions

Projective geometry extends affine and Euclidean geometry by adding "points at infinity" where parallel lines meet. This completion eliminates special cases and reveals beautiful dualities in geometric theorems.

DefinitionProjective Space

The $n$ -dimensional projective space $\mathbb{P}^n$ over a field $K$ is the set of equivalence classes of non-zero vectors in $K^{n+1}$ under the equivalence relation:

(x_0, \ldots, x_n) \sim (λx_0, \ldots, λx_n) \quad \text{for all } λ \neq 0

The equivalence class of $(x_0, \ldots, x_n)$ is denoted $[x_0: \cdots : x_n]$ and called homogeneous coordinates.

The key idea is that each point in projective space represents a line through the origin in $(n+1)$ -dimensional space. This construction adds points at infinity to ordinary affine space: when $x_0 \neq 0$ , we can normalize to get $[1: x_1/x_0: \cdots: x_n/x_0]$ , corresponding to the affine point $(x_1/x_0, \ldots, x_n/x_0)$ . Points with $x_0 = 0$ are the "points at infinity."

DefinitionProjective Transformations

A projective transformation (or projectivity) of $\mathbb{P}^n$ is induced by an invertible linear map of $K^{n+1}$ :

[x_0: \cdots : x_n] \mapsto [y_0: \cdots : y_n]

where $(y_0, \ldots, y_n)^T = A(x_0, \ldots, x_n)^T$ for some $(n+1) \times (n+1)$ invertible matrix $A$ .

The group of projective transformations is $PGL(n+1, K) = GL(n+1, K) / K^*$ , where scalar matrices act trivially.

In projective space, parallel lines meet at a point at infinity, and every pair of distinct lines in the projective plane intersects at exactly one point. This eliminates the special case of parallel lines from Euclidean geometry.

ExampleThe Real Projective Line

The real projective line $\mathbb{P}^1(\mathbb{R})$ can be visualized as the usual real line $\mathbb{R}$ plus one point at infinity: $\mathbb{R} \cup \{\infty\}$ . Topologically, it's a circle.

Points have homogeneous coordinates $[x:y]$ with $(x,y) \neq (0,0)$ . When $y \neq 0$ , normalize to $[x/y : 1]$ , corresponding to the real number $x/y$ . The point $[1:0]$ is the point at infinity.

The projective plane $\mathbb{P}^2$ consists of lines through the origin in $\mathbb{R}^3$ . We can visualize it as the unit sphere with antipodal points identified (since $[x:y:z]$ and $[-x:-y:-z]$ represent the same point). Alternatively, it's the ordinary plane plus a "line at infinity."

Remark

Projective geometry arose from studying perspective in Renaissance art. When painters draw parallel railroad tracks converging at a vanishing point, they're implicitly using projective geometry. The mathematical theory was developed rigorously in the 19th century by Poncelet, Steiner, and others.

Homogeneous coordinates provide elegant equations for geometric objects. A line in $\mathbb{P}^2$ has equation $ax + by + cz = 0$ where $[a:b:c]$ are homogeneous coefficients. A point $[x:y:z]$ lies on the line if the equation is satisfied. This representation treats points and lines symmetrically, foreshadowing projective duality.