Geometry often begins by measuring where points are, but many geometric questions are really about directions. Two parallel railway tracks do not meet in the affine plane, yet in a perspective drawing they converge toward a single point on the horizon. Projective space is the geometry obtained by adding those missing directions as genuine points, so that incidence statements become cleaner and linear equations behave uniformly.

[example: Parallel Lines Acquire a Point at Infinity] In the affine plane $\mathbb{R}^2$, the horizontal lines $y = 0$ and $y = 1$ do not meet: a common point would have second coordinate both $0$ and $1$, which is impossible in $\mathbb{R}$. In the temporary coordinate notation used in this example, $[a:b:c]$ means the projective point represented by the nonzero vector $(a,b,c)$, and multiplying all three entries by the same nonzero scalar gives the same projective point. Embed the affine plane into projective space by sending \begin{align*} (x,y) \mapsto [x:y:1]. \end{align*} The line $y = 0$ becomes the set of points $[x:0:1]$, and the line $y = 1$ becomes the set of points $[x:1:1]$. For $x \neq 0$, rescale each representative by $1/x$. On the first line, \begin{align*} [x:0:1] = [1:0:1/x]. \end{align*} On the second line, \begin{align*} [x:1:1] = [1:1/x:1/x]. \end{align*} As $|x| \to \infty$, both [real numbers](/page/Real%20Numbers) $1/x$ tend to $0$, so the first family tends to $[1:0:0]$ and the second family also tends to $[1:0:0]$. Thus the two parallel affine lines acquire the same projective point at infinity, represented by the nonzero vector $(1,0,0)$, which records their common horizontal direction. [/example]

This example explains the guiding rule: a projective point is not a vector, but a one-dimensional subspace of a [vector space](/page/Vector%20Space). Scaling a nonzero vector changes its length, not the direction it spans. Projective space remembers the direction and forgets the scale.

## Definition

The most economical construction starts with a vector space and asks which objects should count as directions. A direction is carried by a whole one-dimensional line through the origin, so the construction must discard the vector with no direction and remember only the line spanned by a nonzero vector.

[definition: Projective Space] Let $K$ be a field and let $V$ be a nonzero vector space over $K$. The projective space associated to $V$ is the set \begin{align*} \mathbb{P}(V) = \{ L \subset V : L \text{ is a one-dimensional } K\text{-linear subspace} \}. \end{align*} [/definition]

Most calculations need a coordinate model rather than an arbitrary vector space. This motivates the standard projective space, where the ambient vector space is $K^{n+1}$ and the resulting projective dimension is $n$.

[definition: Standard Projective Space] For a field $K$ and an integer $n \geq 0$, the standard projective $n$-space over $K$ is \begin{align*} \mathbb{P}^n_K = \mathbb{P}(K^{n+1}). \end{align*} [/definition]

The standard model still needs notation that remembers scaling has been discarded. Homogeneous coordinates provide a coordinate language in which any nonzero representative may be used without changing the point.

[definition: Homogeneous Coordinates] Let $K$ be a field. A point of $\mathbb{P}^n_K$ represented by a nonzero vector $(x_0,\ldots,x_n) \in K^{n+1}$ is written \begin{align*} [x_0:\cdots:x_n]. \end{align*} [/definition]

Two coordinate lists represent the same point precisely when there is a scalar $\lambda \in K^\times$ such that

\begin{align*} (y_0,\ldots,y_n) = (\lambda x_0,\ldots,\lambda x_n). \end{align*}

The colons in homogeneous coordinates are a warning that ratios, not absolute coordinates, are intrinsic. The expression $[2:4:6]$ is the same point as $[1:2:3]$ over any field in which $2$ is invertible.

[example: The Projective Line] The projective line $\mathbb{P}^1_K$ consists of homogeneous coordinate points $[x_0:x_1]$ with $(x_0,x_1) \neq (0,0)$, where multiplying both coordinates by the same nonzero scalar does not change the point. We separate the points according to whether the first coordinate vanishes. If $x_0 \neq 0$, then $x_0^{-1} \in K^\times$, so rescaling the representative $(x_0,x_1)$ by $x_0^{-1}$ gives \begin{align*} [x_0:x_1] = [x_0^{-1}x_0:x_0^{-1}x_1]. \end{align*} Since $x_0^{-1}x_0 = 1$ and $x_0^{-1}x_1 = x_1/x_0$, this becomes \begin{align*} [x_0:x_1] = [1:x_1/x_0]. \end{align*} Writing \begin{align*} t = \frac{x_1}{x_0} \end{align*} therefore puts every point with $x_0 \neq 0$ in the form $[1:t]$ for a unique $t \in K$: if $[1:t] = [1:s]$, then there is $\lambda \in K^\times$ with $(1,s) = (\lambda,\lambda t)$, so $\lambda = 1$ and hence $s=t$. If $x_0 = 0$, then $x_1 \neq 0$, because the pair $(x_0,x_1)$ is not allowed to be $(0,0)$. Rescaling by $x_1^{-1}$ gives \begin{align*} [0:x_1] = [x_1^{-1}0:x_1^{-1}x_1]. \end{align*} Thus \begin{align*} [0:x_1] = [0:1]. \end{align*} This point is not equal to any $[1:t]$, since equality would require $(0,1)=(\lambda,\lambda t)$ for some $\lambda \in K^\times$, but the first coordinate would then force $\lambda=0$, a contradiction. Hence $\mathbb{P}^1_K$ is exactly one affine coordinate line $\{[1:t]:t \in K\}$ together with the single extra point $[0:1]$, the point at infinity for this coordinate. [/example]

The example suggests that the central operation is not merely choosing representatives, but identifying all representatives of the same direction. The [equivalence relation](/page/Equivalence%20Relation) below isolates exactly the identification that turns nonzero vectors into projective points.

[definition: Projective Equivalence Relation] Let $V$ be a vector space over a field $K$. The projective equivalence relation is the binary relation $\sim$ on $V \setminus \{0\}$ defined by declaring $v \sim w$ when there exists $\lambda \in K^\times$ such that $w = \lambda v$. [/definition]

Once this relation is named, the line-based and quotient-based pictures should agree. The next statement is the compatibility check that lets projective space be used as a quotient in [topology](/page/Topology) and as a set of lines in linear algebra.

[quotetheorem:9466]

This quotient description is the source of many habits in projective geometry. It says that the line spanned by a nonzero vector and the equivalence class of all its nonzero scalar multiples are not two competing constructions; they are the same projective point described in two languages. From this point on, we can choose whichever representative makes a calculation easier, provided the result does not change after simultaneous rescaling of all coordinates. This is why affine charts can normalize one coordinate to $1$, and why later maps between projective spaces must be checked for compatibility with scalar multiples.

## Affine Charts and Points at Infinity