A line in the affine plane can miss another line even when the geometry wants an intersection: the equations $y=0$ and $y=1$ never meet in affine $2$-space, but they have the same horizontal direction. Projective geometry repairs this by adding directions as points, and homogeneous coordinates are the notation that lets us compute with those added points.

The main surprise is that a projective point is not represented by a unique tuple. A nonzero scalar multiple represents the same point. That ambiguity is not a defect; it is what allows affine points and points at infinity to be handled by the same equations.

A first example shows the repair in its most concrete form. We need it before the definition because it explains why coordinates should be allowed to have a common scaling ambiguity.

[example: Parallel Lines Meet at Infinity] Let $k$ be a field and consider $\mathbb{P}^2_k$ with homogeneous coordinates $[X_0:X_1:X_2]$. On the affine chart $X_0\neq 0$, the point $(x,y)\in k^2$ is represented by $[1:x:y]$, so $x=X_1/X_0$ and $y=X_2/X_0$. The affine line $y=0$ is therefore described on this chart by \begin{align*} X_2/X_0=0. \end{align*} Since $X_0\neq 0$ on the chart, this is equivalent to \begin{align*} X_2=0. \end{align*} The affine line $y=1$ is described on the same chart by \begin{align*} X_2/X_0=1. \end{align*} Multiplying by the nonzero coordinate $X_0$ gives \begin{align*} X_2=X_0. \end{align*} Equivalently, its projective equation is \begin{align*} X_2-X_0=0. \end{align*} A common projective point of the two lines must satisfy both equations \begin{align*} X_2=0 \end{align*} and \begin{align*} X_2-X_0=0. \end{align*} Substituting $X_2=0$ into the second equation gives \begin{align*} 0-X_0=0. \end{align*} Thus \begin{align*} X_0=0. \end{align*} So any common point has the form $[0:X_1:0]$. Because homogeneous coordinates cannot all be zero, we must have $X_1\neq 0$. Rescaling the nonzero tuple $(0,X_1,0)$ by $X_1^{-1}$ gives \begin{align*} (0,X_1,0)\sim (0,1,0). \end{align*} Hence the two projective lines meet in the single point $[0:1:0]$. This point has $X_0=0$, so it lies outside the affine chart $X_0\neq 0$ and records the horizontal direction shared by the affine lines $y=0$ and $y=1$. [/example]

This example gives the central rule: the coordinate tuple matters only up to a common nonzero scalar. To make that rule precise, we define the coordinate notation and the [projective space](/page/Projective%20Space) it describes at the same time.

## Definition

An affine coordinate system measures displacement after an origin has been chosen. Homogeneous coordinates keep a different datum: the one-dimensional subspace spanned by a nonzero vector. This is the natural setting when two proportional tuples should describe the same geometric point.

[definition: Homogeneous Coordinates] Let $k$ be a field and let $n \geq 0$. Homogeneous coordinates on projective $n$-space over $k$ are equivalence classes of nonzero tuples in $k^{n+1}$ under the relation \begin{align*} (a_0,\ldots,a_n) \sim (b_0,\ldots,b_n) \end{align*} if there exists $\lambda \in k^\times$ such that \begin{align*} (b_0,\ldots,b_n)=(\lambda a_0,\ldots,\lambda a_n). \end{align*} The set of all such classes is denoted $\mathbb{P}^n_k$, and the class of $(a_0,\ldots,a_n)$ is written \begin{align*} [a_0:\cdots:a_n]. \end{align*} [/definition]

The notation $[a_0:\cdots:a_n]$ denotes a class, not a single tuple. To use the notation safely, the reader must internalize how rescaling changes representatives without changing the point.

[example: Many Representatives for One Point] Over $\mathbb{R}$, the tuple $(2,4,6)$ is obtained from $(1,2,3)$ by multiplying every coordinate by the nonzero scalar $2 \in \mathbb{R}^\times$: \begin{align*} (2,4,6)=(2\cdot 1,2\cdot 2,2\cdot 3)=2(1,2,3). \end{align*} Therefore these two tuples represent the same homogeneous coordinate class: \begin{align*} [2:4:6]=[1:2:3]. \end{align*} Similarly, $(-1,-2,-3)$ is obtained from $(1,2,3)$ by multiplying every coordinate by the nonzero scalar $-1 \in \mathbb{R}^\times$: \begin{align*} (-1,-2,-3)=((-1)\cdot 1,(-1)\cdot 2,(-1)\cdot 3)=(-1)(1,2,3). \end{align*} Thus \begin{align*} [-1:-2:-3]=[1:2:3]. \end{align*} So $(1,2,3)$, $(2,4,6)$, and $(-1,-2,-3)$ are three representatives of one point of $\mathbb{P}^2_{\mathbb{R}}$. The tuple $(0,0,0)$ defines no projective point because homogeneous coordinates are equivalence classes of nonzero tuples. The tuple $(0,1,3)$ does define a point, since \begin{align*} (0,1,3)\neq (0,0,0). \end{align*} Its first coordinate is allowed to vanish; only the simultaneous vanishing of all coordinates is forbidden. [/example]

## Affine Charts and Normalization

### Standard Affine Charts

Rescaling explains equality, but it does not yet explain how ordinary affine coordinates are recovered. For calculations, we need regions where one coordinate is nonzero and can be normalized to $1$.

[definition: Standard Affine Chart] Let $k$ be a field and let $0 \leq i \leq n$. The standard affine chart $U_i$ in $\mathbb{P}^n_k$ is \begin{align*} U_i=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : a_i \neq 0\}. \end{align*} Its chart map is \begin{align*} \varphi_i: U_i \to k^n. \end{align*} It sends each point by the rule \begin{align*} \varphi_i([a_0:\cdots:a_n])=(a_0/a_i,\ldots,a_{i-1}/a_i,a_{i+1}/a_i,\ldots,a_n/a_i). \end{align*} [/definition]

The complement of $U_0$ contains exactly the new directions added to [affine space](/page/Affine%20Space). We need a name for this complement because it is where parallel affine phenomena become ordinary intersections.

[definition: Hyperplane at Infinity] In the standard affine chart $U_0 \subset \mathbb{P}^n_k$, the hyperplane at infinity is \begin{align*} H_\infty=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : a_0=0\}. \end{align*} [/definition]

The hyperplane at infinity is not intrinsic to $\mathbb{P}^n_k$ alone; it depends on the chosen affine chart. It is useful because it separates the familiar affine part from the added directions.

### Equality and Normalization

The most common computational error is to compare homogeneous coordinates coordinate by coordinate. That ignores the [equivalence relation](/page/Equivalence%20Relation). We need an equality criterion that asks whether two tuples span the same line through the origin.

[quotetheorem:9486]