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 Projective space is built from linear directions, but it contains ordinary [affine space](/page/Affine%20Space) in many overlapping ways. Each coordinate that is not zero can be normalized to $1$, producing a chart that looks affine. [definition: Standard Affine Chart] For $0 \leq i \leq n$, the standard affine chart $U_i \subset \mathbb{P}^n_K$ is \begin{align*} U_i = \{ [x_0:\cdots:x_n] \in \mathbb{P}^n_K : x_i \neq 0 \}. \end{align*} [/definition] The chart $U_i$ should really be an affine coordinate system, not just a subset with a suggestive name. The possible obstruction is that homogeneous coordinates are only defined up to common rescaling, so a proposed coordinate tuple must be independent of the representative chosen. Normalizing by the nonzero coordinate $x_i$ removes that ambiguity and should leave exactly $n$ free affine coordinates. [quotetheorem:9467] No single affine chart contains every projective point, because some representatives have the chosen normalizing coordinate equal to zero. Still, every projective point $[x_0:\cdots:x_n]$ has at least one nonzero coordinate, so it belongs to at least one of the subsets $U_i$. Thus the standard charts cover $\mathbb{P}^n_K$ in the elementary set-theoretic sense: \begin{align*} \mathbb{P}^n_K = U_0 \cup U_1 \cup \cdots \cup U_n. \end{align*} This covering statement is the reason the local coordinate descriptions above are useful: even though no one normalization works globally, the finitely many normalizations together describe every projective direction. The chart $U_0$ identifies most of $\mathbb{P}^n_K$ with $K^n$ by writing points as $[1:x_1:\cdots:x_n]$. The only points missing from this affine view are those for which $x_0=0$; naming this complement lets us separate the ordinary affine part from the projective directions added at infinity. This complement is not just a leftover set from the covering argument. It is the standard place where projective geometry records directions of affine lines, so it needs its own name before we use $U_0$ as the affine part of projective space. [definition: Hyperplane at Infinity] Relative to the standard affine chart $U_0 \subset \mathbb{P}^n_K$, the hyperplane at infinity is the complement \begin{align*} H_\infty = \mathbb{P}^n_K \setminus U_0 = \{ [x_0:\cdots:x_n] \in \mathbb{P}^n_K : x_0 = 0 \}. \end{align*} [/definition] [example: The Projective Plane] The affine plane $K^2$ sits inside $\mathbb{P}^2_K$ as the chart $U_0$ by sending $(a,b)$ to $[1:a:b]$. This parametrization is injective: if $[1:a:b]=[1:c:d]$, then there is $\lambda \in K^\times$ such that \begin{align*} (1,c,d)=(\lambda,\lambda a,\lambda b). \end{align*} The first coordinate gives $\lambda=1$, so $c=a$ and $d=b$. Every point of $U_0$ has first coordinate nonzero, so if $[x_0:x_1:x_2]\in U_0$, then rescaling by $x_0^{-1}$ gives \begin{align*} [x_0:x_1:x_2]=[1:x_1/x_0:x_2/x_0]. \end{align*} Thus $U_0$ is exactly the affine plane written in homogeneous coordinates. Its complement consists of the points whose first coordinate is zero: \begin{align*} H_\infty=\{[0:x_1:x_2]:(x_1,x_2)\neq(0,0)\}. \end{align*} A nonzero affine direction is represented by a vector $(u,v)\in K^2\setminus\{(0,0)\}$, with $(u,v)$ and $(\lambda u,\lambda v)$ representing the same direction when $\lambda\in K^\times$. Send this direction to $[0:u:v]$. This is well-defined because \begin{align*} [0:\lambda u:\lambda v]=[0:u:v] \end{align*} for every $\lambda\in K^\times$. It is injective because $[0:u:v]=[0:u':v']$ means $(0,u',v')=(\lambda 0,\lambda u,\lambda v)$ for some $\lambda\in K^\times$, hence $(u',v')=\lambda(u,v)$. It is surjective onto $H_\infty$ because every point of $H_\infty$ has the form $[0:x_1:x_2]$ with $(x_1,x_2)\neq(0,0)$. Therefore the line at infinity records exactly the directions in the affine plane. [/example] This compact way of adding directions is the reason projective geometry simplifies incidence. The distinction between ordinary intersections and limiting intersections becomes a coordinate artifact rather than a separate case. ## Linear Subspaces and Incidence ### Subspaces and Dimension Affine subspaces can be parallel, skew, or intersecting depending on their positions. Projective subspaces are more rigid because they are inherited directly from linear subspaces of the ambient vector space. [definition: Projective Subspace] Let $V$ be a vector space over a field $K$. A projective subspace of $\mathbb{P}(V)$ is a subset of the form \begin{align*} \mathbb{P}(W) = \{ L \in \mathbb{P}(V) : L \subset W \}, \end{align*} where $W \subset V$ is a nonzero linear subspace. [/definition] The word "line" in projective geometry refers to the dimension after quotienting by scalar multiplication. To measure projective subspaces without confusing $W$ with $\mathbb{P}(W)$, we need the following dimension convention. [definition: Projective Dimension] Let $W$ be a nonzero finite-dimensional vector space over a field $K$. The projective dimension of $\mathbb{P}(W)$ is \begin{align*} \dim \mathbb{P}(W) = \dim_K W - 1. \end{align*} [/definition] The most important large subspaces are the projective analogues of codimension-one linear equations. To formulate ordinary affine hyperplanes and hyperplanes at infinity in the same language, we need projective hyperplanes. [definition: Projective Hyperplane] Let $V$ be a finite-dimensional vector space over a field $K$ with $\dim_K V \geq 2$. A projective hyperplane in $\mathbb{P}(V)$ is a projective subspace $\mathbb{P}(W)$ where $W \subsetneq V$ is a nonzero linear subspace of codimension $1$. [/definition] This definition turns a single linear equation upstairs in $V$ into a codimension-one projective object downstairs. The next question is how to record all such equations without remembering an arbitrary scalar multiple of the same equation. ### Dual Projective Space A hyperplane in a vector space can be described as the kernel of a nonzero linear functional. Multiplying that functional by a nonzero scalar does not change its kernel, so hyperplanes are naturally parametrized by projective points in the dual vector space rather than by individual equations. [definition: Dual Projective Space] Let $V$ be a nonzero finite-dimensional vector space over a field $K$. The dual projective space of $\mathbb{P}(V)$ is \begin{align*} \mathbb{P}(V^*), \end{align*} whose points are one-dimensional subspaces of the dual vector space $V^*$. [/definition] This definition packages linear equations in the same way that ordinary projective space packages vectors: it keeps the equation up to nonzero scalar multiple. The next statement is the formal reason dual projective space is the correct parameter space for projective hyperplanes. [quotetheorem:9468] For $V = K^{n+1}$, a point $[a_0:\cdots:a_n] \in \mathbb{P}((K^{n+1})^*)$ corresponds to the hyperplane \begin{align*} a_0x_0 + \cdots + a_nx_n = 0 \end{align*} in $\mathbb{P}^n_K$. This is the beginning of projective duality: points and hyperplanes live in paired projective spaces, and incidence can be read from the vanishing of a linear form. Incidence questions reduce to the dimensions of sums and intersections upstairs in the vector space. The next formula is the linear-algebra bookkeeping rule translated into projective dimension. [quotetheorem:9469] In the projective plane the dimension formula answers the most basic incidence question: what should replace the affine distinction between intersecting and parallel lines? The answer is that distinct projective lines always have one common direction. [quotetheorem:9470] The theorem says that the point at infinity is not an optional add-on; it is the place where an affine exception is repaired. The most familiar repair occurs for parallel affine lines, whose missing intersection becomes visible once their equations are homogenized. [example: Parallel Affine Lines in Homogeneous Coordinates] In the affine chart $U_0 \subset \mathbb{P}^2_K$, write an affine point $(a,b)\in K^2$ as $[1:a:b]$, so the affine coordinate $y$ is $b=x_2/x_0$. The line $y=0$ is therefore described inside $U_0$ by \begin{align*} \frac{x_2}{x_0}=0. \end{align*} Since $x_0\neq 0$ on $U_0$, multiplying by $x_0$ gives the homogeneous equation \begin{align*} x_2=0. \end{align*} Similarly, the line $y=1$ is described inside $U_0$ by \begin{align*} \frac{x_2}{x_0}=1. \end{align*} Multiplying by $x_0$ gives \begin{align*} x_2=x_0. \end{align*} Equivalently, its homogeneous equation is \begin{align*} x_2-x_0=0. \end{align*} A projective intersection point must satisfy both equations. From the first equation, \begin{align*} x_2=0. \end{align*} Substituting this into the second equation gives \begin{align*} 0-x_0=0. \end{align*} Hence \begin{align*} x_0=0. \end{align*} Thus every common point has the form $[0:x_1:0]$. Homogeneous coordinates cannot all be zero, so $x_1\neq 0$, and rescaling by $x_1^{-1}$ gives \begin{align*} [0:x_1:0]=[0:1:0]. \end{align*} The intersection is therefore the single projective point $[0:1:0]$. Since its first coordinate is $0$, it lies on the line at infinity $H_\infty$; under the direction notation $[0:u:v]$, it is the direction $[0:1:0]$ of the horizontal vector $(1,0)$. [/example] Projective incidence is therefore not a decorative extension of affine geometry. It is a way to write theorems without exceptions caused by a chosen coordinate chart. ## Projective Maps Maps between projective spaces must respect the rule that scalar multiples represent the same point. Linear maps are the natural source, but only when they do not send the represented line to zero. [definition: Projectivization of an Injective Linear Map] Let $T: V \to W$ be an injective [linear map](/page/Linear%20Map) of vector spaces over a field $K$. The projectivization of $T$ is the map \begin{align*} \mathbb{P}(T): \mathbb{P}(V) &\to \mathbb{P}(W). \end{align*} \begin{align*} \mathbb{P}(T)([v]) &= [T(v)]. \end{align*} [/definition] The definition uses a representative $v$, so there is a real well-definedness issue: the same projective point may be written as $[v]$ or as $[\lambda v]$ for any $\lambda \in K^\times$. If applying $T$ to these two representatives produced different projective points, then $\mathbb{P}(T)$ would not be a function on projective space at all. The needed check is therefore not a technical aside but the bridge from linear algebra to projective geometry: a linear map should descend to projective points precisely because it carries scalar-equivalent representatives to scalar-equivalent images. This is the first place where the equivalence relation defining projective space has to interact correctly with an ordinary linear construction. Before using projectivized maps as geometric transformations, we need a formal guarantee that injectivity prevents nonzero representatives from being sent to zero and that linearity respects the scalar ambiguity in the notation $[v]$. [quotetheorem:9471] When $T$ is invertible, the projectivization is a projective change of coordinates. There is still redundancy in the linear description: multiplying the whole operator by a nonzero scalar changes every vector image by the same scalar, so it fixes every projective line. To name the genuine coordinate symmetries, we must package this redundancy into the definition of the group itself: the linear automorphisms are kept, but scalar automorphisms are treated as acting identically on projective space. The object we want is not the full group of invertible matrices or linear maps, since that remembers choices invisible to projective points. The following definition records exactly the transformations that projective geometry can detect: two linear automorphisms represent the same projective symmetry when they differ only by a nonzero scalar multiple of the identity. [definition: Projective Linear Group] Let $V$ be a nonzero finite-dimensional vector space over a field $K$. The projective linear group of $V$ is \begin{align*} \operatorname{PGL}(V) = \operatorname{GL}(V)/\{\lambda \operatorname{id}_V : \lambda \in K^\times\}. \end{align*} [/definition] The quotient by scalar maps is visible even in dimension one: different matrices can produce the same transformation of projective points. The familiar fractional linear formulas are the affine shadow of this projective action. [example: Fractional Linear Transformations] Let $A$ be the invertible $2$ by $2$ matrix whose first row is $(a,b)$ and whose second row is $(c,d)$. It sends a nonzero vector $(x_0,x_1)$ to \begin{align*} (ax_0+bx_1,cx_0+dx_1). \end{align*} The two new coordinates cannot both be zero: if $ax_0+bx_1=0$ and $cx_0+dx_1=0$, then $A(x_0,x_1)^\top=0$, so $(x_0,x_1)^\top=0$ because $A$ is invertible, contradicting that $(x_0,x_1)$ represents a projective point. Hence $A$ acts on $\mathbb{P}^1_K$ by \begin{align*} [x_0:x_1]\mapsto [ax_0+bx_1:cx_0+dx_1]. \end{align*} On the affine chart $x_0\neq 0$, put \begin{align*} t=\frac{x_1}{x_0}. \end{align*} Then $x_1=x_0t$, so the image of $[x_0:x_1]$ is \begin{align*} [ax_0+bx_1:cx_0+dx_1]=[ax_0+bx_0t:cx_0+dx_0t]. \end{align*} Factoring out $x_0$ gives \begin{align*} [ax_0+bx_0t:cx_0+dx_0t]=[x_0(a+bt):x_0(c+dt)]. \end{align*} Since $x_0\in K^\times$, homogeneous coordinates allow rescaling by $x_0^{-1}$, and therefore \begin{align*} [x_0(a+bt):x_0(c+dt)]=[a+bt:c+dt]. \end{align*} If $a+bt\neq 0$, the image remains in the affine chart with first coordinate nonzero, and its new affine coordinate is \begin{align*} \frac{c+dt}{a+bt}. \end{align*} Thus, on the part where the denominator is nonzero, the induced affine formula is \begin{align*} t\mapsto \frac{c+dt}{a+bt}. \end{align*} If $a+bt=0$, the image is $[0:c+dt]$. The second coordinate is nonzero, because $a+bt=0$ and $c+dt=0$ would imply that $A(1,t)^\top=0$, contradicting invertibility. Hence rescaling by $(c+dt)^{-1}$ gives \begin{align*} [0:c+dt]=[0:1]. \end{align*} So the denominator-zero case is exactly the point at infinity for this affine chart. The formula is the transpose of the common convention $t\mapsto (at+b)/(ct+d)$ because here the affine coordinate is $t=x_1/x_0$, and the first row of $A$ supplies the new $x_0$-coordinate. [/example] This is the first sign that projective geometry is the natural domain for rational formulas whose denominators sometimes vanish. Instead of stopping at a pole, the projective target records where the value has gone. [remark: Projective Linear Maps, Morphisms, and Rational Maps] Projectivizing an invertible linear map gives a projective linear automorphism, and projectivizing an injective linear map gives an embedding of one projective space into another. This is narrower than the later notion of a morphism given by homogeneous forms: projective linear maps are degree-one homogeneous formulas with no common zero, while general projective morphisms may use homogeneous polynomials of higher common degree. It is narrower still than the notion of a rational map, where homogeneous formulas may have a common zero locus and therefore define a map only away from that base locus. This page treats everywhere-defined projective maps first; rational maps and base loci belong to the algebraic-geometric continuation. [/remark] ## Topology and Smooth Structure Over $\mathbb{R}$ or $\mathbb{C}$, projective space is not only a set. Give $\mathbb{P}^n_K$ the [quotient topology](/page/Quotient%20Topology) induced by the projection $K^{n+1} \setminus \{0\} \to \mathbb{P}^n_K$, where $K^{n+1} \setminus \{0\}$ has its usual Euclidean topology when $K = \mathbb{R}$ and its usual complex Euclidean topology when $K = \mathbb{C}$. With this topology fixed, the standard affine charts become candidates for a smooth or complex manifold structure. [definition: Real Projective Space] The real projective $n$-space is \begin{align*} \mathbb{R}\mathbb{P}^n = \mathbb{P}^n_{\mathbb{R}}. \end{align*} [/definition] The complex version keeps the same projective construction but changes the scalars, so each projective point is now a complex line. This is the model needed for complex manifolds and projective algebraic geometry. [definition: Complex Projective Space] The complex projective $n$-space is \begin{align*} \mathbb{C}\mathbb{P}^n = \mathbb{P}^n_{\mathbb{C}}. \end{align*} [/definition] The affine charts should not merely cover the set; to construct a manifold, we need to specify which charts are being used and then test their overlaps. The following atlas makes the construction precise enough for that compatibility check. [definition: Standard Projective Atlas] For $K = \mathbb{R}$ or $K = \mathbb{C}$, the standard projective atlas on $\mathbb{P}^n_K$ is the collection of charts $(U_i,\phi_i)$ for $0 \leq i \leq n$, where $U_i$ is the standard affine chart and \begin{align*} \phi_i: U_i &\to K^n. \end{align*} \begin{align*} \phi_i([x_0:\cdots:x_n]) &= \left(\frac{x_0}{x_i},\ldots,\frac{x_{i-1}}{x_i},\frac{x_{i+1}}{x_i},\ldots,\frac{x_n}{x_i}\right). \end{align*} [/definition] A list of charts is not yet a smooth or complex structure. The missing question is whether changing from the $i$-th normalization to the $j$-th normalization gives transition functions of the correct kind on $U_i \cap U_j$. On an overlap, one coordinate used for normalization is divided by another, so the compatibility depends on those denominators staying nonzero and the resulting formulas being smooth over $\mathbb{R}$ or holomorphic over $\mathbb{C}$. [quotetheorem:9472] This compatibility result is the point at which the coordinate description becomes a genuine geometric structure rather than just a convenient notation for lines. Over $\mathbb{R}$ it gives $\mathbb{P}^n_{\mathbb{R}}$ a smooth manifold structure, and over $\mathbb{C}$ it gives $\mathbb{P}^n_{\mathbb{C}}$ a complex manifold structure. The restriction to $K=\mathbb{R}$ or $K=\mathbb{C}$ matters: the transition maps use division in $K$ and are judged by smoothness or holomorphicity, notions that are specific to these analytic fields. With that atlas in place, we can compare the same projective space with a different, more topological construction. The quotient viewpoint raises a concrete topological question over $\mathbb{R}$: which points of the unit sphere represent the same projective line? Every real line through the origin meets the sphere in two antipodal points, leading to the following model. [quotetheorem:9473] This model turns the abstract scaling relation into a concrete topological operation. In the first nonzero dimension, the quotient can be seen directly on the circle. [example: The Real Projective Line Is a Circle] View $S^1$ as the unit circle in $\mathbb{C}$, and define \begin{align*} p:S^1\to S^1,\qquad p(z)=z^2. \end{align*} If $z\in S^1$, then $-z\in S^1$ and \begin{align*} p(-z)=(-z)^2=z^2=p(z). \end{align*} Conversely, if $p(z)=p(w)$, then $z^2=w^2$, so \begin{align*} z^2-w^2=(z-w)(z+w)=0. \end{align*} Since $\mathbb{C}$ has no zero divisors, either $z-w=0$ or $z+w=0$, hence $w=z$ or $w=-z$. Thus the fibers of $p$ are exactly the antipodal pairs $\{z,-z\}$, so $p$ induces a bijection \begin{align*} S^1/(z\sim -z)\to S^1. \end{align*} The map $p$ is continuous, and the quotient map $S^1\to S^1/(z\sim -z)$ is continuous by definition of the quotient topology, so this induced bijection is the standard circle obtained by identifying antipodal points. In homogeneous coordinates, every point of $\mathbb{R}\mathbb{P}^1$ with first coordinate nonzero has the form $[1:t]$ because \begin{align*} [x_0:x_1]=[1:x_1/x_0] \end{align*} when $x_0\neq 0$. The remaining case has $x_0=0$, and then $x_1\neq 0$, so \begin{align*} [0:x_1]=[0:1]. \end{align*} Thus the affine coordinate $t$ gives all finite points $[1:t]$, while the single extra point $[0:1]$ is the point at infinity that closes the affine line into the circle described by the antipodal quotient. [/example] Complex projective space is central in [complex geometry](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature) because its coordinate changes are holomorphic on overlaps. The next compactness statement is one reason it replaces affine complex space in global algebraic and analytic geometry. [quotetheorem:9474] Compactness changes the behaviour of complex coordinates: a chart may look like ordinary affine complex space, but the missing values are organized into lower-dimensional projective pieces. For $n = 1$, this produces the classical compactification of the complex plane. [example: The Complex Projective Line] The space $\mathbb{C}\mathbb{P}^1$ has two standard affine charts. On the chart $U_0$, where $x_0\neq 0$, every point can be written uniquely as \begin{align*} [x_0:x_1]=[1:x_1/x_0]. \end{align*} Write \begin{align*} z=\frac{x_1}{x_0}. \end{align*} Thus the points of $U_0$ have the form $[1:z]$ with $z\in\mathbb{C}$. On the chart $U_1$, where $x_1\neq 0$, every point can be written uniquely as \begin{align*} [x_0:x_1]=[x_0/x_1:1]. \end{align*} Write \begin{align*} w=\frac{x_0}{x_1}. \end{align*} Thus the points of $U_1$ have the form $[w:1]$ with $w\in\mathbb{C}$. On the overlap $U_0\cap U_1$, both $x_0$ and $x_1$ are nonzero, so both $z=x_1/x_0$ and $w=x_0/x_1$ are defined and nonzero. Multiplying the two chart coordinates gives \begin{align*} zw=\frac{x_1}{x_0}\frac{x_0}{x_1}=1. \end{align*} Since $z\neq 0$, this is equivalent to \begin{align*} w=\frac{1}{z}. \end{align*} The chart $U_0$ supplies the ordinary complex coordinate $z\in\mathbb{C}$, while the only point not in $U_0$ has $x_0=0$ and hence is $[0:1]$; this single missing point is the point at infinity, so $\mathbb{C}\mathbb{P}^1$ is the complex plane compactified by one extra point, the Riemann sphere. [/example] The same chart construction is used in [differential geometry](/page/Cambridge%20III%20Differential%20Geometry), [Riemannian geometry](/page/Cambridge%20III%20Riemannian%20Geometry), and algebraic geometry, but with different structures placed on the same underlying projective set. ## Homogeneous Equations and Projective Varieties ### Equations Compatible with Scaling Affine polynomial equations do not behave well under scaling, because substituting $\lambda x$ changes different degree terms by different powers of $\lambda$. Homogeneous polynomials fix this problem. [definition: Homogeneous Polynomial] Let $K$ be a field. Here $K[x_0,\ldots,x_n]$ denotes the ring of polynomials in the variables $x_0,\ldots,x_n$ with coefficients in $K$. A polynomial $F \in K[x_0,\ldots,x_n]$ is homogeneous of degree $d$ if every monomial appearing in $F$ has total degree $d$. [/definition] Homogeneity makes the vanishing condition independent of the chosen representative. If $F$ has degree $d$, then \begin{align*} F(\lambda x_0,\ldots,\lambda x_n) = \lambda^d F(x_0,\ldots,x_n). \end{align*} Thus the equation $F = 0$ is a condition on projective points. To construct the projective analogue of affine algebraic sets, we need to impose collections of homogeneous equations at once. [definition: Projective Algebraic Set] Let $K$ be a field and let $S \subset K[x_0,\ldots,x_n]$ be a set whose members are homogeneous polynomials, with degrees allowed to depend on the polynomial. The projective algebraic set defined by $S$ is \begin{align*} V(S) = \{ [x_0:\cdots:x_n] \in \mathbb{P}^n_K : F(x_0,\ldots,x_n) = 0 \text{ for every } F \in S \}. \end{align*} [/definition] Because every polynomial in $S$ is homogeneous, replacing $(x_0,\ldots,x_n)$ by a nonzero scalar multiple multiplies each equation by a power of that scalar. Vanishing is therefore independent of the chosen representative of the projective point. A nonzero constant polynomial is homogeneous of degree $0$, and including one simply gives the empty algebraic set. [example: A Projective Conic] Over a field $K$ with characteristic not equal to $2$, the polynomial \begin{align*} F=x_0^2+x_1^2-x_2^2 \end{align*} is homogeneous of degree $2$, since each term has total degree $2$. If $\lambda\in K^\times$, then \begin{align*} F(\lambda x_0,\lambda x_1,\lambda x_2)=(\lambda x_0)^2+(\lambda x_1)^2-(\lambda x_2)^2 \end{align*} and expanding each square gives \begin{align*} F(\lambda x_0,\lambda x_1,\lambda x_2)=\lambda^2x_0^2+\lambda^2x_1^2-\lambda^2x_2^2. \end{align*} Factoring out $\lambda^2$ gives \begin{align*} F(\lambda x_0,\lambda x_1,\lambda x_2)=\lambda^2(x_0^2+x_1^2-x_2^2)=\lambda^2F(x_0,x_1,x_2). \end{align*} Thus $F=0$ is independent of the chosen nonzero homogeneous representative, so it defines a projective conic in $\mathbb{P}^2_K$. On the affine chart $x_2\neq 0$, set \begin{align*} u=\frac{x_0}{x_2} \end{align*} and \begin{align*} v=\frac{x_1}{x_2}. \end{align*} Starting from the projective equation, \begin{align*} x_0^2+x_1^2-x_2^2=0, \end{align*} divide by $x_2^2$, which is allowed because $x_2\neq 0$. This gives \begin{align*} \frac{x_0^2}{x_2^2}+\frac{x_1^2}{x_2^2}-\frac{x_2^2}{x_2^2}=0. \end{align*} Using $\frac{x_0^2}{x_2^2}=(x_0/x_2)^2=u^2$, $\frac{x_1^2}{x_2^2}=(x_1/x_2)^2=v^2$, and $\frac{x_2^2}{x_2^2}=1$, this becomes \begin{align*} u^2+v^2-1=0. \end{align*} Adding $1$ to both sides gives \begin{align*} u^2+v^2=1. \end{align*} The points at infinity for this affine chart are the points with $x_2=0$. Substituting $x_2=0$ into the homogeneous equation gives \begin{align*} x_0^2+x_1^2-0^2=0, \end{align*} hence \begin{align*} x_0^2+x_1^2=0. \end{align*} Therefore the projective conic consists of the affine curve $u^2+v^2=1$ in the chart $x_2\neq 0$, together with any projective points $[x_0:x_1:0]$ satisfying $x_0^2+x_1^2=0$; whether such points exist depends on the field $K$. [/example] Projective algebraic geometry studies these homogeneous zero sets and their maps. To make this into a geometry, the zero sets of homogeneous equations are taken as the closed sets of the appropriate projective topology. In this elementary form, the underlying set is only the set of $K$-rational projective points, and the equations are homogeneous polynomials with coefficients in $K$. Over a field that is not algebraically closed, this convention can be much coarser than the usual variety-theoretic or scheme-theoretic projective space, because it only sees the $K$-points. [definition: Projective Zariski Topology] Let $K$ be a field. The projective Zariski topology on the set $\mathbb{P}^n_K$ of $K$-rational projective points is the topology whose closed subsets are exactly the sets $V(S)$, where $S \subset K[x_0,\ldots,x_n]$ ranges over collections of homogeneous polynomials. [/definition] With this topology in place, projective algebraic sets are not merely subsets of projective space; they are the closed geometric objects of the theory. This construction belongs naturally with [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry), where projective space is the ambient object for projective varieties, divisors, and intersection theory. [remark: Field Conventions for the Projective Zariski Topology] The definition above is the set-theoretic topology on the $K$-rational projective points. This is the elementary projective algebraic set viewpoint: it is enough for homogeneous equations, affine charts, and first incidence computations. Modern algebraic geometry often changes the ambient object itself to the scheme \begin{align*} \operatorname{Proj} K[x_0,\ldots,x_n]. \end{align*} That scheme-theoretic projective space remembers prime homogeneous ideals, residue fields, nilpotent structure on closed subschemes, and base change in a way that a set of $K$-points cannot. The present page stays with the elementary model until those extra structures become necessary. [/remark] ### Coordinate Rings and Projective Maps A projective algebraic set is cut out by homogeneous equations, so its algebraic functions are recorded by a graded quotient rather than by an ordinary coordinate ring. This grading remembers the scaling action that projective space has divided out. [definition: Homogeneous Ideal of a Projective Algebraic Set] Let $X \subset \mathbb{P}^n_K$ be a projective algebraic set. The homogeneous ideal of $X$ is \begin{align*} I(X) = \{ F \in K[x_0,\ldots,x_n] : F_j(x_0,\ldots,x_n) = 0 \text{ for every } [x_0:\cdots:x_n] \in X \text{ and every homogeneous component } F_j \text{ of } F \}. \end{align*} [/definition] This makes $I(X)$ a homogeneous ideal: if $F = F_0 + \cdots + F_d$ is the decomposition into homogeneous components and $F \in I(X)$, then each $F_j$ also lies in $I(X)$. Equivalently, $I(X)$ is generated by the homogeneous forms that vanish on $X$, but the ideal itself contains all finite sums of such homogeneous vanishing pieces. The ideal cutting out a projective set is part of the algebraic information, not just a disposable choice of equations: different homogeneous ideals can have the same zero set. To attach a graded ring to the visible set of $K$-rational projective points, we need to quotient the [homogeneous polynomial](/page/Homogeneous%20Polynomial) ring by the vanishing ideal $I(X)$. [definition: Homogeneous Coordinate Ring] Let $X \subset \mathbb{P}^n_K$ be a projective algebraic set, interpreted as a set of $K$-rational projective points. The homogeneous coordinate ring of $X$ in the set-theoretic $K$-rational convention of this page is the graded ring \begin{align*} K[X] = K[x_0,\ldots,x_n]/I(X). \end{align*} [/definition] This definition gives a clean algebraic object attached to the visible set of projective points, but it also exposes a convention that matters once the ground field is not algebraically closed. The next warning separates this elementary set-theoretic ring from the richer coordinate rings used when a projective object is specified by equations, ideals, or schemes rather than only by its $K$-rational points. [remark: Coordinate Ring Convention over Non-Algebraically Closed Fields] If $X = V(S)$ is defined by homogeneous equations $S$, the ring $K[X]$ above is formed from the vanishing ideal of the resulting set of $K$-rational points. Over an [algebraically closed field](/page/Algebraically%20Closed%20Field), after passing to the radical homogeneous ideal, this agrees with the usual reduced projective coordinate ring attached to the zero set. Over a non-algebraically closed field, and especially over a finite field, many different homogeneous ideals can have the same $K$-rational points. The scheme-theoretic or variety-theoretic homogeneous coordinate ring is therefore usually built from the chosen saturated homogeneous ideal defining the projective object, not from the set of $K$-points alone. [/remark] The grading in this coordinate ring is what lets algebraic formulas survive the scaling ambiguity of homogeneous coordinates. Suppose we try to define a map into $\mathbb{P}^m_K$ by writing down $m+1$ polynomial coordinates. If the coordinates have different degrees, rescaling a representative of the source point changes the target coordinates by different powers of the scalar, so the projective point may change. To formulate the standard global presentation of a projective-space-valued map, we need homogeneous forms of one common degree that never vanish all at once. [definition: Projective-Space-Valued Map Given by Homogeneous Forms] Let $X \subset \mathbb{P}^n_K$ be a projective algebraic set. A projective-space-valued map given by homogeneous forms from $X$ to $\mathbb{P}^m_K$ is a map \begin{align*} f: X &\to \mathbb{P}^m_K. \end{align*} \begin{align*} f([x_0:\cdots:x_n]) &= [F_0(x_0,\ldots,x_n):\cdots:F_m(x_0,\ldots,x_n)], \end{align*} where $F_0,\ldots,F_m \in K[x_0,\ldots,x_n]$ are homogeneous polynomials of the same positive degree and do not vanish simultaneously at any point of $X$. [/definition] The notation in this formula means that a homogeneous representative $(x_0,\ldots,x_n)$ of the projective point is chosen before the polynomials are evaluated. If that representative is multiplied by $\lambda \in K^\times$, then each value $F_i(x_0,\ldots,x_n)$ is multiplied by the same scalar power of $\lambda$, so the resulting point of $\mathbb{P}^m_K$ is unchanged. This definition is deliberately phrased as a global homogeneous-form presentation of a map into projective space, the case most often used when first meeting the subject. It is not meant to replace the full definition of a morphism of projective varieties or schemes. For a map whose target is a projective algebraic set $Y \subset \mathbb{P}^m_K$, the same homogeneous tuple defines a morphism to $Y$ only when its image satisfies the homogeneous equations of $Y$. In the broader theory, morphisms are local on affine charts and are tracked algebraically by compatible maps of graded or sheaf-theoretic coordinate data; a single global tuple of forms is a useful presentation in common embedded cases, not the whole concept. ## Beyond and Connected Topics Projective space is a meeting point for several parts of geometry. In algebraic geometry, $\mathbb{P}^n_K$ is the basic complete ambient space, and homogeneous coordinate rings turn projective subvarieties into graded algebraic objects. Projective morphisms refine the idea of homogeneous formulas that are compatible with scaling, while dual projective space organizes hyperplanes as points of another projective space. The natural continuation is [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry), where projective varieties, morphisms, homogeneous coordinate rings, and duality are developed systematically. In differential geometry, real and complex projective spaces provide compact manifolds with rich symmetry. They are standard examples for quotient manifolds, covering spaces, characteristic classes, and metrics induced from linear algebra. This connects naturally to [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). In Riemannian geometry, projective spaces carry canonical metrics: the real projective space inherits a metric from the sphere quotient, and complex projective space carries the Fubini-Study metric. These examples are central test cases in [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). In complex geometry, $\mathbb{C}\mathbb{P}^n$ is the model compact complex manifold and the target of projective embeddings. It appears throughout [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature), especially when curvature and positivity are studied through holomorphic line bundles. ## References Androma, [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). Androma, [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). Hartshorne, *Algebraic Geometry* (1977). Harris, *Algebraic Geometry: A First Course* (1992). Griffiths and Harris, *Principles of Algebraic Geometry* (1978).