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] This criterion turns projective equality into linear algebra. In $\mathbb{P}^1_k$, it becomes a determinant test, which is often the fastest way to compare two points. [example: Equality in $\mathbb{P}^1_k$] In $\mathbb{P}^1_{\mathbb{Q}}$, the points $[2:6]$ and $[1:3]$ are equal because the second representative is obtained from the first by a common nonzero scalar. Indeed, with $\lambda=2\in \mathbb{Q}^{\times}$, \begin{align*} 2(1,3)=(2\cdot 1,2\cdot 3)=(2,6). \end{align*} Thus $(2,6)\sim (1,3)$, so \begin{align*} [2:6]=[1:3]. \end{align*} The points $[2:6]$ and $[1:4]$ are not equal. If they were equal, there would be some $\lambda\in \mathbb{Q}^{\times}$ such that \begin{align*} (1,4)=\lambda(2,6). \end{align*} Comparing the first coordinates gives \begin{align*} 1=2\lambda. \end{align*} Since $2\neq 0$ in $\mathbb{Q}$, this forces \begin{align*} \lambda=\frac{1}{2}. \end{align*} Comparing the second coordinates with this value gives \begin{align*} 4=6\cdot \frac{1}{2}=3, \end{align*} which is false in $\mathbb{Q}$. Therefore no common nonzero scalar carries $(2,6)$ to $(1,4)$, and hence \begin{align*} [2:6]\neq [1:4]. \end{align*} Equivalently, the determinant test detects the same failure: \begin{align*} 2\cdot 4-6\cdot 1=8-6=2\neq 0. \end{align*} The nonzero determinant means the two ordered pairs are not scalar multiples, so they span different one-dimensional subspaces of $\mathbb{Q}^2$. [/example] The equality test tells us when two representatives describe the same point, but it does not choose a preferred representative. Local calculations often require such a choice. The safe way to choose is to first select a nonzero coordinate and divide every coordinate by it. [definition: Normalized Homogeneous Coordinates] Let $P=[a_0:\cdots:a_n] \in \mathbb{P}^n_k$ and suppose $a_i \neq 0$. The normalized homogeneous coordinates of $P$ relative to $a_i$ are \begin{align*} [a_0/a_i:\cdots:a_{i-1}/a_i:1:a_{i+1}/a_i:\cdots:a_n/a_i]. \end{align*} [/definition] Normalization is a local operation, not a global convention. A point with $a_i=0$ cannot be normalized in the $i$th chart, and that failure often carries geometric information. ## Homogeneous Equations Equations on projective space must respect rescaling. If a formula changes unpredictably when all coordinates are multiplied by the same scalar, it cannot define a projective condition. The algebraic remedy is to use polynomials whose monomials all have the same total degree. [definition: Homogeneous Polynomial] Let $k$ be a field, and let $k[X_0,\ldots,X_n]$ denote the [polynomial ring](/page/Polynomial%20Ring) over $k$ in the variables $X_0,\ldots,X_n$. A polynomial $F \in k[X_0,\ldots,X_n]$ is homogeneous of degree $d \geq 0$ if every monomial appearing in $F$ with nonzero coefficient has total degree $d$. [/definition] A [homogeneous polynomial](/page/Homogeneous%20Polynomial) still does not give a well-defined numerical value on a projective point, because changing representatives rescales the value. The essential question is narrower: does changing representatives preserve whether the value is zero? This is the invariant needed for equations to cut out subsets of projective space. [quotetheorem:9487] The scalar value of $F(a_0,\ldots,a_n)$ is not generally an invariant of the point. Its vanishing is invariant, and that is enough to define projective zero sets. [example: A Non-Homogeneous Equation Fails] In $\mathbb{P}^1_{\mathbb{R}}$, the tuples $(1,1)$ and $(2,2)$ define the same point because \begin{align*} (2,2)=(2\cdot 1,2\cdot 1)=2(1,1), \end{align*} and $2\in \mathbb{R}^{\times}$. Now let \begin{align*} F(X_0,X_1)=X_0+X_1^2. \end{align*} Evaluating at the first representative gives \begin{align*} F(1,1)=1+1^2=1+1=2. \end{align*} Evaluating at the second representative gives \begin{align*} F(2,2)=2+2^2=2+4=6. \end{align*} The value did not change by a fixed power of the scaling factor $2$: multiplying $2$ by $2$ gives $4$, while multiplying $2$ by $2^2$ gives $8$, and neither equals $6$. This happens because $X_0$ has degree $1$ while $X_1^2$ has degree $2$, so $F$ is not homogeneous. By contrast, let \begin{align*} G(X_0,X_1)=X_0X_1+X_1^2. \end{align*} Both terms have total degree $2$. At $(1,1)$, \begin{align*} G(1,1)=1\cdot 1+1^2=1+1=2. \end{align*} At $(2,2)$, \begin{align*} G(2,2)=2\cdot 2+2^2=4+4=8. \end{align*} Since $8=4\cdot 2=2^2G(1,1)$, rescaling the representative by $2$ rescales the value of $G$ by $2^2$. More generally, \begin{align*} G(\lambda X_0,\lambda X_1)=(\lambda X_0)(\lambda X_1)+(\lambda X_1)^2=\lambda^2X_0X_1+\lambda^2X_1^2=\lambda^2G(X_0,X_1). \end{align*} Thus $G$ may change value under rescaling, but its vanishing does not: $G(\lambda X_0,\lambda X_1)=0$ holds exactly when $G(X_0,X_1)=0$. [/example] The example shows why the zero set alone is not the whole story: the degree controls how an equation responds to rescaling. Projective calculations therefore need an algebraic home that keeps linear forms, quadrics, cubics, and higher-degree forms in separate layers while still allowing them to be multiplied. That bookkeeping is the role of the homogeneous coordinate ring. [definition: Homogeneous Coordinate Ring] Let $k$ be a field. The homogeneous coordinate ring of projective $n$-space over $k$ is the graded polynomial ring \begin{align*} k[X_0,\ldots,X_n]=\bigoplus_{d=0}^{\infty} k[X_0,\ldots,X_n]_d, \end{align*} where $k[X_0,\ldots,X_n]_d$ is the $k$-[vector space](/page/Vector%20Space) of homogeneous polynomials of degree $d$. [/definition] The ring is not merely a collection of equations. Its grading is part of the geometry, because projective constructions remember degree. The rest of the page uses only the coordinate-level consequences of this grading. Projective algebraic sets, hypersurfaces, maps, and base loci are included here as applications of homogeneous coordinates, not as a substitute for a full treatment of projective varieties. ## Projective Varieties and Affine Charts ### Projective Zero Sets A projective equation defines a set of classes of nonzero tuples. Since the all-zero tuple is not a point, and since all representatives of a point must give the same zero condition, the defining equations must be homogeneous. This turns a family of homogeneous polynomials into a geometric object. [definition: Projective Algebraic Set] Let $k$ be a field and let $S \subset k[X_0,\ldots,X_n]$ be a set of homogeneous polynomials. The point-set projective algebraic set defined by $S$ is \begin{align*} V(S)=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : F(a_0,\ldots,a_n)=0 \text{ for every } F \in S\}. \end{align*} [/definition] This definition records the $k$-rational points cut out by homogeneous equations, so it is a coordinate-level point-set definition. Over a non-[algebraically closed field](/page/Algebraically%20Closed%20Field), this point set should not be mistaken for the whole geometric object. For example, a homogeneous equation may have few or no $k$-rational solutions while acquiring many points after extending scalars to an [algebraic closure](/page/Algebraic%20Closure). The equations, and later the homogeneous ideal or scheme they define, preserve information that the set $V(S) \subset \mathbb{P}^n_k$ can hide. The next few notions belong to the coordinate toolkit rather than to a full theory of projective varieties. We introduce only the pieces needed to show how homogeneous coordinates translate global projective equations into affine calculations. When a projective algebraic set is cut out by many equations, its geometry can be hard to see from the notation $V(S)$. The first case where degree, dimension, and local affine equations can all be read directly is the one-equation case. We give it a separate name because projective lines, conics, plane curves, projective surfaces, and their higher-dimensional analogues are all hypersurfaces, and because later computations often begin by reducing a general question to this model situation. [definition: Projective Hypersurface] Let $k$ be a field and let $F \in k[X_0,\ldots,X_n]$ be a nonzero homogeneous polynomial. The projective hypersurface defined by $F$ is \begin{align*} V(F)=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : F(a_0,\ldots,a_n)=0\}. \end{align*} [/definition] A global homogeneous equation can be studied locally by entering a chart, but this requires a controlled translation from projective variables to affine variables. The required operation is to set one nonzero coordinate equal to $1$ and keep the remaining ratios. [definition: Dehomogenization] Let $k[X_0,\ldots,X_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$, and fix $0 \leq i \leq n$. The dehomogenization map on the chart $U_i$ is \begin{align*} (-)_{(i)}: k[X_0,\ldots,X_n]_d \to k[x_1,\ldots,x_n]. \end{align*} It is given by \begin{align*} F_{(i)}(x_1,\ldots,x_n)=F(x_1,\ldots,x_i,1,x_{i+1},\ldots,x_n), \end{align*} with the variable $x_j$ substituted for $X_{j-1}$ when $j \leq i$ and for $X_j$ when $j>i$. [/definition] Dehomogenization converts a projective equation into an affine equation. The next example shows how an affine parabola becomes a projective conic with an added point at infinity. [example: Homogenizing a Parabola] Let $f(x,y)=y-x^2 \in \mathbb{R}[x,y]$. To make both terms have total degree $2$, replace $y$ by $X_2/X_0$ and $x$ by $X_1/X_0$, then multiply by $X_0^2$: \begin{align*} X_0^2\left(\frac{X_2}{X_0}-\left(\frac{X_1}{X_0}\right)^2\right)=X_0^2\cdot \frac{X_2}{X_0}-X_0^2\cdot \frac{X_1^2}{X_0^2}=X_0X_2-X_1^2. \end{align*} Thus the degree-two homogenization is \begin{align*} F(X_0,X_1,X_2)=X_0X_2-X_1^2. \end{align*} On the chart $X_0\neq 0$, set \begin{align*} x=\frac{X_1}{X_0} \end{align*} and \begin{align*} y=\frac{X_2}{X_0}. \end{align*} The projective equation $F=0$ becomes \begin{align*} X_0X_2-X_1^2=0. \end{align*} Dividing both sides by $X_0^2$, which is nonzero on this chart, gives \begin{align*} \frac{X_0X_2-X_1^2}{X_0^2}=0. \end{align*} The left-hand side is \begin{align*} \frac{X_0X_2}{X_0^2}-\frac{X_1^2}{X_0^2}=\frac{X_2}{X_0}-\left(\frac{X_1}{X_0}\right)^2=y-x^2. \end{align*} So the chart equation is exactly \begin{align*} y-x^2=0. \end{align*} At infinity, points satisfy $X_0=0$. Substituting $X_0=0$ into the homogeneous equation gives \begin{align*} 0\cdot X_2-X_1^2=0. \end{align*} Since $0\cdot X_2=0$, this is \begin{align*} -X_1^2=0. \end{align*} Over $\mathbb{R}$, this forces \begin{align*} X_1=0. \end{align*} Thus any point at infinity on the projective parabola has the form $[0:0:X_2]$. Homogeneous coordinates cannot all vanish, so $X_2\neq 0$, and rescaling by $X_2^{-1}$ gives \begin{align*} [0:0:X_2]=[0:0:1]. \end{align*} The affine parabola therefore acquires exactly one point at infinity, namely $[0:0:1]$. [/example] Examples suggest that setting one homogeneous coordinate equal to $1$ recovers an affine equation, but this must be independent of the chosen representative. The issue is to compare the projective zero condition with the ordinary affine zero condition on a chart where the chosen coordinate is nonzero. Here $\mathbb{A}^n_k$ denotes affine $n$-space over $k$, whose points are ordinary coordinate tuples $(a_1,\ldots,a_n)$. An affine variety is a common zero set of polynomials in affine coordinates. A homogeneous ideal is an ideal generated, or equivalently graded, by homogeneous polynomials; this condition is what makes its projective zero set well-defined under rescaling of homogeneous coordinates. With this language, the chart question becomes precise: when $X_i\neq 0$, does the projective variety cut out by homogeneous equations become an affine variety in the coordinates obtained by setting $X_i=1$? [quotetheorem:2138] The hypotheses matter because arbitrary nonhomogeneous equations do not define subsets of projective space invariant under rescaling. The theorem is also local in nature: it does not replace the whole projective variety by one affine variety, but identifies each standard chart with an affine piece. This is the mechanism used below whenever a projective computation is reduced to ordinary polynomial equations after choosing a nonzero homogeneous coordinate. ### Incidence in the Projective Plane Projective coordinates simplify incidence. Parallel affine lines become intersecting projective lines, and the distinction between ordinary intersections and intersections at infinity disappears. [definition: Projective Hyperplane] Let $k$ be a field. A projective hyperplane in $\mathbb{P}^n_k$ is a subset of the form $V(L)$ for a nonzero homogeneous linear polynomial $L \in k[X_0,\ldots,X_n]_1$, written \begin{align*} L(X_0,\ldots,X_n)=c_0X_0+\cdots+c_nX_n. \end{align*} The subset is \begin{align*} V(L)=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : L(a_0,\ldots,a_n)=0\}. \end{align*} [/definition] The definition turns a line in the projective plane into the kernel of a linear functional. This setup makes it possible to prove the central incidence fact: two distinct projective lines have one common point. [quotetheorem:9470] This theorem is the formal version of the opening example. Homogeneous coordinates replace the affine exception of parallelism with one uniform linear-algebraic statement. ## Maps in Homogeneous Coordinates A map into projective space cannot be given by arbitrary coordinate functions. The output list must transform by a common scalar when the input representative is rescaled, and it must not become the all-zero tuple. [definition: Projective Map Given by Homogeneous Forms] Let $k$ be a field, let $X \subset \mathbb{P}^n_k$ be a projective algebraic set, and let $F_0,\ldots,F_m \in k[X_0,\ldots,X_n]$ be homogeneous polynomials of the same degree. If, for every $P=[a_0:\cdots:a_n] \in X$, the tuple \begin{align*} (F_0(a_0,\ldots,a_n),\ldots,F_m(a_0,\ldots,a_n)) \end{align*} is nonzero, then the associated projective map is \begin{align*} \varphi: X \to \mathbb{P}^m_k. \end{align*} It is defined by \begin{align*} \varphi([a_0:\cdots:a_n])=[F_0(a_0,\ldots,a_n):\cdots:F_m(a_0,\ldots,a_n)]. \end{align*} [/definition] The same-degree hypothesis is the map version of homogeneous equations. Without it, rescaling the input would rescale different output coordinates by different powers. [example: The Veronese Map] The degree-two Veronese formula sends a point of $\mathbb{P}^1_k$ to $\mathbb{P}^2_k$ by \begin{align*} [S:T]\mapsto [S^2:ST:T^2]. \end{align*} The three coordinate forms $S^2$, $ST$, and $T^2$ are homogeneous of degree $2$, since each monomial has total degree $2$. To check that the formula is independent of the representative, let $\lambda\in k^\times$ and replace $(S,T)$ by $(\lambda S,\lambda T)$. The three output coordinates become \begin{align*} (\lambda S)^2=\lambda^2S^2. \end{align*} \begin{align*} (\lambda S)(\lambda T)=\lambda^2ST. \end{align*} \begin{align*} (\lambda T)^2=\lambda^2T^2. \end{align*} Therefore \begin{align*} [(\lambda S)^2:(\lambda S)(\lambda T):(\lambda T)^2]=[\lambda^2S^2:\lambda^2ST:\lambda^2T^2]. \end{align*} Since $\lambda\neq 0$, also $\lambda^2\neq 0$, so the last point is the same homogeneous coordinate class as \begin{align*} [S^2:ST:T^2]. \end{align*} The output tuple is never $(0,0,0)$: if $S^2=0$ and $T^2=0$, then $S=0$ and $T=0$, contradicting that $(S,T)$ represents a point of $\mathbb{P}^1_k$. Thus the formula gives a well-defined projective map, and the common degree $2$ is exactly what makes every output coordinate rescale by the same factor $\lambda^2$. [/example] The preceding definition evaluates the coordinate forms on a representative of the input point; equal degrees make the resulting target point independent of that representative. A proposed projective formula may still fail where all output coordinates vanish. Those points need a name because they are the source of indeterminacy in rational maps. [definition: Base Locus] Let $F_0,\ldots,F_m \in k[X_0,\ldots,X_n]$ be homogeneous polynomials of the same degree. The base locus of the list is \begin{align*} \operatorname{Bs}(F_0,\ldots,F_m)=\{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : F_0(a_0,\ldots,a_n)=\cdots=F_m(a_0,\ldots,a_n)=0\}. \end{align*} [/definition] A base locus is not a minor technicality. It records exactly where the formula tries to produce the forbidden projective tuple $[0:\cdots:0]$. When the base locus is empty, the formula gives an everywhere-defined projective map; when the base locus is nonempty, the same list of forms is better regarded as a rational map, defined only away from its indeterminacy points. [example: A Formula with a Base Point] The formula \begin{align*} [X:Y:Z]\mapsto [X:Y] \end{align*} uses the two homogeneous linear forms $X$ and $Y$. If the representative $(X,Y,Z)$ is rescaled by $\lambda\in k^\times$, then the proposed output changes to \begin{align*} [\lambda X:\lambda Y]. \end{align*} Since $\lambda\neq 0$, this is the same point of $\mathbb{P}^1_k$ as $[X:Y]$ whenever $(X,Y)\neq (0,0)$. The only obstruction is the forbidden target tuple $(0,0)$. The base locus is the set of points of $\mathbb{P}^2_k$ for which both coordinate forms vanish: \begin{align*} \operatorname{Bs}(X,Y)=\{[X:Y:Z]\in \mathbb{P}^2_k:X=0\text{ and }Y=0\}. \end{align*} Thus every base point has the form $[0:0:Z]$. Homogeneous coordinates require the representative $(0,0,Z)$ to be nonzero, so $Z\neq 0$. Rescaling by $Z^{-1}\in k^\times$ gives \begin{align*} [0:0:Z]=[0\cdot Z^{-1}:0\cdot Z^{-1}:Z\cdot Z^{-1}]=[0:0:1]. \end{align*} Conversely, at $[0:0:1]$ the two target coordinates are \begin{align*} X=0 \end{align*} and \begin{align*} Y=0. \end{align*} So the formula would produce $[0:0]$, which is not a point of $\mathbb{P}^1_k$. Therefore \begin{align*} \operatorname{Bs}(X,Y)=\{[0:0:1]\}. \end{align*} The projection formula is an everywhere-defined projective map on $\mathbb{P}^2_k\setminus\{[0:0:1]\}$, and the missing point is exactly where all proposed target coordinates vanish. [/example] ## Beyond and Connected Topics Homogeneous coordinates are the entry point to [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry), where projective varieties, homogeneous ideals, and graded rings are developed systematically. The next conceptual step is to replace a single homogeneous equation by a homogeneous ideal, then study how the grading records projective information that ordinary affine coordinate rings forget. They also connect to [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry) through projective manifolds, projective transformations, and coordinate charts on embedded varieties. Here the same notation $[a_0:\cdots:a_n]$ becomes a coordinate system on smooth manifolds such as complex projective space, while chart changes become geometric transition maps rather than just algebraic normalizations. In [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry), related projective ideas appear in spaces of directions and geometric structures where scale is not the primary datum. For complex geometry, homogeneous coordinates describe complex projective space $\mathbb{P}^n_{\mathbb{C}}$, the basic compact complex manifold. This leads naturally to [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature), where projective embeddings, line bundles, and curvature interact. Computationally, homogeneous coordinates support intersection calculations, resultants, rational parametrizations, and the systematic study of points at infinity. This is why they remain useful far beyond synthetic projective geometry: they turn geometric completion into polynomial algebra. ## References Androma, [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). Hartshorne, *Algebraic Geometry* (1977). Shafarevich, *Basic Algebraic Geometry 1* (1994). Harris, *Algebraic Geometry: A First Course* (1992).