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Use the cross-ratio convention compatible with the permutation identities in this paper. Represent a projective point $P_i\in {\mathbb{CP}}^1$ by a nonzero vector $p_i\in {\mathbb{C}}^2$, and write $[p_i,p_j]=\det (p_i,p_j)$. DefineThe denominator is nonzero because the four projective points are distinct. Rescaling any one representative multiplies numerator and denominator by the same nonzero scalar, so the definition is independent of the chosen homogeneous coordinates. In finite affine coordinates it readswith the point at infinity of the complex projective line handled by the homogeneous formula.

A projective transformation is represented by an invertible matrix $A\in {\mathrm{GL}}_2(\mathbb{C})$. Since $[A{p}_i,A{p}_j]=\det (A)[p_i,p_j]$, the two factors of $\det (A)$ in the numerator cancel the two in the denominator. Therefore every projective transformation preserves this cross-ratio. This also shows that the formula descends from matrices to the projective linear group, whose representatives differ by a nonzero common scalar.

The real projective plane is the set of one-dimensional linear subspaces of ${\mathbb{R}}^3$. Equivalently,A projective point is written in homogeneous coordinates $[x:y:z]$. A projective line is the set of projective points whose representatives belong to one fixed two-dimensional linear subspace of ${\mathbb{R}}^3$. Such a linear subspace is the kernel of a linear map of a nonzero linear functional, so its equation has the form $\alpha x+\beta y+\gamma z=0$ with $(\alpha ,\beta ,\gamma )\ne 0$.

Two distinct projective points have linearly independent representatives $p,q$. Their linear span has dimension two and hence defines a projective line containing both. Conversely, any two-dimensional linear subspace containing both representatives contains their entire linear span and therefore equals it. Thus exactly one projective line passes through two distinct projective points.

Choose nonzero vectors $p_i\in {\mathbb{F}}^2$ representing $P_i$, and write $[u,v]=\det (u,v)$. Use the cross-ratio conventionDistinctness makes the denominator nonzero. Every rescaling of a representative occurs once in numerator and denominator, so the value depends only on the projective points. Swapping both pairs simply exchanges the numerator's factors and the denominator's factors:Under an invertible linear map $A$, $[Au,Av]=\det (A)[u,v]$. Two factors of $\det (A)$ occur above and below the fraction, proving projective transformation invariance.

Finally choose homogeneous coordinates making $P_0=[1:0]$, $P_1=[0:1]$, $P_2=[1:1]$. This normalization exists by the allowed theorem about a projective frame. A point distinct from $P_0$ is uniquely $[t:1]$, and the displayed cross-ratio is $t$. Thus equality of the two cross-ratios forces equal coordinates andThis also shows that for fixed first three distinct points the fourth point is determined by its cross-ratio.

The four points form a projective frame: no three are collinear. By the allowed uniqueness theorem and cross-ratio invariance, useIntersecting the indicated projective lines givesA point of $Q_0{Q}_2$ has coordinates $[a:b:a+b]$. Intersecting with $P_0{P}_1$, whose equation is $z=0$, gives $Q_3=[1:-1:0]$. On that line use its first two homogeneous coordinates. The cross-ratio from (a)(ii) isThe characteristic of a field being zero ensures that $Q_3\ne Q_1$. The construction is the diagonal-point property of a complete quadrangle, producing harmonic conjugate points.

With column-vector homogeneous coordinates, the translation matrix isIt maps $(x,y,1{)}^T$ to $(x+a,y+b,1{)}^T$. The final coordinate incorporates the constant displacement into matrix multiplication.

The line passes through $p=(0,1{)}^T$ and has the same direction as in part (i). Translate by $-p$, reflect by $H$, then translate back. In homogeneous coordinates the affine reflection isAct on column $(x,y,1{)}^T$, reading the product from right to left. Every point with $y=4x/3+1$ is fixed. The translation column is $p-Hp$, as for any reflection in an affine hyperplane.

Conversion of a transformed column of homogeneous coordinates $(X,Y,Z,W)$ with $W\ne 0$ to $(X/W,Y/W,Z/W)$. This is the nonlinear step in an otherwise matrix-based perspective projection pipeline.

In the affine coordinate $z\mapsto [z:1]$, the complex projective line consists of the complex plane together with the single extra projective point $[1:0]$, called its point at infinity. It is the only point whose second homogeneous coordinate vanishes. This coordinate choice distinguishes the point; it is not intrinsically special under projective transformations. The map $[u:v]\mapsto [v:u]$ exchanges zero and infinity. The resulting one-point compactification is the Riemann sphere.

For a homogeneous polynomial $F$ of degree $d$, its first polar with respect to $[a:b:c]$ is the equation $a{F}_x+b{F}_y+c{F}_z=0$, of degree $d-1$ when this polynomial is nonzero. Every singular point of $V(F)$ lies on every such polar. Polars turn the Bézout's theorem intersection count into bounds on singularities and tangencies. The equation is covariant under a linear change of homogeneous coordinates, by the chain rule.

For a local ring $A$, every map from $\mathrm{Spec}A$ into projective space lands in one standard chart: the inverse image of a chart containing the closed point is the whole spectrum. This gives homogeneous coordinates with at least one unit entry, unique up to a common unit. For a nonlocal ring, coordinates may instead merely generate the unit ideal, and general maps can involve a nontrivial invertible sheaf quotient.

In the real projective plane, represent a projective point by a nonzero vector and a projective line by its nonzero coefficient covector, identified with a vector in chosen coordinates. The join of distinct projective points represented by $p,q$ has coefficient vector $p\times q$, using the cross product. It is nonzero because the representatives are linearly independent, and its dot product with each representative vanishes. Dually, two distinct projective lines with coefficients $\ell ,m$ meet in the projective point represented by $\ell \times m$. Scaling an input scales the output and leaves the projective object unchanged. This makes joins, intersections, concurrency, and collinearity exact calculations with homogeneous coordinates.

For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]

For a nonzero real normal vector $n$, orthogonal reflection in $n^{T}x=d$ has the displayed affine map. It is $x\mapsto Hx+t$ with $H=I-2n{n}^T/(n^{T}n)$ and $t=2dn/(n^{T}n)$. In homogeneous coordinates its matrix isEvery point on the hyperplane is fixed, and the signed normal displacement changes sign. Choosing any point $p$ on the hyperplane gives $t=p-Hp$, so translating by $-p$, applying the reflection matrix $H$ and translating back gives the same transformation.

For an integer $d\ge 1$, the degree-$d$ Veronese map sends a projective point to all degree-$d$ monomials in its homogeneous coordinates:Multiplying the original coordinates by $\lambda \ne 0$ multiplies every output by ${\lambda }^d$, so the map is well-defined on projective points. It is a closed immersion, given by the complete linear system of $\mathcal{O}(d)$.

For $n=1,d=2$, the formula is $[s:t]\mapsto [s^2:st:t^2]$. Its image in the projective plane satisfies $XZ=Y^2$, a smooth plane conic. On the chart $X\ne 0$, the ratio $Y/X$ recovers $t/s$; the chart $Z\ne 0$ recovers $s/t$. The degree-$d$ construction similarly makes homogeneous degree-$d$ expressions into linear coordinates in a larger projective space.