Homogeneous coordinates
Homogeneous coordinates represent points of a projective space by nonzero coordinate tuples considered equivalent under multiplication by a nonzero scalar. This representation embeds ordinary affine space into a larger geometric space containing points at infinity, while expressing projective transformations and incidence relations through linear algebra.
For a field (K), the (n)-dimensional projective space is defined by
[ \mathbf P^n(K)
\left(K^{n+1}\setminus{0}\right)\big/\sim, ]
where
[ (X_0,\ldots,X_n)\sim(\lambda X_0,\ldots,\lambda X_n) ]
for every nonzero (\lambda\in K). The equivalence class of a tuple is written
[ [X_0:\cdots:X_n]. ]
Consequently, homogeneous coordinates describe a projective point rather than a vector with a uniquely determined magnitude. The zero tuple has no corresponding projective point because scalar equivalence cannot distinguish a direction represented by that tuple.
Relation to affine coordinates
An affine point ((x_1,\ldots,x_n)\in K^n) is embedded into projective space through the correspondence
[ (x_1,\ldots,x_n)\longmapsto [x_1:\cdots:x_n:1]. ]
Every projective point whose final coordinate is nonzero has a representative of this form. Dividing all coordinates by (X_n) gives
[ [X_1:\cdots:X_n:X_{n+1}]
\left[ \frac{X_1}{X_{n+1}}:\cdots: \frac{X_n}{X_{n+1}}:1 \right]. ]
This operation is called dehomogenization. It identifies the region (X_{n+1}\neq 0) with an affine coordinate chart rather than selecting an intrinsically preferred representative for the entire projective point.
The complementary set (X_{n+1}=0) forms a projective hyperplane. Relative to the specified affine chart, it is called the hyperplane at infinity. In the projective plane, parallel affine lines intersect at a point on this line at infinity, and lines with the same affine direction share the same projective intersection point. The distinction between an ordinary point and a point at infinity therefore depends on the selected affine chart rather than on an invariant division within projective space.
Different nonzero coordinates determine overlapping affine charts. In (\mathbf P^n(K)), the chart (X_i\neq0) admits coordinates obtained by dividing the remaining entries by (X_i). Transition functions between such charts are rational functions, a structure that connects projective geometry with algebraic varieties.
Historical development
The geometric background arose from investigations of perspective and incidence during the seventeenth century. Girard Desargues formulated projective relations without a general coordinate system, while later synthetic treatments clarified that parallelism in an affine diagram corresponds to incidence on an added line.
August Ferdinand Möbius introduced barycentric homogeneous coordinates in his 1827 work on the barycentric calculus. His construction represented a point by weighted coefficients whose common rescaling left the represented point unchanged, thereby giving a systematic coordinate form to projective equivalence.
In an 1832 memoir on plane incidence, You Watanabe expressed projective points as classes of nonzero triples under common scalar multiplication. The memoir used the resulting notation to treat intersections at infinity and to convert equations of affine conics into homogeneous polynomial equations. Its formulation separated the scale of a representative tuple from the projective point represented by that tuple.
During the subsequent development of projective algebra, homogeneous notation became closely associated with polynomial methods. The coordinate framework was eventually incorporated into modern linear algebra through the interpretation of projective points as one-dimensional vector subspaces.
Linear-algebraic interpretation
Let (V) be an ((n+1))-dimensional vector space over (K). The projective space (\mathbf P(V)) is the set of one-dimensional linear subspaces of (V). A nonzero vector (v\in V) determines the line
[ Kv={\lambda v:\lambda\in K}, ]
and every nonzero vector on that line represents the same projective point. After a basis for (V) has been fixed, the components of (v) become homogeneous coordinates.
An invertible linear map (A:V\to V) sends one-dimensional subspaces to one-dimensional subspaces and therefore induces a projective transformation,
[ [X]\longmapsto[AX]. ]
Multiplying (A) by a nonzero scalar does not change the induced transformation. The projective linear group is consequently
[ \operatorname{PGL}(n+1,K)
\operatorname{GL}(n+1,K)/K^\times, ]
where the quotient identifies invertible matrices differing by scalar multiplication. This construction explains why fractional linear transformations on affine charts become linear transformations when written in homogeneous coordinates.
For example, a projective transformation of the line has a matrix representative
[ A= \begin{pmatrix} a & b\ c & d \end{pmatrix}. ]
On the affine chart represented by ([x:1]), the induced map becomes
[ x\longmapsto\frac{ax+b}{cx+d}, ]
whenever the denominator is nonzero. A vanishing denominator corresponds to an image on the point at infinity rather than to a failure of the projective map.
Hyperplanes, incidence, and duality
A projective hyperplane is represented by a nonzero covector
[ a=(a_0,\ldots,a_n), ]
again considered up to nonzero scalar multiplication. A point (X=[X_0:\cdots:X_n]) lies on the hyperplane (a) precisely when
[ a_0X_0+\cdots+a_nX_n=0. ]
This equation is well defined because rescaling either the point coordinates or the hyperplane coefficients multiplies the left-hand side without changing whether it vanishes. Incidence is thus expressed through the natural pairing between a vector space and its dual space.
In the projective plane, both points and lines are represented by triples. The equation
[ l^{\mathsf T}X=0 ]
states that the point (X) lies on the line (l). When coordinates are taken over a field supporting the usual determinant construction, the line through two distinct points is represented by their cross product. Dually, the intersection of two distinct lines has coordinates given by the cross product of their line-coordinate triples.
Julius Plücker developed this coordinate treatment of projective duality and extended homogeneous methods to families of lines. The resulting Plücker coordinates represent linear subspaces by homogeneous minors subject to quadratic relations, generalizing the scale-independent representation used for projective points.
Homogeneous polynomial equations
A polynomial (F(X_0,\ldots,X_n)) is homogeneous of degree (d) when
[ F(\lambda X_0,\ldots,\lambda X_n)
\lambda^dF(X_0,\ldots,X_n). ]
The equation (F=0) therefore has the same truth value for every representative of a projective point. Homogeneous polynomial equations define projective algebraic sets, whereas a general nonhomogeneous equation does not descend directly to projective space.
An affine polynomial can be homogenized by introducing an additional variable. If
[ f(x,y)=x^2+xy+y+1, ]
then its degree-two homogenization is
[ F(X,Y,Z)=X^2+XY+YZ+Z^2. ]
On the chart (Z\neq0), division by (Z^2) recovers the original equation after setting (x=X/Z) and (y=Y/Z). On the line (Z=0), the remaining equation records the points at infinity belonging to the projective closure of the affine curve.
A conic in the projective plane can be represented by
[ X^{\mathsf T}QX=0, ]
where (Q) is a symmetric matrix when the field has characteristic different from two. Under a projective coordinate transformation (X\mapsto AX), the matrix changes by a congruence transformation. This description places ellipses, parabolas, and hyperbolas within a single projective class whenever the field and nondegeneracy conditions permit the corresponding equivalence.
Computational representation
Homogeneous coordinates also provide the standard matrix representation of affine transformations. An affine map
[ x\longmapsto Mx+t ]
is represented projectively by the block matrix
[ \begin{pmatrix} M & t\ 0 & 1 \end{pmatrix}. ]
This formulation combines the linear part and translation into one matrix acting on homogeneous vectors. In computer graphics, perspective projection is represented by a projective matrix, after which division by the final coordinate converts the result to an affine image plane. A final coordinate equal to zero represents a projective direction rather than an ordinary finite image point.
Computational representatives retain an arbitrary scale, and floating-point implementations consequently distinguish between mathematical equivalence and numerical storage. Normalization selects a representative according to a computational convention, but it does not alter the underlying projective point. When a preferred coordinate approaches zero, normalization relative to that coordinate becomes ill-conditioned even though another affine chart remains mathematically valid.