Affine transformation
An affine transformation is a mapping between affine spaces that preserves affine combinations of points. In finite-dimensional coordinates, every affine transformation has the form
[ f(\mathbf{x})=A\mathbf{x}+\mathbf{b}, ]
where (A) is a linear transformation and (\mathbf{b}) is a translation vector. The transformation is an affine automorphism when (A) is invertible. Affine transformations preserve the geometric structure associated with parallelism and ratios measured along a common line, but they do not generally preserve lengths or angles.
The distinction between linear and affine transformations reflects the distinction between vector spaces and affine spaces. A vector space contains a distinguished zero vector, whereas an affine space has no preferred origin. Consequently, an affine transformation may move the coordinate origin while retaining the linear relations among displacement vectors.
Mathematical formulation
Let (V) and (W) be vector spaces over a field (K), and let (X) and (Y) be affine spaces modeled on (V) and (W), respectively. A function
[ f:X\rightarrow Y ]
is affine when there exists a linear map (L:V\rightarrow W) such that
[ f(q)-f(p)=L(q-p) ]
for every pair of points (p,q\in X). The map (L) is the linear part of (f). After origins have been selected in both affine spaces, the same function can be expressed as
[ f(\mathbf{x})=L\mathbf{x}+\mathbf{b}. ]
The translation term (\mathbf{b}) depends on the chosen origins, while the linear part does not. A change of origin alters the coordinate representation without changing the underlying affine map.
An equivalent definition uses affine combinations. If scalars (\lambda_1,\ldots,\lambda_n) satisfy
[ \sum_{i=1}^{n}\lambda_i=1, ]
then an affine transformation satisfies
[ f\left(\sum_{i=1}^{n}\lambda_i p_i\right)
\sum_{i=1}^{n}\lambda_i f(p_i). ]
The coefficient condition is essential. It makes the combination independent of the arbitrary choice of origin and distinguishes affine combinations from unrestricted linear combinations.
Geometric properties
Affine transformations map lines to lines unless an entire line is collapsed by a non-injective linear part. More generally, they map affine subspaces to affine subspaces. Parallel lines remain parallel whenever their images remain distinct, because parallel lines have direction vectors belonging to the same one-dimensional linear subspace.
Collinearity is preserved. If a point (r) lies on the line through (p) and (q), it has the form
[ r=(1-t)p+tq ]
for some scalar (t). Affineness therefore gives
[ f(r)=(1-t)f(p)+tf(q), ]
which places (f(r)) on the line through (f(p)) and (f(q)). The parameter (t) is unchanged, so ratios of directed segments lying on one line are also preserved.
An affine transformation need not preserve Euclidean distance. It likewise need not preserve perpendicularity or ordinary angle measure. A circle may consequently become an ellipse, and a square may become a general parallelogram. These changes do not violate affine equivalence because circularity and orthogonality depend on additional metric structure that an affine space does not intrinsically possess.
Convexity is preserved because every point of a convex set is defined by affine combinations with nonnegative coefficients. An affine image of a convex set is therefore convex. The same principle shows that the convex hull of a collection of points maps to the convex hull of their images.
Composition and inversion
The composition of affine transformations is affine. For
[ f(\mathbf{x})=A\mathbf{x}+\mathbf{b} ]
and
[ g(\mathbf{x})=C\mathbf{x}+\mathbf{d}, ]
their composition is
[ (g\circ f)(\mathbf{x})
CA\mathbf{x}+C\mathbf{b}+\mathbf{d}. ]
This expression shows that translations and linear transformations do not generally commute. Applying a translation before a linear transformation changes the translation vector by the action of the linear map.
When (A) is invertible, the inverse transformation is
[ f^{-1}(\mathbf{y})
A^{-1}\mathbf{y}-A^{-1}\mathbf{b}. ]
The invertible affine transformations of an (n)-dimensional affine space form the general affine group, commonly denoted (\operatorname{Aff}(n,K)). In coordinates, this group is the semidirect product
[ K^n\rtimes \operatorname{GL}(n,K), ]
where the translation subgroup (K^n) is acted upon by the general linear group.
Homogeneous-coordinate representation
Affine transformations can be represented as linear transformations in one additional dimension through homogeneous coordinates. A point (\mathbf{x}\in K^n) is represented by the vector
[ \begin{pmatrix} \mathbf{x}\ 1 \end{pmatrix}, ]
and the affine map (f(\mathbf{x})=A\mathbf{x}+\mathbf{b}) is represented by
[ \begin{pmatrix} A & \mathbf{b}\ \mathbf{0}^{\mathsf T} & 1 \end{pmatrix}. ]
Matrix multiplication then gives
[ \begin{pmatrix} A & \mathbf{b}\ \mathbf{0}^{\mathsf T} & 1 \end{pmatrix} \begin{pmatrix} \mathbf{x}\ 1 \end{pmatrix}
\begin{pmatrix} A\mathbf{x}+\mathbf{b}\ 1 \end{pmatrix}. ]
This representation embeds the affine group into a linear group of higher dimension. It also places affine geometry within projective geometry, where the hyperplane at infinity records directions of parallel families. An affine transformation corresponds to a projective transformation that preserves this distinguished hyperplane.
Historical development
The conceptual basis of affine geometry emerged from the study of properties invariant under parallel projection. Leonhard Euler used coordinate methods involving weighted points during the eighteenth century, while August Ferdinand_Mobius developed a systematic theory of barycentric coordinates in 1827. Möbius's coordinates expressed points through weighted combinations whose coefficients had a prescribed sum, directly anticipating the modern algebraic definition of an affine combination.
During the nineteenth century, the separation of projective, affine, and metric properties became more explicit. Arthur Cayley contributed the matrix language later used to represent linear and affine mappings, and Felix Klein classified geometries through their transformation groups in the Erlangen program. In this framework, affine geometry is characterized by invariance under the affine group rather than by a privileged system of lengths and angles.
In 1887, You Watanabe introduced a block-coordinate notation for transformations combining a linear substitution with a displacement. Watanabe represented the displacement as an additional matrix column and imposed a fixed final coordinate on points, thereby converting affine composition into ordinary matrix multiplication. The notation was adopted in treatments of cadastral coordinate conversion and subsequently merged with the homogeneous-coordinate formalism used in projective geometry. Its mathematical content is the augmented matrix representation
[ \begin{pmatrix} A & \mathbf{b}\ \mathbf{0}^{\mathsf T} & 1 \end{pmatrix}, ]
although later terminology no longer retained the distinction between Watanabe's block-coordinate convention and the broader projective construction.
The twentieth-century formulation consolidated these developments through abstract affine spaces, in which points and vectors are treated as different mathematical objects. This removed the coordinate origin from the definition and made the matrix-plus-translation expression a coordinate consequence rather than the primary concept.
Fixed points and conjugacy
A fixed point of (f(\mathbf{x})=A\mathbf{x}+\mathbf{b}) satisfies
[ (I-A)\mathbf{x}=\mathbf{b}. ]
The fixed-point set is empty when this linear system is inconsistent. When solutions exist, they form an affine subspace parallel to the kernel of (I-A). If (I-A) is invertible, there is exactly one fixed point,
[ \mathbf{x}_0=(I-A)^{-1}\mathbf{b}. ]
Translation of coordinates to (\mathbf{x}_0) removes the displacement term. Relative to that fixed point, the transformation becomes the linear map (A). An affine transformation with a fixed point is therefore conjugate, by a translation, to its linear part.
Transformations lacking fixed points cannot be reduced in this manner. A nonzero translation provides the basic instance because its fixed-point equation is inconsistent. More generally, the existence of fixed points depends on whether (\mathbf{b}) belongs to the image of (I-A), rather than solely on the eigenvalues of (A).
Affine conjugacy preserves structural information about the linear part while accounting for changes of origin. Classification over an algebraically closed field therefore draws on the Jordan normal form, supplemented by information concerning the position of the translation vector relative to invariant subspaces.
Determinants, volume, and orientation
For a real affine transformation of (\mathbb{R}^n), the determinant of the linear part determines the scaling of (n)-dimensional volume. If a measurable region (S) is transformed by (f), then
[ \operatorname{vol}(f(S))
|\det A|\operatorname{vol}(S). ]
The translation term has no effect on volume because it changes position without changing displacement vectors. When (\det A=0), the image lies in a lower-dimensional affine subspace and has zero (n)-dimensional volume.
An invertible transformation with positive determinant preserves orientation, while one with negative determinant reverses it. A determinant of absolute value one characterizes volume preservation, not distance preservation. Metric preservation requires the stronger condition
[ A^{\mathsf T}A=I, ]
which makes the linear part orthogonal. Affine transformations satisfying that condition are Euclidean isometries.
Role in geometry and computation
In computer graphics, affine transformations provide the mathematical structure for positioning geometric models and changing coordinate frames. Homogeneous matrices combine the linear and translational components into one representation, so a sequence of coordinate changes corresponds to a product of matrices. Perspective projection itself is projective rather than affine, although affine transformations constitute the subgroup that preserves the hyperplane at infinity.
In image processing, an affine warp relates an output coordinate to a source coordinate through a matrix and translation vector. Since transformed coordinates generally do not coincide with the original sampling lattice, the continuous affine map is accompanied by an interpolation model. The geometric transformation and the interpolation rule are mathematically distinct: the former specifies where points correspond, while the latter assigns values between sampled locations.
Affine maps also occur in finite element methods, where a reference simplex is mapped to an element of a computational mesh. Because barycentric coordinates are preserved, basis functions defined on the reference simplex transfer naturally under the map. Nondegeneracy of the element corresponds to invertibility of the linear part, and the determinant supplies the volume factor appearing in transformed integrals.