Affine transformation

Structure

By the definition of an affine space, V acts on X, so that, for every pair ${\displaystyle (x,\mathbf{v})}$ in X × V there is associated a point y in X. We can denote this action by ${\displaystyle \overrightarrow{v}(x)=y}$. Here we use the convention that ${\displaystyle \overrightarrow{v}=\mathbf{\text{v}}}$ are two interchangeable notations for an element of V. By fixing a point c in X one can define a function m_c : X → V by m_c(x) = cx→. For any c, this function is one-to-one, and so, has an inverse function m_c⁻¹ : V → X given by ${\displaystyle m_c^{-1}(\mathbf{\text{v}})=\overrightarrow{v}(c)}$. These functions can be used to turn X into a vector space (with respect to the point c) by defining:

• ${\displaystyle x+y=m_c^{-1}\left(m_c(x)+m_c(y)\right),\text{ for all }x,y\text{ in }X,}$ and
• ${\displaystyle rx=m_c^{-1}\left(r{m}_c(x)\right),\text{ for all }r\text{ in }k\text{ and }x\text{ in }X.}$

This vector space has origin c and formally needs to be distinguished from the affine space X, but common practice is to denote it by the same symbol and mention that it is a vector space after an origin has been specified. This identification permits points to be viewed as vectors and vice versa.

For any linear transformation λ of V, we can define the function L(c, λ) : X → X by $${\displaystyle L(c,\lambda )(x)=m_c^{-1}\left(\lambda (m_c(x))\right)=c+\lambda (\overrightarrow{cx}).}$$

Then L(c, λ) is an affine transformation of X which leaves the point c fixed. It is a linear transformation of X, viewed as a vector space with origin c.

Let σ be any affine transformation of X. Pick a point c in X and consider the translation of X by the vector ${\displaystyle \mathbf{w}=\overrightarrow{c\sigma (c)}}$, denoted by T_w. Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of c, there exists a unique linear transformation λ of V such that $${\displaystyle \sigma (x)=T_{\mathbf{w}}\left(L(c,\lambda )(x)\right).}$$ That is, an arbitrary affine transformation of X is the composition of a linear transformation of X (viewed as a vector space) and a translation of X.

This representation of affine transformations is often taken as the definition of an affine transformation (with the choice of origin being implicit).

Representation

As shown above, an affine map is the composition of two functions: a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, and vector addition to represent translations. Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by an invertible matrix ${\displaystyle A}$ and the translation as the addition of a vector ${\displaystyle \mathbf{b}}$, an affine map ${\displaystyle f}$ acting on a vector ${\displaystyle \mathbf{x}}$ can be represented as

$${\displaystyle \mathbf{y}=f(\mathbf{x})=A\mathbf{x}+\mathbf{b}.}$$

Augmented matrix

Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors be augmented with a "1" at the end, and all matrices be augmented with an extra row of zeros at the bottom, an extra column—the translation vector—to the right, and a "1" in the lower right corner. If ${\displaystyle A}$ is a matrix,

$${\displaystyle \left[\begin{array}{c}\mathbf{y}\\ 1\end{array}\right]=\left[\begin{array}{cccc}& A& & \mathbf{b}\\ 0& \cdots & 0& 1\end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ 1\end{array}\right]}$$

is equivalent to the following

$${\displaystyle \mathbf{y}=A\mathbf{x}+\mathbf{b}.}$$

The above-mentioned augmented matrix is called an affine transformation matrix. In the general case, when the last row vector is not restricted to be ${\displaystyle \left[\begin{array}{cccc}0& \cdots & 0& 1\end{array}\right]}$, the matrix becomes a projective transformation matrix (as it can also be used to perform projective transformations).

This representation exhibits the set of all invertible affine transformations as the semidirect product of ${\displaystyle K^n}$ and ${\displaystyle \mathrm{GL}(n,K)}$. This is a group under the operation of composition of functions, called the affine group.

Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate "1" to every vector, one essentially considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the additional coordinate is 1. Thus the origin of the original space can be found at ${\displaystyle (0,0,\dots ,0,1)}$. A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean, the higher dimensional space is a real projective space.

The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices. This property is used extensively in computer graphics, computer vision and robotics.

Example augmented matrix

Suppose you have three points that define a non-degenerate triangle in a plane, or four points that define a non-degenerate tetrahedron in 3-dimensional space, or generally n + 1 points x₁, ..., x_n+1 that define a non-degenerate simplex in n-dimensional space. Suppose you have corresponding destination points y₁, ..., y_n+1, where these new points can lie in a space with any number of dimensions. (Furthermore, the new points need not form a non-degenerate simplex, nor even be distinct from each other.) The unique augmented matrix M that achieves the affine transformation $${\displaystyle \left[\begin{array}{c}{\mathbf{y}}_i\\ 1\end{array}\right]=M\left[\begin{array}{c}{\mathbf{x}}_i\\ 1\end{array}\right]}$$ for every i is $${\displaystyle M=\left[\begin{array}{ccc}{\mathbf{y}}_1& \cdots & {\mathbf{y}}_{n+1}\\ 1& \cdots & 1\end{array}\right]{\left[\begin{array}{ccc}{\mathbf{x}}_1& \cdots & {\mathbf{x}}_{n+1}\\ 1& \cdots & 1\end{array}\right]}^{-1},}$$ using matrix inversion.

Properties

Properties preserved

An affine transformation preserves:

1. collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation.
2. parallelism: two or more lines which are parallel, continue to be parallel after the transformation.
3. convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set.
4. ratios of lengths of parallel line segments: for distinct parallel segments defined by points ${\displaystyle p_1,p_2}$ and ${\displaystyle p_3,p_4}$, respectively, the ratio of ${\displaystyle \overrightarrow{p_1{p}_2}}$ to ${\displaystyle \overrightarrow{p_3{p}_4}}$ is the same as that of ${\displaystyle \overrightarrow{f(p_1)f(p_2)}}$ to ${\displaystyle \overrightarrow{f(p_3)f(p_4)}}$.
5. barycenters of weighted collections of points.

Groups

As an affine transformation is invertible, the square matrix ${\displaystyle A}$ appearing in its matrix representation is invertible. The matrix representation of the inverse transformation is thus $${\displaystyle \left[\begin{array}{cccc}& A^{-1}& & -A^{-1}\overrightarrow{b}\\ 0& \dots & 0& 1\end{array}\right].}$$

The invertible affine transformations (of an affine space onto itself) form the affine group, which has the general linear group of degree ${\displaystyle n}$ as subgroup and is itself a subgroup of the general linear group of degree ${\displaystyle n+1}$.

The similarity transformations form the subgroup where ${\displaystyle A}$ is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of ${\displaystyle A}$ is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.

Each of these groups has a subgroup of orientation-preserving or positive affine transformations: those where the determinant of ${\displaystyle A}$ is positive. In the last case this is in 3D the group of rigid transformations (proper rotations and pure translations).

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.

Affine maps

An affine map ${\displaystyle f:\mathcal{A}\to \mathcal{B}}$ between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, ${\displaystyle f}$ determines a linear transformation ${\displaystyle \phi }$ such that, for any pair of points ${\displaystyle P,Q\in \mathcal{A}}$:

$${\displaystyle \overrightarrow{f(P)f(Q)}=\phi (\overrightarrow{PQ})}$$ or $${\displaystyle f(Q)-f(P)=\phi (Q-P).}$$

We can interpret this definition in a few other ways, as follows.

If an origin ${\displaystyle O\in \mathcal{A}}$ is chosen, and ${\displaystyle B}$ denotes its image ${\displaystyle f(O)\in \mathcal{B}}$, then this means that for any vector ${\displaystyle \overrightarrow{x}}$:

$${\displaystyle f:(O+\overrightarrow{x})\mapsto (B+\phi (\overrightarrow{x})).}$$

If an origin ${\displaystyle O^{\prime }\in \mathcal{B}}$ is also chosen, this can be decomposed as an affine transformation ${\displaystyle g:\mathcal{A}\to \mathcal{B}}$ that sends ${\displaystyle O\mapsto O^{\prime }}$, namely

$${\displaystyle g:(O+\overrightarrow{x})\mapsto (O^{\prime }+\phi (\overrightarrow{x})),}$$ followed by the translation by a vector ${\displaystyle \overrightarrow{b}=\overrightarrow{O^{\prime }B}}$.

The conclusion is that, intuitively, ${\displaystyle f}$ consists of a translation and a linear map.

Alternative definition

Given two affine spaces ${\displaystyle \mathcal{A}}$ and ${\displaystyle \mathcal{B}}$, over the same field, a function ${\displaystyle f:\mathcal{A}\to \mathcal{B}}$ is an affine map if and only if for every family ${\displaystyle \{(P_i,{\lambda }_i){\}}_{i\in I}}$ of weighted points in ${\displaystyle \mathcal{A}}$ such that $${\displaystyle \sum _{i\in I}{\lambda }_i=1,}$$ we have $${\displaystyle f\left(\sum _{i\in I}{\lambda }_i{P}_i\right)=\sum _{i\in I}{\lambda }_{i}f(P_i).}$$ In other words, ${\displaystyle f}$ preserves barycenters.

Example

Let ${\displaystyle \mathcal{A}}$ be the three-dimensional Euclidean space, ${\displaystyle \mathcal{B}\subseteq \mathcal{A}}$ a plane, and both be equipped with a Cartesian coordinate system. If ${\displaystyle f:\mathcal{A}\to \mathcal{B}}$ is a parallel projection or, more generally, is generated by an axonometry, then ${\displaystyle f}$ is affine and surjective. Hence it can be represented by $${\displaystyle \left[\begin{array}{c}x\\ y\\ z\end{array}\right]⟼\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]+b}$$ with a matrix ${\displaystyle A\in {\mathbb{R}}^{2\times 3}}$ of rank 2 and a column vector ${\displaystyle b\in {\mathbb{R}}^2.}$ In case ${\displaystyle b=(0,0{)}^{\mathsf{T}},}$ this is treated in more detail in the section Coordinate calculation at Axonometry.

History

The word "affine" as a mathematical term is defined in connection with tangents to curves in Euler's 1748 Introductio in analysin infinitorum. Felix Klein attributes the term "affine transformation" to Möbius and Gauss.

Image transformation

In their applications to digital image processing, the affine transformations are analogous to printing on a sheet of rubber and stretching the sheet's edges parallel to the plane. This transform relocates pixels requiring intensity interpolation to approximate the value of moved pixels, bicubic interpolation is the standard for image transformations in image processing applications. Affine transformations scale, rotate, translate, mirror and shear images as shown in the following examples:

The affine transforms are applicable to the registration process where two or more images are aligned (registered). An example of image registration is the generation of panoramic images that are the product of multiple images stitched together.

Affine warping

The affine transform preserves parallel lines. However, the stretching and shearing transformations warp shapes, as the following example shows:

This is an example of image warping. However, the affine transformations do not facilitate projection onto a curved surface or radial distortions.

In the plane

Every affine transformations in a Euclidean plane is the composition of a translation and an affine transformation that fixes a point; the latter may be

• a homothety,
• rotations around the fixed point,
• a scaling, with possibly negative scaling factors, in two directions (not necessarily perpendicular); this includes reflections,
• a shear mapping
• a squeeze mapping.

Given two non-degenerate triangles ABC and A′B′C′ in a Euclidean plane, there is a unique affine transformation T that maps A to A′, B to B′ and C to C′. Each of ABC and A′B′C′ defines an affine coordinate system and a barycentric coordinate system. Given a point P, the point T(P) is the point that has the same coordinates on the second system as the coordinates of P on the first system.

Affine transformations do not respect lengths or angles; they multiply areas by the constant factor

area of A′B′C′ / area of ABC.

A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by comparing the orientations of the triangles.

Examples

Over the real numbers

The functions ${\displaystyle f:\mathbb{R}\to \mathbb{R},f(x)=mx+c}$ with ${\displaystyle m}$ and ${\displaystyle c}$ in ${\displaystyle \mathbb{R}}$ and ${\displaystyle m\ne 0}$, are precisely the affine transformations of the real line.

In plane geometry

In ${\displaystyle {\mathbb{R}}^2}$, the transformation shown at left is accomplished using the map given by:

$${\displaystyle \left[\begin{array}{c}x\\ y\end{array}\right]\mapsto \left[\begin{array}{cc}0& 1\\ 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}-100\\ -100\end{array}\right]}$$

Transforming the three corner points of the original triangle (in red) gives three new points which form the new triangle (in blue). This transformation skews and translates the original triangle.

In fact, all triangles are related to one another by affine transformations. This is also true for all parallelograms, but not for all quadrilaterals.