Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.

More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

If $X$ is the point set of an affine space, then every affine transformation on $X$ can be represented as the composition of a linear transformation on $X$ and a translation of $X$ . Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear.

Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane.

A generalization of an affine transformation is an affine map^[1] (or affine homomorphism or affine mapping) between two (potentially different) affine spaces over the same field $k$ . Let $(X, V, k)$ and $(Z, W, k)$ be two affine spaces with $X$ and $Z$ the point sets and $V$ and $W$ the respective associated vector spaces over the field $k$ . A map $f : X \to Z$ is an affine map if there exists a linear map $m f : V \to W$ such that $m f (x - y) = f (x) - f (y)$ for all $x, y$ in $X$ .^[2]

Definition

Let $X$ be an affine space over a field $k$ , and $V$ be its associated vector space. An affine transformation is a bijection $f$ from $X$ onto itself that is an affine map; this means that a linear map $g$ from $V$ to $V$ is well defined by the equation $g(y-x)=f(y)-f(x);$ here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that $y-x=y'-x'$ implies that $f(y)-f(x)=f(y')-f(x').$

If the dimension of $X$ is at least two, a semiaffine transformation $f$ of $X$ is a bijection from $X$ onto itself satisfying:^[3]

For every $d$ -dimensional affine subspace $S$ of $X$ , then $f (S)$ is also a $d$ -dimensional affine subspace of $X$ .
If $S$ and $T$ are parallel affine subspaces of $X$ , then $f (S)$ and $f (T)$ are parallel.

These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that " $f$ preserves parallelism".

These conditions are not independent as the second follows from the first.^[4] Furthermore, if the field $k$ has at least three elements, the first condition can be simplified to: $f$ is a collineation, that is, it maps lines to lines.^[5]

Structure

By the definition of an affine space, $V$ acts on $X$ , so that, for every pair $(x,\mathbf {v} )$ in $X \times V$ there is associated a point $y$ in $X$ . We can denote this action by ${\vec {v}}(x)=y$ . Here we use the convention that ${\vec {v}}={\textbf {v}}$ are two interchangeable notations for an element of $V$ . By fixing a point $c$ in $X$ one can define a function $m c : X \to V$ by $m c (x) = cx \to$ . For any $c$ , this function is one-to-one, and so, has an inverse function $m c -1 : V \to X$ given by $m_{c}^{-1}({\textbf {v}})={\vec {v}}(c)$ . These functions can be used to turn $X$ into a vector space (with respect to the point $c$ ) by defining:^[6]

$x+y=m_{c}^{-1}\left(m_{c}(x)+m_{c}(y)\right),{\text{ for all }}x,y{\text{ in }}X,$ and
$rx=m_{c}^{-1}\left(rm_{c}(x)\right),{\text{ for all }}r{\text{ in }}k{\text{ and }}x{\text{ in }}X.$

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 $L (c, λ) : X \to X$ by

L(c,\lambda )(x)=m_{c}^{-1}\left(\lambda (m_{c}(x))\right)=c+\lambda ({\vec {cx}}).

Then $L (c, λ)$ is an affine transformation of $X$ which leaves the point $c$ fixed.^[7] 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 ${\mathbf {w}}={\overrightarrow {c\sigma (c)}}$ , 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^[8]

\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).^[9]^[10]^[11]

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 $A$ and the translation as the addition of a vector $\mathbf {b}$ , an affine map $f$ acting on a vector $\mathbf {x}$ can be represented as

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

Augmented matrix

Affine transformations on the 2D plane can be performed by linear transformations in three dimensions. Translation is done by shearing along over the z axis, and rotation is performed around the z axis.

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 $A$ is a matrix,

{\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}}=\left{\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}

is equivalent to the following

\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 $\left$

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