Imagine a vector space where two points $P$ and $P’$ exist. Then, there’s a unique translation of the plane that maps $P$ to $P’$. This means that the space of translations in the plane can be identified with a set of vectors that exist in the plane. The composition of translations corresponds to the addition of vectors, e.g., $\v{PP’’} = \v{PP’} + \v{P’P’’}$.

affine space

affine space

An affine space is a space where translation is defined. Formally, an affine space is a set $E$ (of points) that admits a free transitive action of a vector space $\v{E}$ (of translations) whose action results in an element of the set $E$. That is, there’s a map $E \times \v{E} \rightarrow E: (a,\mathbf{v}) \mapsto a + \mathbf{v}$ such that

  1. The zero vector acts as an identity, i.e., for all $a \in E$, $a + \mathbf{0} = a$.
  2. Addition of vectors corresponds to translations, i.e., for all $a \in E$ and $\mathbf{u,v} \in \v{E}$, $x + (\mathbf{u} + \mathbf{v}) = (x + \mathbf{u}) + \mathbf{v}$.
  3. For any $a,b \in E$, there’s a unique free vector $\mathbf{u} \in \v{E}$ such that $a + \mathbf{u} = b.

The affine space is commonly represented by the triple $\left \langle E, \v{E}, + \right \rangle$, where $E$ is a set of points, $\v{E}$ is a vector space acting on $E$, and $+$ is an action $E \times \v{E} \rightarrow E$.

Consider a subset $L$ of $\mathbb{A}^2$ consisting of points satisfying

$$ -x + y - 2 = 0 $$

Where any point has the form $(x, f(x)) = (x, 2 + x)$, the line can be made into an affine space by defining $+: L \times V \rightarrow L$ (note that $V$ is a vector space) such that for any $u \in V$

$$ (x, 2 + x) + u = (x + u, 2 + x + u) $$

For example, the point $(-2,0)$ added to the vector $u = [1,1]$ results in the point $(-1, 1)$, which belongs to the set $L$. Note that for the example above, the vector space $V$ has only vectors parallel to $u = [1,1]$.

Chasles’s Identity

Given any three points $a,b,c \in E$, we know that $c = a + \mathbf{ac}$, $b = a + \mathbf{ab}$, and $c = b + \mathbf{bc}$ by Axiom 3. Therefore,

$$ c = b + \mathbf{bc} = (a + \mathbf{ab}) + \mathbf{bc} = a + (\mathbf{ab} + \mathbf{bc}) $$

And thus

$$ \mathbf{ab} + \mathbf{bc} = \mathbf{ac} $$

Which is known as Chasles’s identity

Affine combinations

Consider $\mathbb{R}^2$ as an affine space with its origin at $(0,0)$ and basis vectors $\mathbf{b_1} = [1, 0]$ and $\mathbf{b_2} = [0,1]$. Given any two points $a,b \in \mathbb{R}^2$ with coordinates $a = (a_1,a_2)$ and $b = (b_1,b_2)$, we can define the affine combination $\lambda a + \mu b$ as the point with coordinates

$$ (\lambda a_1 + \mu b_1, \lambda a_2 + \mu b_2) $$

Let $\lambda = 1, \mu = 1$, $a = (-1,1)$, and $b = (2, 2)$. Then $a + b = (1, 1)$.

If we change the coordinate system to have an origin at $(1,1)$ with the same basis vectors, then the coordinates of the given points are $a=(-2,-2)$ and $b=(1,1)$. The linear combination is then $a + b = (-1,-1)$, which is the same as the point $(0,0)$ of the first coordinate system. Therefore, $a+b$ corresponds to two different points depending on the coordinate system used.

A restriction is needed for affine combinations to make sense: the scalars must add up to 1.

Lemma: Given an affine space $E, \v{E}, +$, let $a_i, i \in I$ be a family of points in $E$ and let $\lambda_i, i \in I$ be a family of scalars. Then, for any two points $a,b \in E$, the following properties hold:

$$ \begin{equation} \label{lemma-1} a + \sum_{i \in I} \lambda_i \mathbf{aa_i} = b + \sum_{i \in I} \lambda_i \mathbf{ba_i} \quad \text{if $\sum_{i \in I} \lambda_i = 1$} \end{equation} $$
> > and >
$$ \begin{equation} \label{lemma-2} \sum_{i \in I} \lambda_i \mathbf{aa_i} = \sum_{i \in I} \lambda_i \mathbf{ba_i} \quad \text{if $\sum_{i \in I} \lambda_i = 0$} \end{equation} $$

To prove \eqref{lemma-1}, we apply Chasles’s identity:

$$ \begin{align*} a + \sum_{i \in I} \lambda_i \mathbf{aa_i} &= a + \sum_{i \in I} \lambda_i (\mathbf{ab} + \mathbf{ba_i}) \\ &= a + (\sum_{i \in I} \lambda_i) \mathbf{ab} + \sum_{i \in I} \lambda_i \mathbf{ba_i} \\ &= a + \mathbf{ab} + \sum_{i \in I} \lambda_i \mathbf{ba_i} \quad \text{since $\sum_{i \in I} \lambda_i = 1$} \\ &= b + \sum_{i \in I} \lambda_i \mathbf{ba_i} \quad \text{since $b = a + \mathbf{ab}$} \\ \end{align*} $$

For \eqref{lemma-2}, we also have:

$$ \begin{align*} \sum_{i \in I} \lambda_i \mathbf{aa_i} &= \sum_{i \in I} \lambda_i (\mathbf{ab} + \mathbf{ba_i}) \\ &= (\sum_{i \in I} \lambda_i) \mathbf{ab} + \sum_{i \in I} \lambda_i \mathbf{ba_i} \\ &= \sum_{i \in I} \lambda_i \mathbf{ba_i} \quad \text{since $\sum_{i \in I} \lambda_i = 0$} \end{align*} $$

Formally, for any family of points $a_i, i \in I$ in $E$ and for any family $\lambda_i, i \in I$ of scalars such that $\sum_{i \in I} \lambda_i = 1$, the point

$$ \begin{equation} \label{affine-combination} x = a + \sum_{i \in I} \lambda_i \mathbf{aa_i} \end{equation} $$

is independent of $a \in E$ and is called the barycenter or affine combination of the points $a_i$ with weights $\lambda_i$. It is denoted as

$$ \sum_{i \in I} \lambda_i a_i $$

Affine maps

An affine map between two affine spaces $X$ and $Y$ is a map $f: X \rightarrow Y$ that preserves affine combinations, i.e.,

$$ f \left (\sum_{i \in I} \lambda_i a_i \right ) = \sum_{i \in I} \lambda_i f(a_i) $$