Image a vector space where two points \(P\) and \(P'\) exist, then there's a *unique translation of the plane* that maps \(P\) to \(P'\) which means that the space of *translations* in the plane can be identified with a set of vectors that exist in the plane, composition of translation correspond to addition of vectors e.g. \(\v{PP''} = \v{PP'} + \v{P'P''}\)

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

- The zero vector acts as an identity i.e. for all \(a \in E\), \(a + \mathbf{0} = a\)
- Addition of vectors correspond to translations i.e. for all \(a \in E\) and \(\mathbf{u,v} \in \v{E}\), \(x + (\mathbf{u} + \mathbf{v}) = (x + \mathbf{u}) + \mathbf{v}\)
- 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 with the triple \(\left \langle E, \v{E}, + \right \rangle\) where \(E\) is a set of points, \(\v{E}\) a vector space acting on \(E\) and 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) so that for any \(u \in V\)

\[ (x, 2 + x) + u = (x + u, 2 + x + u) \]

For example the point \((-2,0)\) added with 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 parallels 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 the 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 \(\mathbb{R}^2\) 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 of 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 and the restriction is that the scalar 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 \(\lambda_i, i \in I\) a family of scalars then 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 \(E\), 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 \(\lambda_i\)*, and 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.

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

- Bærentzen, J. A., Gravesen, J., Anton François, & Aanæs, H. (2012). Guide to computational geometry processing: foundations, algorithms, and methods. London: Springer.