Definition:Affine Frame

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Let $\mathcal E$ be an affine space with difference space $V$.

Definition 1

An affine frame in $\mathcal E$ is an ordered tuple $\left({p_0, e_1, \ldots, e_n}\right)$, where:

$p_0$ is an element of $\mathcal E$ called the origin
$\left({e_1, \ldots, e_n}\right)$ is an ordered basis for $V$.

Definition 2

An affine frame may be given by the set of $n+1$ points:

$\left({q_0,\ldots, q_n}\right) = \left({p_0, p_0 + e_1, \ldots, p_0 + e_n}\right)$

The frame $\left({p_0, e_1, \ldots, e_n}\right)$ is then recovered by:

$\left({q_0, q_1 - q_0 ,\ldots, q_n - q_0}\right)$