Definition:Affine Frame

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Let $\EE$ be an affine space with difference space $V$.

Definition 1

An affine frame in $\EE$ is an ordered tuple $\tuple {p_0, e_1, \ldots, e_n}$, where:

$p_0$ is an element of $\EE$ called the origin
$\tuple {e_1, \ldots, e_n}$ is an ordered basis for $V$.

Definition 2

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

$\tuple {q_0, \ldots, q_n} = \tuple {p_0, p_0 + e_1, \ldots, p_0 + e_n}$

The frame $\tuple {p_0, e_1, \ldots, e_n}$ is then recovered by:

$\tuple {q_0, q_1 - q_0, \ldots, q_n - q_0}$