# Definition:Affine Frame

From ProofWiki

## Definition

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

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 $V$.

Equivalently, 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:

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