Category:Affine Transformations

This category contains results about Affine Transformations.
Definitions specific to this category can be found in Definitions/Affine Transformations.

Let $\EE$ and $\FF$ be affine spaces with difference spaces $E$ and $F$ respectively.

Let $\LL: \EE \to \FF$ be a mapping.

Definition 1

$\LL$ is an affine transformation if and only if there exists a linear transformation $L: E \to F$ such that for every pair of points $p, q \in \EE$:

$\map \LL q = \map \LL p + \map L {\vec {p q} }$

Definition 2

$\LL$ is an affine transformation if and only if:

$\forall v_1, v_2 \in \EE: \map \LL {s v_1 + t v_2} = s \map \LL {v_1} + t \map \LL {v_2}$

for some $s, t \in \R$ such that $s + t = 1$.