Definition:Affine Transformation

Definition

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 $s + t = 1$.

Also known as

An affine transformation is also known as an affine mapping.

Some sources refer to it as an affinity.

Also see

• Equivalence of Definitions of Affine Transformation

• Definition:Tangent Map of Affine Transformation

• Results about affine transformations can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): affine transformation (or affinity)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): affine transformation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): affine transformation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): affine transformation