Definition:Affine Transformation
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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
- 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