Definition:Affine Transformation

From ProofWiki
(Redirected from Definition:Affine Mapping)
Jump to navigation Jump to search

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


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: Make sure these are properly covered
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Affine_Transformation&oldid=738653"