Composition of Affine Transformations is Affine Transformation
Jump to navigation
Jump to search
Theorem
Let $\EE$, $\FF$ and $\GG$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively.
Let $\LL: \EE \to \FF$ and $\MM: \FF \to \GG$ be affine transformations.
Let $L$ and $M$ be the tangent maps of $\LL$ and $\MM$ respectively.
Then the composition $\MM \circ \LL: \EE \to \FF$ is an affine transformation with tangent map $M \circ L$.
Proof
Let $\NN = \MM \circ \LL : \EE \to \GG$ be the composition.
We want to show that for any $p, q \in \EE$
- $\map \GG Q = \map \GG p + \map {M \circ L} {\vec {p q} }$
We find that:
\(\ds \map \GG q\) | \(=\) | \(\ds \map {\MM \circ \LL} q\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \MM {\map \LL p} + \map L {\vec{p q} }\) | $\LL$ is an Affine Transformation |
Now let:
- $p' = \map \LL p$
and:
- $q' = \map \LL p + \map L {\vec {p q} }$
so:
- $\vec {p' q'} = \map L {\vec {p q} }$
Then:
\(\ds \map \GG q\) | \(=\) | \(\ds \map \MM {q'}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \MM {p'} + \map M {\vec {p' q'} }\) | $\MM$ is an Affine Transformation | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \MM {\map \LL p} + \map M {\map L {\vec {p q} } }\) | Definitions of $p'$ and $q'$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\MM \circ \LL} p + \map {M \circ L} {\vec {p q} }\) |
as required.
$\blacksquare$