Similarity Mapping on Plane Commutes with Half Turn about Origin

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Theorem

Let $\beta \in \R_{>0}$ be a (strictly) positive real number.

Let $s_{-\beta}: \R^2 \to \R^2$ be the similarity mapping on $\R^2$ whose scale factor is $-\beta$.


Then $s_{-\beta}$ is the same as:

a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn

and:

a rotation one half turn followed by a stretching or contraction of scale factor $\beta$.


Proof

Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.

From Similarity Mapping on Plane with Negative Parameter, $s_{-\beta}$ is a stretching or contraction of scale factor $\beta$ followed by a rotation one half turn.


Thus:

\(\ds \map {s_{-\beta} } P\) \(=\) \(\ds \map {s_{-1} } {\map {s_\beta} P}\) Similarity Mapping on Plane with Negative Parameter
\(\ds \) \(=\) \(\ds \paren {-1} \map {s_\beta} P\) Definition of $s_{-1}$
\(\ds \) \(=\) \(\ds \paren {-1} \tuple {\beta x, \beta y}\) Definition of $s_\beta$
\(\ds \) \(=\) \(\ds \tuple {-\beta x, -\beta y}\)
\(\ds \) \(=\) \(\ds \beta \tuple {-x, -y}\)
\(\ds \) \(=\) \(\ds \beta \map {s_{-1} } P\) Definition of $s_{-1}$
\(\ds \) \(=\) \(\ds \map {s_\beta} {\map {s_{-1} } P}\) Definition of $s_\beta$

That is:

$s_\beta$ is a rotation one half turn followed by a stretching or contraction of scale factor $\beta$.

$\blacksquare$


Sources

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