Similarity Mapping on Plane Commutes with Half Turn about Origin

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 $\beta$.

$\blacksquare$