Similarity Mapping on Plane with Negative Parameter

Theorem
Let $\beta \in \R_{<0}$ be a (strictly) negative 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 a stretching or contraction followed by a rotation one half turn.

Proof
Let $\beta = -\gamma$ where $\gamma \in \R_{>0}$.

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

Then:

Because $\gamma > 0$ we have by definition that $s_\gamma$ is a stretching or contraction.

From Similarity Mapping on Plane with Scale Factor Minus 1, $s_{-1}$ is the plane rotation of the plane about the angle $\pi$.

Hence, by definition of half turn:
 * $s_\beta$ is a stretching or contraction followed by a rotation one half turn.

Also see

 * Similarity Mapping on Plane Commutes with Half Turn about Origin, where it is seen that $s_\beta$ is also the same as a rotation one half turn followed by a stretching or contraction.