Similarity Mapping on Plane with Negative Parameter

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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:

\(\ds \map {s_\beta} P\) \(=\) \(\ds \tuple {\paren {-\gamma} x, \paren {-\gamma} y}\) Definition of $\beta$
\(\ds \) \(=\) \(\ds \paren {-1} \tuple {\gamma x, \gamma y}\)
\(\ds \) \(=\) \(\ds \paren {-1} \map {s_\gamma} P\) Definition of $s_\gamma$
\(\ds \) \(=\) \(\ds \map {s_{-1} } {\map {s_\gamma} P}\) Definition of $s_{-1}$

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.

$\blacksquare$


Also see


Sources

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