Equations defining Plane Reflection/Cartesian

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Theorem

Let $\LL$ be a straight line through the origin $O$ of a cartesian plane.

Let the angle between $\LL$ and the $x$-axis be $\alpha$.

Let $\phi_\alpha$ denote the reflection in the plane whose axis is $\LL$.


Let $P = \tuple {x, y}$ be an arbitrary point in the plane.

Then:

$\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$


Proof

Reflection-equations-origin.png

Let $\LL$ reflect $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.

Let $OP$ form an angle $\theta$ with the $x$-axis.

We have:

$OP = OP'$

Thus:

\(\ds x\) \(=\) \(\ds OP \cos \theta\)
\(\ds y\) \(=\) \(\ds OP \sin \theta\)


Then:

\(\ds x'\) \(=\) \(\ds OP \map \cos {\theta + 2 \paren {\alpha - \theta} }\) from the geometry
\(\ds \) \(=\) \(\ds OP \map \cos {2 \alpha - \theta}\) simplifying
\(\ds \) \(=\) \(\ds OP \paren {\cos 2 \alpha \cos \theta + \sin 2 \alpha \sin \theta}\) Cosine of Difference
\(\ds \) \(=\) \(\ds OP \cos \theta \cos 2 \alpha + OP \sin \theta \sin 2 \alpha\) factoring
\(\ds \) \(=\) \(\ds x \cos 2 \alpha + y \sin 2 \alpha\) substituting $x$ and $y$


and:

\(\ds y'\) \(=\) \(\ds OP \map \sin {\theta + 2 \paren {\alpha - \theta} }\) from the geometry
\(\ds \) \(=\) \(\ds OP \map \sin {2 \alpha - \theta}\) simplifying
\(\ds \) \(=\) \(\ds OP \paren {\sin 2 \alpha \cos \theta - \cos 2 \alpha \sin \theta}\) Sine of Difference
\(\ds \) \(=\) \(\ds OP \cos \theta \sin 2 \alpha - OP \sin \theta \cos 2 \alpha\) factoring
\(\ds \) \(=\) \(\ds x \sin 2 \alpha - y \cos 2 \alpha\) substituting $x$ and $y$


The result follows.

$\blacksquare$


Examples

$x$-Axis

Let $\phi_x$ denote the reflection in the plane whose axis is the $x$-axis.


Let $P = \tuple {x, y}$ be an arbitrary point in the plane

Then:

$\map {\phi_x} P = \tuple {x, -y}$


$y$-Axis

Let $\phi_y$ denote the reflection in the plane whose axis is the $y$-axis.


Let $P = \tuple {x, y}$ be an arbitrary point in the plane

Then:

$\map {\phi_y} P = \tuple {-x, y}$


Sources

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