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
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): reflection: 2. (in a line)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): reflection: 2. (in a line)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): reflection (of the plane)