Equations defining Plane Rotation/Cartesian

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Theorem

Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.

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

Then:

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


Proof

Rotation-equations-origin.png

Let $r_\alpha$ rotate $P = \tuple {x, y}$ 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:

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


and:

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


The result follows.

$\blacksquare$


Examples

Right Angle

Let $r_\Box$ be the rotation of the plane about the origin through a right angle.


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

Then:

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


Sources