Equations defining Plane Rotation/Cartesian
Jump to navigation
Jump to search
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
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.2$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 21$