Login
This is a place holder for potential sponsor logos.
derivation of rotation matrix using polar coordinates
We derive formally the expression for the rotation of a two-dimensional vector $\vv = a\vx + b\vy$ by an angle $\phi$ counter-clockwise. Here $\vx$ and $\vy$ are perpendicular unit vectors that are oriented counter-clockwise (the usual orientation).
In terms of polar coordinates, $\vv$ may be rewritten:
 |
 |
|
|
for some angle  and radius  . To rotate a vector
 by  really means to shift its polar angle by a constant amount  but leave its polar radius fixed. Therefore, the result of the rotation must be:
|
||
 |
 |
|
|
Expanding using the angle addition formulae, we obtain |
||
 |
 |
|
 |
||
When this transformation is written out in $[\vx, \vy]$ -coordinates, we obtain the formula for the rotation matrix:$$ \vv' = \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} \,.$$
derivation of rotation matrix using polar coordinates is owned by Steve Cheng.
None.