# Rotation Transformation

• Oct 16th 2012, 11:21 AM
bugatti79
Rotation Transformation
Folks,

I am struggling to understand how the attached expressions were obtained for the simple rotation of 2 axes normal to each other by an angle $\displaystyle \theta$

This is different to what is shown the wiki link Rotation matrix - Wikipedia, the free encyclopedia (See in 'two dimensions')

Ie, $\displaystyle x'= x\cos \theta - y \sin \theta$ and $\displaystyle y'=x \sin \theta + y \cos \theta$

Any clues?

Regards
bugatti
• Oct 16th 2012, 12:00 PM
TheEmptySet
Re: Rotation Transformation
Quote:

Originally Posted by bugatti79
Folks,

I am struggling to understand how the attached expressions were obtained for the simple rotation of 2 axes normal to each other by an angle $\displaystyle \theta$

This is different to what is shown the wiki link Rotation matrix - Wikipedia, the free encyclopedia (See in 'two dimensions')

Ie, $\displaystyle x'= x\cos \theta - y \sin \theta$ and $\displaystyle y'=x \sin \theta + y \cos \theta$

Any clues?

Regards
bugatti

Here is a picture

Attachment 25252

Note that

$\displaystyle T(e_1)=\cos(\theta)e_1+\sin(\theta)e_1$

and

$\displaystyle T(e_2)=-\sin(\theta)e_1+\cos(\theta)e_2$

So the general transformation is

$\displaystyle A=\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$

Now for any arbitary x and y you get

$\displaystyle A \begin{bmatrix} x \\ y \end{bmatrix} =\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

$\displaystyle \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \cos(\theta)-y \sin(\theta) \\ x \sin(\theta)+y \cos(\theta) \end{bmatrix}$
• Oct 16th 2012, 12:16 PM
bugatti79
Re: Rotation Transformation
Thank you sir, just what I was looking for.

Regards
• Oct 16th 2012, 12:18 PM
johnsomeone
Re: Rotation Transformation
I wrote this out for someone last month, if you want to take a look: