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[parent] derivation of rotation matrix using polar coordinates (Derivation)

We derive formally the expression for the rotation of a two-dimensional vector $ \boldsymbol{v}= a\boldsymbol{x}+ b\boldsymbol{y}$ by an angle $ \phi$ counter-clockwise. Here $ \boldsymbol{x}$ and $ \boldsymbol{y}$ are perpendicular unit vectors that are oriented counter-clockwise (the usual orientation).

\includegraphics{rotation-matrix.eps}

In terms of polar coordinates, $ \boldsymbol{v}$ may be rewritten:

$\displaystyle \boldsymbol{v}$ $\displaystyle = r(\cos \theta \boldsymbol{x}+ \sin \theta \boldsymbol{y}) , \quad a = r\cos \theta ; b = r\sin\theta ,$    

for some angle $ \theta$ and radius $ r \geq 0$. To rotate a vector $ \boldsymbol{v}$ by $ \phi$ really means to shift its polar angle by a constant amount $ \phi$ but leave its polar radius fixed. Therefore, the result of the rotation must be:


$\displaystyle \boldsymbol{v}'$ $\displaystyle = r\bigl(\cos(\theta + \phi) \boldsymbol{x}+ \sin(\theta + \phi) \boldsymbol{y}\bigr)$    

Expanding using the angle addition formulae, we obtain


$\displaystyle \boldsymbol{v}'$ $\displaystyle = r\bigl(\cos \theta \cos \phi - \sin \theta \sin \phi) \boldsymbol{x}+ (\sin \theta \cos \phi + \cos \theta \sin \phi) \boldsymbol{y}\bigr)$    
  $\displaystyle = (a \cos \phi - b \sin \phi) \boldsymbol{x}+ (b \cos \phi + a \sin \phi) \boldsymbol{y} .$    

When this transformation is written out in $ [\boldsymbol{x}, \boldsymbol{y}]$-coordinates, we obtain the formula for the rotation matrix:

$\displaystyle \boldsymbol{v}' = \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 stevecheng.
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See Also: rotation matrix, polar coordinates, decomposition of orthogonal operators as rotations and reflections, derivation of 2D reflection matrix

Keywords:  rotation matrix

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Cross-references: rotation matrix, formula, transformation, polar coordinates, terms, orientation, oriented, unit vectors, perpendicular, angle, vector, rotation, expression

This is version 6 of derivation of rotation matrix using polar coordinates, born on 2005-07-25, modified 2006-04-11.
Object id is 7261, canonical name is DerivationOfRotationMatrixUsingPolarCoordinates.
Accessed 9746 times total.

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AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
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