# Transformation of Coordinates Involving Pure Translation

$\begin{cases}x = x' + x_0 \\ y = y' + y_0 \end{cases}$   or   $\begin{cases}x' = x - x_0 \\ y' = y - y_0 \end{cases}$

where (x, y) are old coordinates [i.e. coordinates relative to xy system], (x',y') are new coordinates [relative to x'y' system] and (x0, y0) are the coordinates of the new origin 0' relative to the old xy coordinate system.

#### Transformation of Coordinates Involving Rotation

$\begin{cases}x = x' \cos\alpha - y' \sin\alpha \\ y = x' \sin\alpha + y' \cos\alpha \end{cases}$

or

$\begin{cases}x' = x \cos\alpha + y \sin\alpha \\ y' = y \cos\alpha - x \sin\alpha \end{cases}$

where the origins of the old [xy] and new [x'y'] coordinate systems are the same but the x' axis makes an angle α with the positive x axis.

#### Transformation of Coordinates Involving Translation and Rotation

$x = x' \cos\alpha - y' \sin\alpha + x_0 \\ y = x' \sin\alpha + y' \cos\alpha + y_0$

or

$x' = (x - x_0)\cos\alpha + (y - y_0)\sin\alpha \\ y' = (y - y_0)\cos\alpha - (x - x_0)\sin\alpha$

where the new origin O' of x'y' coordinate system has coordinates (x0, y0) relative to the old xy coordinate system and the x' axis makes an angle α with the positive x axis.

#### Polar Coordinates(r, θ)

A point P can be located by rectangular coordinates (x, y) or polar coordinates (r, θ). The transformation between these coordinates

$\begin{cases}x = r \cos\theta \\ y = r \sin \theta\end{cases}$  or  $\begin{cases} r = \sqrt{x^2 + y^2} \\ \theta = \tan^{-1}\big(\frac{y}{x}\big) \end{cases}$

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