Conversion Between Cartesian, Cylindrical and Spherical Coordinates

Coordinate System	Notation
Cartesian	$(x, y, z)$
Cylindrical	$(\rho, \phi, z)$
Spherical	$(r, \theta, \phi)$

Cartesian to Cylindrical

$\begin{eqnarray*} x &=& \rho * \cos(\phi) \\ y &=& \rho * \sin(\phi) \\ z &=& z \end{eqnarray*}$

Cylindrical to Cartesian

$\begin{eqnarray*} \rho &=& \sqrt(x^{2} + y^{2}) \\ \phi &=& \tan^{-1}{\frac{x}{y}} \\ z &=& z \end{eqnarray*}$

Cartesian to Spherical

$\begin{eqnarray*} x &=& r * \sin{\theta} * \cos{\phi} \\ y &=& r * \sin{\theta} * \sin{\phi} \\ z &=& r * \cos{\theta} \end{eqnarray*}$

Spherical to Cartesian

$\begin{eqnarray*} r &=& \sqrt{x^{2} + y^{2} + z^{2}} \\ \theta &=& \tan^{-1}{\frac{\sqrt{x^{2}+y^{2}}}{z} \\ \phi &=& \tan^{-1}{\frac{y}{x}} \end{eqnarray*}$

Rotation of a Point along an Arc

The task is to rotate a point $A(x, y)$ along an arc described by an angle $\alpha$ . In order to do that, we express the point $A$ in polar coordinates:

$\begin{eqnarray*} x &\mapsto& r*\cos{\theta} \\ y &\mapsto& r*\sin{\theta} \end{eqnarray*}$

Then, rotating by angle $\alpha$ in the trigonometric sense (counter-clockwise) will change the angle $\theta$ and leave $r$ unchanged giving us the new point $B(u, v)$ , where:

$\begin{eqnarray*} u &=& r*\cos{\theta + \alpha} \\ v &=& r*\sin{\theta + \alpha} \end{eqnarray*}$

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