3D Coordinates Systems

From Cartesian coordinates $ (x, y, z) $, the base / referential change to spherical coordinates $ (\rho,\theta,\varphi) $ follows the equations: $$ \rho = \sqrt{x^2 + y^2 + z^2} \\ \theta = \arccos \left( \frac{z}{\sqrt{x^2 + y^2 + z^2}} \right) = \arccos \left( \frac{z}{\rho} \right) \\ \varphi = \arctan \left( \frac{y}{x} \right) $$

Example: Le point in space in position $ (0,\sqrt{2},\sqrt{2}) $ from cartesian coordinates is defined by spherical coordinates $ \rho = 1 $, $ \theta = \pi/4 $ and $ \varphi = \pi/2 $

The conversion can be seen as two consecutive Cartesian to Polar coordinates conversions, first one in the $ xy $ plane to convert $ (x, y) $ to $ (R, \varphi) $ (with $ R $ the projection of $ \rho $ on the $ xy $ plane, then a second conversion but in the $ zR $ plane to change $ (z, R) $ to $ (\rho, \theta) $

NB: by convention, the value of $ \rho $ is positive, the value of $ \theta $ is included in the invervalle $ ] 0, \pi [ $ and the value of $ \varphi $ is included in the inverval $ ] -\pi, \pi [ $

If $ \rho = 0 $ then the angles can be defined by any real numbers of the interval

From cartesian coordinates $ (x, y, z) $ the base / referential change to cylindrical coordinates $ (r, \theta, z) $ follows the equations: $$ r = \sqrt{x^2 + y^2} \\ \theta = \arctan \left( \frac {y}{x} \right) \\ z = z $$

NB: by convention, the value of $ \rho $ is positive, the value of $ \theta $ is included in the invervalle $ ] -\pi, \pi [ $ and the $ \varphi $ is a real number

From spherical coordinates $ (\rho,\theta,\varphi) $ the base / referential change to cartesian coordinates $ (x,y,z) $ follows the equations: $$ x = \rho \sin\theta \cos\varphi \\ y = \rho \sin\theta \sin\varphi \\ z = \rho \cos\varphi $$

From spherical coordinates $ (\rho,\theta,\varphi) $ the base / referential change to cylindrical coordinates $ (r,\theta^*,z) $ follows the equations: $$ r = \rho \sin \theta \\ \theta^* = \varphi \\ z = \rho \cos \theta $$

From cylindrical coordinates $ (r,\theta,z) $ the base / referential change to cartesian coordinates $ (x,y,z) $ follows the equations: $$ x = r \cos\theta \\ y = r \sin\theta \\ z = z $$

From cylindrical coordinates $ (r,\theta^*,z) $ the base / referential change to spherical coordinates $ (\rho,\theta,\varphi) $ follows the equations: $$ \rho = \sqrt{r^2 + z^2} \\ \theta = \arctan \left( \frac{r}{z} \right) = \arccos \left( \frac{z}{\sqrt{r^2 + z^2}} \right) \\ \varphi = \theta^* $$