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3D Coordinates Systems

Tool for making coordinates changes system in 3d-space (Cartesian, spherical, cylindrical, etc.), geometric operations to represent elements in different referentials.

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3D Coordinates Systems -

Tag(s) : Geometry

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3D Coordinates Systems

Change of 3D Coordinates (space)

From Cartesian Coordinates




From Cartesian to Spherical Coordinates

From Cartesian to Cylindrical Coordinates

From spherical coordinates




From Spherical to Cartesian Coordinates

From Spherical to Cylindrical Coordinates

From cylindrical coordinates




From Cylindrical to Cartesian Coordinates

From Cylindrical to Spherical Coordinates

Change of 2D Coordinates (plane)

Tool for making coordinates changes system in 3d-space (Cartesian, spherical, cylindrical, etc.), geometric operations to represent elements in different referentials.

Answers to Questions

How to convert cartesian coordinates to spherical?

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 interval $ ] 0, \pi [ $ and the value of $ \varphi $ is included in the interval $ ] -\pi, \pi [ $

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

How to convert cartesian coordinates to cylindrical?

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 interval $ ] -\pi, \pi [ $ and the $ \varphi $ is a real number

How to convert spherical coordinates to cartesian?

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

How to convert spherical coordinates to cylindrical?

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

How to convert cylindrical coordinates to cartesian?

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

How to convert cylindrical coordinates to spherical?

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

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