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

3D Coordinates Systems - dCode

Tag(s) : Geometry

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

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: $$ x = r \sin\theta \cos\varphi \\ y = \rho \sin\theta \sin\varphi \\ z = \rho $$

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: $$ \rho = \sqrt{r^2 + z^2} \\ \theta = \arctan \left( \frac{r}{z} \right) = \arccos \left( \frac{z}{\sqrt{r^2 + z^2}} \right) \\ \varphi = \theta^* $$

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- Change of 3D Coordinates (space)
- How to convert cartesian coordinates to spherical?
- How to convert cartesian coordinates to cylindrical?
- How to convert spherical coordinates to cartesian?
- How to convert spherical coordinates to cylindrical?
- How to convert cylindrical coordinates to cartesian?
- How to convert cylindrical coordinates to spherical?

change,coordinate,cartesian,spherical,cylindrical,x,y,z,r,theta,phi

Source : https://www.dcode.fr/change-coordinates-3d

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