Search for a tool
3D Coordinates Systems

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

Results

3D Coordinates Systems -

Tag(s) : Geometry

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best 3D Coordinates Systems tool. Thank you.

# 3D Coordinates Systems

## Change of 3D Coordinates (space)

### 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.

### 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 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

### 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 invervalle $] -\pi, \pi [$ 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: $$x = r \sin\theta \cos\varphi \\ y = \rho \sin\theta \sin\varphi \\ z = \rho$$

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

## Source code

dCode retains ownership of the source code of the script 3D Coordinates Systems online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online 3D Coordinates Systems script for offline use on PC, iPhone or Android, ask for price quote on contact page !