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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/Rectangular Coordinates

 Convert From Cartesian to Spherical Coordinates From Cartesian to Cylindrical Coordinates

### From Spherical Coordinates

 Convert From Spherical to Cartesian Coordinates From Spherical to Cylindrical Coordinates

### From Cylindrical Coordinates

 Convert From Cylindrical to Cartesian Coordinates From Cylindrical to Spherical Coordinates

### What is a 3D coordinate system? (Definition)

A 3D coordinate system is a mathematical framework for describing the position of points in three-dimensional space.

The main types of 3D coordinate systems are:

— Cartesian coordinate system: Uses the $x$, $y$ and $z$ axes to specify the position of a point, each coordinate represents the perpendicular distance of the point from the plane formed by the other two axes.

— Cylindrical coordinate system: Uses a radial coordinate $r$, an angular coordinate $\theta$, and a height $z$. Position is determined by the distance $r$ from a central axis (usually the $z$ axis), the angle $\theta$ around this axis, and the height $z$ along the central axis.

— Spherical coordinate system: Uses radial distance $\rho$, azimuth angle $\theta$ and colatitude angle $\varphi$. The position is determined by $\rho$ the distance from the point to the origin, $\theta$ is the angle in the $xy$ plane from the $x$ axis, and $\varphi$ is the angle relative to the $z$ axis.

dCode uses the ISO standard for spherical coordinates $(\rho,\theta,\varphi)$

### 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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