# 3d coordinate systems

Transforms 3d coordinate from / to Cartesian, Cylindrical and Spherical coordinate systems.

### This page exists due to the efforts of the following people:

#### Timur

Created: 2018-10-22 12:49:06, Last updated: 2021-02-13 11:39:00

This calculator is intended for coordinates transformation from/to the following 3d coordinate systems:

• Cartesian
• Cylindrical
• Spherical

### Cartesian coordinate system

A point can be defined in the Cartesian coordinate system with 3 real numbers: x, y, z. Each number corresponds to the signed minimal distance along with one of the axis (x, y, or z) between the point and plane, formed by the remaining two axes. The coordinate is negative if the point is behind the coordinate system origin.

### Cylindrical coordinate system

This coordinate system defines a point in 3d space with radius r, azimuth angle φ, and height z. Height z directly corresponds to the z coordinate in the Cartesian coordinate system. Radius r - is a positive number, the shortest distance between point and z-axis. Azimuth angle φ is an angle value in range 0..360. It is an angle between positive semi-axis x and radius from the origin to the perpendicular from the point to the XY plane.

### Spherical coordinate system

This system defines a point in 3d space with 3 real values - radius ρ, azimuth angle φ, and polar angle θ. Azimuth angle φ is the same as the azimuth angle in the cylindrical coordinate system. Radius ρ - is a distance between coordinate system origin and the point. Positive semi-axis z and radius from the origin to the point forms the polar angle θ.

#### Three-dimensional space cartesian coordinate system

Digits after the decimal point: 2

#### Cylindrical coordinates

Azimuth (φ), degrees

Height (z)

#### Spherical coordinates

Azimuth (φ), degrees

Polar angle (θ), degrees

### Cartesian coordinates transformation formulas:

$r = \sqrt{x^2+y^2}$
$\rho = \sqrt{x^2+y^2+z^2}$
Azimuth angle:
$\varphi=Arctan(y,x)$, see Two arguments arctangent

Polar angle:
$\theta=\arctan\frac{\sqrt{x^2+y^2}}{z}$

#### Cylindrical coordinates

Digits after the decimal point: 2

x

y

z

#### Spherical coordinates

Azimuth (φ), degrees

Polar angle (θ), degrees

### Cylindrical coordinates conversion formulas:

To cartesian coordinates:
$x=r\cos\varphi$,
$y=r\sin\varphi$

$\rho = \sqrt{r^2+z^2}$
Polar angle:
$\theta=Arctan(z,r)$, see Two arguments arctangent

#### Spherical coordinates

Digits after the decimal point: 2

x

y

z

#### Cylindrical coordinates

Azimuth (φ), degrees

Height (z)

### Spherical coordinates transformation formulas

Cartesian coordinates:
$x=\rho\sin\theta\cos\varphi$,
$y=\rho\sin\theta\sin\varphi$,
$z=\rho\cos\theta$

$r = \rho\sin\theta$

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PLANETCALC, 3d coordinate systems