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# 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:

#### Anton

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

• Cartesian
• Cylindrical
• Spherical

### Cartesian coordinate system

In Cartesian coordinate system a point can be defined with 3 real numbers : x, y, z. Each number corresponds to the signed minimal distance along one of axis (x, y or z) between the point and plane, formed by remaining two axis. 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 z coordinate in Cartesian coordinate system. Radius r - is a positive number, shortest distance between point and z axis. Azimuth angle φ is angle value in range 0..360, is an angle between positive semi axis x and radius from the origin to 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 azimuth angle in 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 argument arc tangent

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 argument arc tangent

#### Spherical coordinates

Digits after the decimal point: 2

x

y

z

#### Cylindrical coordinates

Azimuth (φ), degrees

Height (z)

### Spherical coordinates transformation formulas

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

$r = \rho\sin\theta$