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3d coordinate systems

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

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

  • Cartesian
  • Cylindrical
  • Spherical
    Cartesian, cylindrical and spherical coordinate systems
    Cartesian, cylindrical and spherical coordinate systems

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.

PLANETCALC, Three-dimensional space cartesian coordinate system

Three-dimensional space cartesian coordinate system

Digits after the decimal point: 2

Cylindrical coordinates

Radius (r)
 
Azimuth (φ), degrees
 
Height (z)
 

Spherical coordinates

Radius (ρ)
 
Azimuth (φ), degrees
 
Polar angle (θ), degrees
 

Cartesian coordinates transformation formulas:

Radius in cylindrical system:
r = \sqrt{x^2+y^2}
Radius in spherical system:
\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}

PLANETCALC, Cylindrical coordinates

Cylindrical coordinates

Digits after the decimal point: 2

Cartesian coordinates

x
 
y
 
z
 

Spherical coordinates

Radius (ρ)
 
Azimuth (φ), degrees
 
Polar angle (θ), degrees
 

Cylindrical coordinates conversion formulas:

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

Radius in spherical coordinate system:
\rho = \sqrt{r^2+z^2}
Polar angle:
\theta=Arctan(z,r), see Two argument arc tangent

PLANETCALC, Spherical coordinates

Spherical coordinates

Digits after the decimal point: 2

Cartesian coordinates

x
 
y
 
z
 

Cylindrical coordinates

Radius (r)
 
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

Radius in cylindrical system:
r = \rho\sin\theta

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