3D-Coordinates Calculator Cartesian↔Spherical↔Cylindrical - Online (2025)

  1. Mathematics
  2. Geometry
  3. 3D Coordinates Systems

Change of 3D Coordinates (space)

Answers to Questions (FAQ)

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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3D-Coordinates Calculator Cartesian↔Spherical↔Cylindrical - Online (2025)

FAQs

How to convert points from Cartesian to cylindrical and spherical coordinates? ›

To convert a point from Cartesian coordinates to spherical coordinates, use equations ρ2=x2+y2+z2,tanθ=yx, and φ=arccos(z√x2+y2+z2). To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,θ=θ, and z=ρcosφ.

How do you convert XYZ to cylindrical coordinates? ›

To convert a point from cylindrical coordinates to Cartesian coordinates, use equations x=rcosθ,y=rsinθ, and z=z. To convert a point from Cartesian coordinates to cylindrical coordinates, use equations r2=x2+y2,tanθ=yx, and z=z.

What is the equation for spherical coordinates calculator? ›

It uses the following equations: x = r × sin θ × cos φ; y = r × sin θ × sin φ; z = r × cos θ.

What is the relationship between Cartesian and spherical coordinates? ›

The spherical coordinates are determined by (ρ,ϕ,θ). The relation between these and the Cartesian coordinates (x,y,z) for a point are as follows. x=ρsin(ϕ)cos(θ), ϕ∈[0,π]y=ρsin(ϕ)sin(θ), θ∈[0,2π)z=ρcosϕ, ρ≥0.

What is the formula for coordinate conversion? ›

To convert the x coordinate from data to normalized coordinates, use the formula Nx = X0 + X1Dx. The same process is used to convert the y and z coordinates using ! Y.S and ! Z.S.

What is the formula for cylindrical and spherical coordinates? ›

To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=√r2+z2,θ=θ, and φ=arccos(z√r2+z2).

How to convert a Cartesian integral to cylindrical? ›

To convert this to cylindrical coordinates, we will set the bounds from zed, then to r, then to theta. However, we have to swap this for our new variables though (ie from x, y to r, theta), so we simply have to use √x2+y2=r per the definition to turn our lower bound and upper bound for z to r2 and r, respectively.

What is the XYZ equation for a cylinder? ›

In Cylindrical Coordinates, the equation r = 1 gives a cylinder of radius 1. x = cosθ y = sinθ z = z. If we restrict θ and z, we get parametric equations for a cylinder of radius 1. gives the same cylinder of radius r and height h.

What do spherical coordinates look like? ›

Spherical coordinates of the system denoted as (r, θ, Φ) is the coordinate system mainly used in three dimensional systems. In three dimensional space, the spherical coordinate system is used for finding the surface area. These coordinates specify three numbers: radial distance, polar angles and azimuthal angle.

What is the R equation for spherical coordinates? ›

Since r=ρsinϕ, these components can be rewritten as x=ρsinϕcosθ and y=ρsinϕsinθ. In summary, the formulas for Cartesian coordinates in terms of spherical coordinates are x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ.

What is the XYZ in cylindrical coordinates? ›

The rectangular coordinates ( x , y , z ) ( x , y , z ) and the cylindrical coordinates ( r , θ , z ) ( r , θ , z ) of a point are related as follows: x = r cos θ These equations are used to convert from y = r sin θ cylindrical coordinates to rectangular z = z coordinates.

How do you convert XYZ to spherical coordinates? ›

In written terms: r is the distance from the origin to the point, ϕ is the angle needed to rotate around z to get to the point, θ is the angle from the positive z-axis, ρ is the distance between the point and the z-axis. r=√ρ2+z2=√x2+y2+z2.

When to use spherical coordinates? ›

Spherical coordinates are preferred over Cartesian and cylindrical coordinates when the geometry of the problem exhibits spherical symmetry. For example, in the Cartesian coordinate system, the surface of a sphere concentric with the origin requires all three coordinates (x, y, and z) to describe.

How to convert rectangular coordinates to cylindrical coordinates? ›

x = r cos θ These equations are used to convert from y = r sin θ cylindrical coordinates to rectangular z = z coordinates. and r 2 = x 2 + y 2 These equations are used to convert from tan θ = y x rectangular coordinates to cylindrical z = z coordinates.

How do you convert Cartesian coordinates? ›

To convert from polar co-ordinates to Cartesian co-ordinates, use the equations x = r cos θ , y = r sin θ . To convert from Cartesian co-ordinates to polar co-ordinates, use the equations r2 = x2 + y2 , tan θ = y x .

How do you make a sphere in cylindrical coordinates? ›

1 Answer
  1. x2+y2+z2=R2 .
  2. Since x2+y2=r2 in cylindrical coordinates, an equation of the same sphere in cylindrical coordinates can be written as.
  3. r2+z2=R2 .
Nov 2, 2014

In what situations would you want to change from rectangular to cylindrical or to spherical coordinates? ›

Final answer:

When dealing with objects like cylinders, spheres, or problems with symmetry about a particular line or point, we would typically choose to switch to cylindrical, spherical, or polar coordinates from the rectangular one.

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