;

Please Support RF Cafe by purchasing my  ridiculously low−priced products, all of which I created.

These Are Available for Free

3-Dimensional Coordinate System Conversions

Rectangular (Cartesian)

...to Cylindrical

r = (x2 + y2)½

θ = tan-1(y/x)

z = z

...to Spherical

ρ = (x2 + y2 + z2)½

θ = tan-1(y/x)

Φ = cos-1[z/(x2 + y2 + z2)]

Cylindrical

...to Rectangular

x = r * cos(θ)

y = r * sin(θ)

z = z

...to Spherical

r = sqrt(x2 + y2)

ρ = tan-1(y/x)

z = z

Spherical

...to Rectangular

x = ρ * sin(Φ) * cos(θ)

y = ρ * sin(Φ) * sin(θ)

z = ρ * sin(Φ)

...to Cylindrical

ρ = (r2 + z2)½

θ = θ

Φ = tan-1(r/z)

There are three fundamental three dimensional (3−D) coordinate systems (rectangular, cylindrical, and spherical), each of which is a more convenient means for calculations depending on the configuration of your model. For example, mapping points on the surface of a sphere using the Cartesian coordinate system requires describing all three coordinates (x, y, and z) in terms distance from each of the threes axis references. Doing the same in the spherical coordinate system requires simply a radius and two angles. When you start doing very complex calculations like those requiring calculus applications, choosing the best coordinate system can make the difference between nightmarish equations and relatively simple ones. Conversion formulas between the three fundamental coordinate systems are as follows. Note that N½ = square root of N.

Rectangular to Cylindrical:

Given rectangular coordinates (x, y, z), convert to cylindrical coordinates (r, θ, z):

r = (x2 + y2)½

θ = tan-1(y/x)

z = z

Rectangular to Spherical:

Given rectangular coordinates (x, y, z), convert to spherical coordinates (ρ, θ, Φ):

ρ = (x2 + y2 + z2)½

θ = tan-1(y/x)

Φ = cos-1[z/(x2 + y2 + z2)]

Cylindrical to Rectangular:

Given cylindrical coordinates (r, θ, z), convert to rectangular coordinates (x, y, z):

x = r * cos(θ)

y = r * sin(θ)

z = z

Spherical to Rectangular:

Given spherical coordinates (ρ, Φ, θ), convert to rectangular coordinates (x, y, z):

x = ρ * sin(Φ) * cos(θ)

y = ρ * sin(Φ) * sin(θ)

z = ρ * sin(Φ)

Spherical to Cylindrical:

Given spherical coordinates (ρ, θ, Φ), convert to cylindrical coordinates (r, θ, z):

r = ρ * sin(Φ)

θ = θ

z = ρ * cos(Φ)

Cylindrical to Spherical:

Given cylindrical coordinates (r, θ, z), convert to spherical coordinates (ρ, θ, Φ):

ρ = (r2 + z2)½

θ = θ

Φ = tan-1(r/z)

Here is a convenient online coordinate system converter on the Random Science Tools and Calculators website.

Posted September 15, 2023
(updated from original post on 2/4/2009)

 About RF Cafe Copyright: 1996 - 2024 Webmaster:     BSEE - KB3UON RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived... Copyright  1996 - 2026 All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged. All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged. My Hobby Website: AirplanesAndRockets.com My Daughter's Website: EquineKingdom