What are the cylindrical coordinates of the point whose spherical coordinates are (4, 2, 1π/6) ?

1 Answer
Jun 16, 2018

Assuming the usual spherical coordinate system, (r,theta,phi)=(4,2,pi/6) equates to (R,psi,Z)=(2,2,2sqrt(3)).

Explanation:

There are several different conventions as to the order of the angles in spherical coordinates. See http://mathworld.wolfram.com/SphericalCoordinates.html for more info.

I will assume that the order of coordinates wanted here is radial, azimuthal, polar, which matches that link, also has the advantage of being a right-handed coordinate system, and is the system most often seen.

Relation of spherical to Cartesian coordinates:

Spherical radius: r=sqrt(x^2+y^2+z^2)
Spherical azimuth: theta=arctan(y/x)
Spherical polar angle: phi=arccos(z/sqrt(x^2+y^2+z^2))

Cylindrical coordinates are less confused, and the standard definition is shown here: http://mathworld.wolfram.com/SphericalCoordinates.html.

Cylindrical radius: R=sqrt(x^2+y^2)
Cylindrical azimuth: psi=arctan(y/x)
Height: Z=z

From these, we may relate the cylindrical coordinates to the spherical ones:

R=rsinphi (via a little trig simplification using the below Z)
psi=theta; the two azimuths are identical
Z=rcosphi

So in this case, (r,theta,phi)=(4,2,pi/6) equates to (R,psi,Z)=(2,2,2sqrt(3)). Assuming I chose the same spherical system as you did - the details are easily enough altered if some other spherical system is wanted.