1. Coordinate Systems

d. Cylindrical and Spherical Coordinates - 3D and nD

2. 3D Spherical Coordinates

In \(3\) dimensional space, in addition to Rectangular Coordinates \((x,y,z)\) and Cylindrical Coordinates \((r,\theta,z)\), there are Spherical Coordinates \((\rho,\phi,\theta)\). Spherical coordinates are useful when you want to study spheres or other objects which are functions of latitude and longitude.

Spherical Coordinates \((\rho,\phi,\theta)\) start from cylindrical coordinates \((r,\theta,z)\) but replace the cylindrical radius \(r\) and the height \(z\) by a new spherical radius \(\rho\) which measures the distance from the point \(P\) to the origin \(O\) and a new angular coordinate \(\phi\) which measures the angle from the positive \(z\)-axis to the ray \(\overrightarrow{OP}\). In this context, \(\phi\) is called the polar angle since it measures the angle down from the north pole, and \(\theta\) is now called the azimuthal angle.

If you think about a point on a sphere, \(\rho\) gives the radius of the sphere, \(\phi\) essentially gives the latitude of the point on the sphere, and \(\theta\) gives the longitude of the point on the sphere.

What is the difference between \(\phi\) and latitude?

The real world latitude is measured north and south of the equator, \(0\) at the equator, positive in the northern hemisphere and negative in the southern hemisphere, \(90^\circ\) at the north pole and \(-90^\circ\) at the south pole.

The polar angle \(\phi\) is measured from the north pole to the south pole, \(0^\circ\) at the north pole and \(180^\circ\) at the south pole.

So \[ \text{latitude}\,=90^\circ-\phi=\dfrac{\pi}{2}-\phi \]

Converting between Cylindrical and Spherical Coordinates

Look at the right triangle with legs \(r\) and \(z\) and hypotenuse \(\rho\). Since \(\phi\) is the angle between \(\rho\) and the \(z\)-axis, we have the relations \[ \rho=\sqrt{r^2+z^2} \qquad \cos\phi=\dfrac{z}{\rho} \qquad \sin\phi=\dfrac{r}{\rho} \]

Consequently, cylindrical and spherical coordinates are related by \[\begin{aligned} r&=\rho\sin\phi \qquad \qquad &\rho&=\sqrt{r^2+z^2} \\[3pt] \theta&=\theta &\phi&=\arccos\dfrac{z}{\sqrt{r^2+z^2}} \\[3pt] z&=\rho\cos\phi &\theta&=\theta \end{aligned}\]

(Since the values for \(\phi\) run from \(0\) at the north pole to \(\pi\) at the south pole, and this is the image of the \(\arccos\) function, there is no ambiguity in the formula for \(\phi\) as there was in the formula for \(\theta\) in polar and cylindrical coordinates.)

Converting between Rectangular and Spherical Coordinates

Recall the formulas relating rectangular and cylindrical coordinates: \[\begin{aligned} x&=r\cos\theta \qquad \qquad &&r=\sqrt{x^2+y^2} \\[3pt] y&=r\sin\theta &&\tan\theta=\dfrac{y}{x} \\[3pt] z&=z &&z=z \end{aligned}\]

Combining these with the formulas relating cylindrical and spherical, we get the formulas relating rectangular and spherical coordinates: \[\begin{aligned} x&=\rho\sin\phi\cos\theta \qquad \qquad &&\rho=\sqrt{x^2+y^2+z^2} \\[3pt] y&=\rho\sin\phi\sin\theta &&\phi=\arccos\dfrac{z}{\sqrt{x^2+y^2+z^2}} \\ z&=\rho\cos\phi &&\tan\theta=\dfrac{y}{x} \end{aligned}\] with the same conditions on solving for theta as there were for polar coordinates.

Write \[ x= \qquad y= \qquad z= \] Add \(\rho,\rho,\rho\) (your boat), to get \[ x=\rho \qquad y=\rho \qquad z=\rho \] Put \(\cos\phi\) next to \(z\) since \(z\) is the polar axis and \(\sin\phi\) next to \(x\) and \(y\), to get \[ x=\rho\sin\phi \qquad y=\rho\sin\phi \qquad z=\rho\cos\phi \] Finally, to distinguish between \(x\) and \(y\), put \(\cos\theta\) next to \(x\) and \(\sin\theta\) next to \(y\) just like for polar coordinates, to get \[ x=\rho\sin\phi\cos\theta \qquad y=\rho\sin\phi\sin\theta \qquad z=\rho\cos\phi \]

Coordinate Surfaces

When you hold one of the coordinates fixed and let the other two vary, the point \(P=(\rho,\phi,\theta)\) traces out a coordinate surface.

When \(\rho\) is constant, you get a sphere. You specify where you are on the surface by giving \(\phi\) and \(\theta\), essentially latitude and longitude.
When \(\phi\) is constant, you get a cone. You specify where you are on the surface by giving \(\rho\) and \(\theta\).
When \(\theta\) is constant, you get a vertical half plane. You specify where you are on the surface by giving \(\rho\) and \(\phi\).

Here are the plots of the coordinate surfaces:

Constant \(\rho\) Surface

Constant \(\phi\) Surface

Constant \(\theta\) Surface

Rotate these plots with your mouse:

Constant \(\rho\) Surface

Constant \(\phi\) Surface

Constant \(\theta\) Surface

Coordinate Curves

When you hold two of the coordinates fixed and let the other one vary, the point \(P=(\rho,\phi,\theta)\) traces out a coordinate curve. The curves are named by the coordinate which is changing. The coordinate curves are the intersections of the coordinate surfaces.

When \(\phi\) and \(\theta\) are constant, you get a radial ray coming out of the origin called a \(\rho\)-curve.
When \(\rho\) and \(\theta\) are constant, you get a semi-circle from the North Pole to the South Pole called a \(\phi\)-curve or line of longitude.
When \(\rho\) and \(\phi\) are constant, you get a horizontal circle called a \(\theta\)-curve or line of latitude.

Here are the plots of the coordinate curves:

\(\rho\)-Curves

\(\phi\)-Curves

\(\theta\)-Curves

Coordinate Grid

When you draw several coordinate curves or surfaces of each type you get a coordinate grid. Here is the spherical coordinate grid as surfaces and as curves. Notice how all points in space can be reached with these coordinates. You can watch an animation of the cylindrical coordinate grid at the top of this page.

Spherical Coordinate Grid as Surfaces
The \(\rho\)-surfaces are red spheres.
The \(\phi\)-surfaces are cyan cones.
The \(\theta\)-surfaces are blue half-planes.

Spherical Coordinate Grid as Curves
The \(\rho\)-curves are red radial lines.
The \(\phi\)-curves are blue longitude circles.
The \(\theta\)-curves are cyan latitude circles.