Once again, if you want to locate a point in 3-dimensional space, you need 3 numbers. In my last post, the 2-D Cartesian coordinate system (sometimes called the rectangular coordinate system) was extended to 3-D by adding another axis that is perpendicular to the other two axes. A point is then located using the coordinates (*x,* *y*, *z*), (1,−2,3) for example. Here are two more ways to locate a 3-D point that uses the rectangular system as a backdrop.

## Spherical Coordinate System

If you remember in the 2-D scenario, polar coordinates used an angle (𝜃) from the positive *x*-axis and a distance (*r*) from the origin to determine the location of a point. And equations to represent a plot of points that satisfied the relationship between these coordinates had *r*‘s and 𝜃’s in them. In 3-D, the *spherical coordinate system* extends this method.

There are different conventions here but they all use two angles and a distance. The mathematical convention is shown below:

Here, a point is located by an angle from the positive *x*-axis, 𝜃, (like in polar coordinates), an angle from the positive *z*-axis, 𝜑, and a distance, *r*, from the origin. A point in this system has coordinates (*r*, 𝜃, 𝜑). As with polar coordinates, there are curves that are more easily expressed in spherical coordinates. For example, a sphere of radius 4 centred at the origin can be easily expressed in spherical coordinates as *r* = 4:

Or how about:

There are other conventions for spherical coordinates, one of which you are very familiar with. Locating a point on the earth is typically done with two numbers, longitude and latitude. Longitude is the angle a location is from the agreed reference meridian that runs through Greenwich England, and latitude is its angle from the equator. If the origin is at the earth’s centre with the *x*-axis going through the reference meridian (called the *prime meridian*) and the *z*-axis going through the north pole, longitude is our 𝜃, latitude is 90° − 𝜑, and *r* is always the radius of the earth.

Another convention is used to locate earth satellites using angles *right ascension* (similar to longitude) and *declination* (similar to latitude) from an agreed earth centred coordinate system where the axes are fixed and do not rotate with the earth.

There are other variations of this coordinate system; these are just a few.

## Cylindrical Coordinate System

You can think of the *cylindrical coordinate system* as the 2D polar system with an added *z* coordinate:

Different letters/Greek symbols can be used, but they all represent the same system. If you look at the *x*–*y* plane above, you see that this is just the polar coordinate system that was explained in a previous post. To add the third dimension, just move up *z* units to the desired point (*r*, 𝜃, *z*).

Cylindrical coordinates are useful in putting objects that are symmetrical with respect to the *z*-axis. For example, a cylinder of radius 4 can be easily described with the equation *r* = 4:

Another example is a cone: *z = r*:

Switching between rectangular, spherical, and cylindrical coordinates is a useful tool in calculus. An equation expressed in one of these systems may be unsolvable but solvable in a different system.

In my next post, I’ll describe a sneaky way to locate a point in 2 or 3-D with one number: parametric equations.