# Coordinate Systems – 3D, part 2

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:

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 xy 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.