Coordinate Systems

Being able to describe something's geometric position is fundamental to a useful and robust mathematical system. So how exactly have we as humans taken to doing this? Well, we use coordinate systems of course! From maps to graphs, and the pixels on the screen you are viewing this on, coordinate systems are everywhere in our everyday lives if you look closely enough, and life just wouldn't be the same without them!

Coordinate System Meaning

There are many types of coordinate systems, some may be very familiar, such as simple number lines, and others you may not be so familiar with, such as polar coordinate systems. The important thing that all coordinate systems have in common, is that they give us a means of describing where a point is in space, relative to some origin, i.e. the zero point of the system.

The number line below is a very simple example of a coordinate system.

Example of a number line, StudySmarter Originals

The position of a point on the number line can be expressed as a single number. This number essentially denotes the horizontal distance of that point from the origin, i.e. where the $0$ is marked. The position of any point in the coordinate system is made in reference to the origin.

For instance, point $A$ on the number line below is 3 units from the origin.

Point A on a number line, StudySmarter Originals

Now, let's take a closer look at some of the main coordinate systems you might come across.

Different Types of Coordinate Systems

There are two primary types of coordinate systems in mathematics: the cartesian coordinate system and the polar coordinate system. The cartesian coordinate system is defined by distances along a set of perpendicular number lines, whereas the polar coordinate system is defined by an angle and radial distance. Let's dig a little deeper into each to get a better understanding.

The Cartesian Coordinate System

Cartesian Coordinate System, StudySmarter Originals

Any point on the coordinate plane can be described using two numbers, one to describe the distance along the horizontal axis, also known as the x-axis, and the other to describe the distance up the vertical axis, also known as the y-axis. Each of these numbers is known as a coordinate.

The notation for the coordinates of a point in the cartesian coordinate system is simply a pair of numbers within brackets, the first representing the distance of a point from the origin along the x-axis, and the other representing the distance of a point from the origin along the y-axis.

For instance, the point $B$ on the below cartesian coordinate system has coordinates $(2,3)$, as it is $2$ steps along the $x$-axis, and $3$ steps up the $y$-axis.

Point in a cartesian coordinate system, StudySmarter Originals

By extending each axis, $x$ and $y$ to include numbers less than zero, i.e. negative numbers, we can reveal the full scope of the cartesian coordinate plane.

Four quadrant cartesian coordinate plane, StudySmarter

1. If the point is located in the upper-right quadrant the two coordinates x and y will be positive.

2. If the point is located in the upper-left quadrant then the x coordinate will be negative and the y coordinate will be positive.

3. If the point is located in the lower-left quadrant then the two coordinates x and y will be negative.

4. Finally, if the point is located in the lower-right quadrant then the x coordinate will be positive and the Y coordinate will be negative.

It is worth mentioning that 2 special cases will occur when the point is located on the x or the y-axes.

This concept can be better explained by the following two examples:

After the two examples given before, we can conclude the following:

Polar Coordinate System

The polar coordinate system bears similarities to the cartesian coordinate system in that the position of any point can be defined by two numbers. However, rather than these two numbers indicating distance along perpendicular axes, in the case of polar coordinates, the two numbers indicate radial distance and angular distance.

What exactly do we mean by this? Let's take a look at a polar coordinate system to find out!

You can see that the polar coordinate system below, rather than being made up of two perpendicular axes, is in fact made up of many concentric circles, with radial lines outward from their common center indicating angles.

Example of a polar coordinate system,StudySmarter Originals

Any point on this coordinate system can be found by first moving along the number line at $0°$ by the desired amount, and then performing a circular rotation. In essence, the two coordinates are a radius, id="2938695" role="math" $r$, and angle id="2938697" role="math" $\theta$. This is written as $(r,\theta ).$

Let's take point A below as an example. To reach point A, we simply move along the number line id="2938701" role="math" $3$ units, and then make a rotation through id="2938699" role="math" $30°.$

We could, therefore, say that the polar coordinates of point A are $(3,30°).$ However, more often coordinates in polar coordinate systems are given in radians rather than degrees.

To convert degrees to radians we simply multiply by $\frac{\mathrm{\pi }}{180}.$ Therefore, $30°$ in radians is

$$\begin{gathered} 30°=30°\times \frac{\mathrm{\pi }}{180} \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$

So our new polar coordinates for point A are $(3,\frac{\mathrm{\pi }}{6}).$

Let's take a look at another example to make sure we've got it!

How exactly do we convert between coordinates in cartesian and polar coordinate systems though? Let's take a look!

Coordinate System Conversion

Converting between the two coordinate systems is simple if we employ some of our knowledge of trigonometry.

Consider the following right-angled triangle in the cartesian coordinate plane. On this triangle, point A's cartesian and polar coordinates have been marked, $(x,y)$ and $(r,\theta )$ respectively.

Converting polar cartesian coordinates, StudySmarter Originals

Using simple trigonometry, we can note the following equations to be true

id="2938793" role="math" $y=r\sin \left(\theta \right)$ and id="2938791" role="math" $x=r\cos \left(\theta \right)$

These equations allow us to convert from polar coordinates to cartesian coordinates.

Considering Pythagoras theorem, we can find the following equation for id="2938795" role="math" $r$

$r=\sqrt{x^2+y^2}$

Finding angular position, $\theta$, is then a simple case of employing trigonometry once again

id="2938796" role="math" $\theta ={\sin }^{-1}\left(\frac{y}{r}\right)$ or $\theta ={\cos }^{-1}\left(\frac{x}{r}\right)$

Let's take a look at a couple of examples to make sure we've got it!

Use of Coordinate Systems

Coordinate systems are frequently used to represent the location of a point, but they can also be used to specify the location of more complex shapes like lines, planes, circles, or spheres.

Perhaps the most important and oldest use of coordinate systems is in maps. World maps use a special geographic coordinate system that bears similarities to the cartesian coordinate system. Positions in world maps are made up of a latitude coordinate, and a longitude coordinate.

As mentioned earlier in the explanation, screens use coordinate systems as well. Each pixel in a screen has a coordinate that specifies its place horizontally and vertically. This gives each pixel a unique identifier with which it can be located and controlled. This is all going on in the background of the device you are reading this on now!

A modified version of polar coordinates is used in many forms of navigation. Polar coordinates are ideal for navigation as relative position can be defined by the desired angle of movement and the distance between points. The velocity-heading model is a guidance system used in missiles intercepting a moving target, and is based on the idea of polar coordinates!