In this article, we describe the two types of **coordinate systems** used in a plane: the
**Cartesian coordinate system** and the **polar coordinate system**. These coordinate
systems are utilized by this graphing calculator as well as
these single-type graphers on this site.

Assuming you are familiar with **one-dimensional coordinate systems**, also called a *number line* or **axis**, we will focus on ** two-dimensional coordinate systems**. A two-dimensional

`(a,b)`

in a plane.
The **Cartesian coordinate system** is formed by two intersecting **axes** with a common
**origin**. The axis containing the values of the first coordinate is usually drawn horizontally,
while the other is drawn vertically. These axes are commonly denoted by the letters `x`

and `y`

and are called the
**x-axis** and **y-axis**, respectively.

On the other hand, the **polar coordinate system** uses one reference **axis**, usually horizontal,
called the **polar axis**. Its origin is called the **pole**. Unlike the Cartesian coordinate system which uses a second axis, the polar coordinate system uses **angles** measured from the polar axis about the pole.

The following figures illustrate the **Cartesian coordinate system** and the **polar coordinate system** used in a plane.

Both the **Cartesian coordinate system** and **polar coordinate system** can be used to locate or plot points. By plotting points one can plot the graph of functions, and graph of parametric equations. The following figures illustrate how the point (`a,b`

) is plotted in each coordinate system for `a=3`

and `b=2`

.

In the **Cartesian coordinate system** with the **x-axis** and **y-axis** as previously described, the first coordinate, ** a**, is referred to as the (or

`b`

In the **polar coordinate system**, ** a** is referred to as the

`b`

`a,b`

).
In the **Cartesian coordinate system**, the point (`a,b`

) can be described as `x = a`

and `y = b`

. In the **polar coordinate system**, it is customary to use `r`

and `θ`

(or `φ`

) to represent the point, where `r = a`

and `θ = b`

(or `φ = b`

).

Every ordered pair (`a,b`

) has a unique location in a coordinate plane. However, a location (point) in the **polar coordinate system** can have multiple **polar representations**. This is because adding multiples of `2π`

(or `360°`

) to an angle does not change its "location". Therefore, the ordered pairs (`a,b+2π`

) and (`a,b`

) determine the same point in the **polar coordinate plane**. Similarly, adding `π`

to the angular coordinate and changing the radial coordinate from `a`

to `-a`

also determines the same point as (`a,b`

), as shown in the following illustrations:

In summary, all the ordered pairs (`a,b`

), (`a,b+2nπ`

) and (`-a,b+(2n+1)π`

) identify the same point in the **polar coordinate system** for integers `n`

.

Learn how to convert between Cartesian and polar Coordinates.

If the axes are **perpendicular**, as shown above, the **Cartesian coordinate system** in a plane is called the **rectangular coordinate system** or **orthogonal coordinate system**. However, the axes need not be perpendicular, in which case, we have an **oblique coordinate system** or **non-orthogonal coordinate system**, as demonstrated in the following figures. Note that the **polar axis** in the **polar coordinate system** need not be horizontal either.