Polar Coordinates

Imagine that you and a friend are making your way toward a coffee shop in the middle of a large city. Looking at a map, your friend says that you are about \(1\) mile (\(1.61 km\)) southwest of the coffee shop. You take a look at the same map and see that you are \(4\) blocks across and \(4\) blocks down from the coffee shop.

Figure 1. Diagram of the coffee shop's location

In this scenario, you and your friend are both communicating the same information about where you are with respect to the coffee shop. The difference is that, while you used a rectangular coordinate system or Cartesian coordinates, your friend used a polar coordinate system. This article will introduce polar coordinates, how to graph polar coordinates, how to convert between polar and rectangular coordinates, and generalizations of polar coordinates in three dimensions.

Polar Coordinate System

You are probably used to talking about points in the plane by referring to their \(x\) and \(y\) coordinates. This system of labeling points is called the rectangular coordinate system, also known as Cartesian coordinates.

Figure 2. The rectangular coordinate system

Polar coordinates are just a different way of labeling points in the plane. Instead of using \(x\) and \(y\) coordinates, the polar coordinate system labels points by how far they are from the origin, \(r\), and their angle \(\theta\) with respect to the positive \(x\)-axis, measured counterclockwise. This angle \(\theta\) is typically measured in radians.

Figure 3. The polar coordinate system

In principle, both the angle \(\theta\) and the radial distance \(r\) can take on any real value. In practice, \(r\) values are expressed using only non-negative values, that is

\[ r \in [0,\infty).\]

If \(r\) is zero, then, regardless of \(\theta\), the point \((r,\theta)\) is at the pole.

Figure 4. The origin, or pole, in polar coordinates

The usual convention is to express it as if \(\theta=0\) to avoid any ambiguity.

As mentioned before, \(r\) can take on negative values, but this is usually not expressed like this. Instead of writing a point \((-r,\theta)\) with a negative \(r\) coordinate, you write the reflection of the point \((r,\theta)\) by adding \(\pi\) to the angle.

Figure 5. A point and its reflection in polar coordinates

The angular coordinate, \(\theta\), is typically expressed using values between \(0\) and \(2\pi\), that is

\[ \theta \in [0,2\pi).\]

Just like the radial coordinate, it is also possible to get a point \((r,-\theta)\) with a negative \(\theta\) coordinate. You can obtain this point by rotating clockwise from the positive x-axis by \(\theta\) instead of counterclockwise.

Figure 6. Two points with opposite \(\theta\) signs

Polar Coordinate Terminology

There are several terms that are important to know when working with polar coordinates. The point we call the origin when working with rectangular coordinates is called the pole when working with polar coordinates. What we call the x-axis when working with rectangular coordinates is called the polar axis when working with polar coordinates.

The polar axis is also known as the reference direction.

The distance from the origin denoted as the \(r\) coordinate, is also referred to as the radius, radial coordinate, or radial distance. The \(\theta\) coordinate is called the angular coordinate, polar angle, or azimuth.

Table 1. Terminology used in polar coordinates

Degrees and Radians

The angular coordinate \(\theta\) is generally specified in either degrees or radians. Degrees and radians are just different ways of measuring angles.

Algebraically, the relationship between degrees and radians is as follows:

\[\begin{align} k\pi\text{ radians } &= 180k \text{ degrees } \\ k\text{ degrees } &= \dfrac{k\pi}{180} \text{ radians }\end{align}\]

Multiple Representations of Points in Polar Coordinates

One of the interesting aspects of working with polar coordinates is that every point has infinitely many polar coordinates that describe it. You noted this earlier for the pole, which we said is represented by \((0,\theta)\) for any value of \(\theta\). For example, the coordinates \((0,0)\), \((0,-\pi)\), and \(\left(0,\tfrac{\sqrt{\pi}}{17}\right)\) all represent the pole.

It is worth noting that this multivalued representation is usually avoided and the coordinates are shown in the range

\[ r \in [0,\infty)\]

and

\[ \theta \in [0,2\pi).\]

However, for some special scenarios, like the description of motion in Physics, these domains are extended so both include all real numbers.

In general, the points \((r,\theta)\) and \((r,\theta+2n\pi)\) for any integer \(n\) describe the same point. The points \((-r,\theta)\) and \((r,\theta+m\pi)\), where \(m\) is an odd integer, also describe the same point.

Geometrically, \((r,\theta)\) and \((r,\theta+2n\pi)\) represent the same point because adding \(2n\pi\) corresponds to rotating the point by \(2n\pi\) radians, or some multiple of 360 degrees. Rotating by a multiple of \(2\pi\) effectively does not change the position of the point, so \((r,\theta)\) and \((r,\theta+2n\pi)\) represent the same point.

Figure 7. Multiple representations of the same point in polar coordinates

How to Convert From Rectangular to Polar Coordinates

By this point, you may already notice that working with polar coordinates has its advantages, so here you can learn how to convert from rectangular coordinates to polar coordinates.

Given a point in Cartesian coordinates, you can find an expression for that same point in polar coordinates. To do so, you can use the formulas

\[\begin{align}r&=\sqrt{x^2+y^2} \\ \theta &= \arctan{\left( \frac{y}{x}\right)}. \end{align} \]

You can also find the arctangent function written as the inverse tangent function, that is

\[ \theta = \tan^{-1}{\left( \frac{y}{x}\right)}.\]

To see where these formulas come from, say you are given a point \((x,y)\) in rectangular coordinates and you want to know its corresponding polar coordinates \((r,\theta)\). If you were to graph the point, you would get something like the following image.

Figure 8. A point on the plane represented in polar and rectangular coordinates

You can use known facts about right triangles to write the relationships between \(x\), \(y\), \(r\), and \(\theta\). First, notice that you can use the Pythagorean Theorem to write

\[x^2+y^2=r^2.\]

From here, you can solve for \(r\) to obtain the first formula,

\[ r = \sqrt{x^2+y^2}.\]

You can also use the fact that the tangent of an angle \(\theta\) in a right triangle is the ratio of its opposite side to its adjacent side so that you can write

\[ \tan{\theta} = \frac{y}{x}.\]

You can now use the inverse tangent function to isolate \(\theta\), that is

\[ \theta = \arctan{\left( \frac{y}{x}\right)}.\]

Since you need to use the inverse tangent function to find the angle \(\theta\), it is better to discuss this a bit further.

Tangent and Arctangent

When using the formula

\[\theta=\arctan{\left(\frac{y}{x}\right)},\]

you have to be aware of certain technicalities related to the tangent and arctangent functions.

First, notice that this expression is undefined when \(x=0\). In this case, the point \((x,y)\) must lie on the \(y\)-axis. So, \(\theta\) must either be \(\tfrac{\pi}{2}\) or \(-\tfrac{\pi}{2}\), depending on whether \(y\) is positive or negative.

Figure 9. Two points in polar coordinates when \(x=0\)

Next, given any \(\tfrac{y}{x}\), there are actually two values of \(\theta\) in \([-\pi,\pi]\) satisfying \(\tan(\theta)=\tfrac{y}{x}\).

To make sure that you are specifying the correct point when writing a point in polar coordinates, you must make sure that the angle \(\theta\) you use is in the correct quadrant. The arctangent function only returns angles between \(-\tfrac{\pi}{2}\) and \(\tfrac{\pi}{2}\), so \(\arctan\left(\tfrac{y}{x}\right)\) only returns the correct value for \(\theta\) if the point \((x,y)\) is in the first or fourth quadrant.

Figure 10. Graph of the arctangent function

Furthermore, since \(\theta\) values are typically given between \(0\) and \(2\pi\), you also need to do a correction if the angle you obtain from the calculator is negative. The following table summarizes how to find \(\theta\).

Table 2. Relation between the sign of \(x\) and the angle \(\theta\)

How to Convert From Polar to Rectangular Coordinates

Now suppose you are given a point in polar coordinates and you want to know how to express this point using rectangular coordinates. For this case, you can also use the same diagram of the right triangle that relates \(x\), \(y\), \(r\), and \(\theta\).

Figure 11. A point on the plane represented in polar and rectangular coordinates

Focus on the angle \(\theta\). Since you have a right triangle, you can write the sine of the angle as:

\[ \sin{\theta} = \frac{y}{r}\]

Likewise, the cosine of the angle is:

\[ \cos{\theta} = \frac{x}{r}\]

Solving each expression for \(x\) and \(y\) will yield you:

\[ x=r\,\cos{\theta}\]

and

\[ y=r\,\sin{\theta}\]

Unlike when passing from rectangular coordinates to polar coordinates, the above expressions do not need you to watch out for any special considerations. These are as straightforward as it gets!

Polar Coordinate Graphers

You can use several electronic resources to graph points in polar coordinates. For example, in Geogebra, you can graph a point \((r,\theta)\) by typing \((r;\theta)\). Many graphing calculators also allow you to plot points using polar coordinates, as do programs and programming languages like Python, Octave, and Matlab. For information on how to graph polar curves, see the article Polar Curves.

Examples of Conversion Between Polar Coordinates and Rectangular Coordinates

Here you can look at some examples of conversion between both coordinate systems!

Examples of Converting from Rectangular to Polar Coordinates

For these examples, suppose you are given a point in rectangular coordinates and you want to find it written in polar coordinates.

Here is another example.

Here comes one last example.