In my last post, I talked about the Cartesian coordinate system where a point or a set of points can be located using the two numbers (*x*, *y*). There is another popular coordinate system that also locates a point in 2D space.

In the graph below, I have plotted the point (5, 3) in the Cartesian coordinate system we now know very well. I have added a line from the origin to that point and noted that the line makes an angle š with the *x*-axis and that the length of the line is *r*. I’ve also added perpendicular lines from the point to the *x* and *y* axes to show that similar right triangles are formed:

From this graph, you can see that the right triangles have sides of lengths 5 and 3 units. From the Pythagorean Theorem,

\[r=\sqrt{3^2+5^2}\approx5.83\]And from trigonometry:

\[\text{tan}(\theta)=\frac{3}{5}\Rightarrow\theta\approx30.96\text{Ā°}\]Why did I do this? Another way to locate that same point is to 1) define a line (also called a ray) from the origin that is 30.96Ā° from the *x*-axis then, 2) go along that line 5.83 units and stop. That is your point. Welcome to polar coordinates.

This system of locating a point in 2D is called “polar” because the origin is a “pole” from which all the rays that you can define radiate from. In the polar coordinate system, you also need two numbers to locate a point: *r* and š. Conventionally, a point in polar coordinates is given in the order (*r*, š).

The variable *r* is a point’s distance from the origin. š is the angle measured from the postive *x*-axis: anti-clockwise is + and clockwise is ā. Because angles repeat every 360Ā° or 2š radians, a particular (*r*, š) for a point is not unique. For example, (2, 25Ā°) locates the same point as (2, 385Ā°).

Graphing relations is usually done by plotting *r* as a function of š. Just as in Cartesian coordinates, the polar graph of an equation between *r* and š is a picture of all the points whose (*r*, š) coordinates satisfy the equation. For example, the graph below are all points that satisfy *r* = 2cos(2š):

Notice how a grid of concentric circles (possible *r* values) and rays (possible š values) is super-imposed on the *x *and *y *axes. This is a polar graph grid.

There are Cartesian graphs that are more easily expressed and plotted in polar coordinates (and vice-versa). One glaring example is a circle. In the Cartesian frame, the equation of a circle, centred at the origin, is

\[x^2+y^2=r^2\]where *r* is the radius. For a circle of radius 2, the above equation would have 4 on the right side and the graph would be a circle of radius 2 centred at the origin. In polar coordinates, the same graph would be *r* = 2. This is a picture of all points that are 2 units away from the origin:

In orbital dynamics, polar plots are most useful plotting a 2-body orbit. What is meant by “2-body” will be the subject of another post. The path of most orbits of satellites around the earth, are approximated by the ellipse. In Cartesian coordinates, the equation of an ellipse is:

\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]The parameters *a* and *b* determine the size and orientation (long side vertical or horizontal) of the ellipse. For example,

The problem with this plot is that the geometric centre of the ellipse is at the origin. The path of an earth satellite is not the path followed in this plot if the earth is at the origin. The earth is at one of two special points associated with an ellipse called *foci* (singular *focus*). It is more useful in orbital dynamics if the ellipse were plotted in polar coordinates. The polar equation of an ellipse (actually any *conic* shape which includes circles, parabolas, and hyperbolas) is

where *p* and *e* are parameters that determine the size and the shape (circle, ellipse, parabola, or hyperbola) of the orbit. The parameter *p* is the *y*-intercept on a superimposed Cartesian frame and we will limit *e* to be strictly between 0 and 1 which makes the equation plot as an ellipse. This equation, by the way, is called the *orbit equation* because it accurately describes the shape of any orbit between two point masses without being perturbed by other masses. An example of an elliptical orbit around the earth with a satellite at a particular position is:

This polar plot is more useful to describe orbits because the earth is at the origin and it shows three of the parameters commonly used to describe a satellite’s position and orbit: *p* (called the *semi-latus rectum*), *e* (called the *eccentricity*), and š (called the *true anomaly*).

Polar plots can generate shapes that would be unwieldy to generate in the Cartesian frame:

There are other less popular 2D coordinate systems like the parabolic coordinate system. Here is what parabolic graph paper looks like:

I personally do not want to go there.