## On the usefulness of spherical coordinates

I noticed students often have problems with polar and spherical coordinates, most notably the problem of not knowing what they are good for. As if cartesian coordinates were not complicated enough. Well, that’s just it, cartesian coordinates are often too complicated! Hence the slightly more obscure coordinate systems.

What are coordinate systems? In their most simple form they define a point in space by a set of numbers, so that by knowing the numbers we know exactly where the point is. (We will stick to the good old real space here, with directions right/left, up/down and forward/backward. No fancy quantum physics phase space and similar in this post.) So in cartesian coordinates you have three axes and the position of the point is determined by three numbers, each of which gives the distance from the origin along one axis. In other words, each number specifies how far up/down, right/left, and forward/backward the point, or perhaps a ball, or even a person, or a building, is. It is up to you where to place the origin, but, to paraphrase the classic, all points being equal, some may be more equal than others. Recall Sherlock Holmes in The Adventure of the Musgrave Ritual and the set of directions he and Watson are following. The choice of the origin is crucial there. But if the origin were in a different place, it would still be possible to write a set of directions leading to the same point, although they would perhaps be more complicated.

Now take polar coordinates. They work in two dimensions, for example on a plane, and you need to select the origin of the coordinate system and one axis. To describe the position of a point you then have to specify the distance from the origin and the angle between a line connecting the point and the origin and the axis. Think about an everyday example of where we use polar coordinates. Think… Think… Got it? Right. The wristwatch. (Not the digital version, obviously!) The distance from the origin tells us how long a watch hand is, that is, whether it is the hour hand or the minute hand, and the angle gives the number the hand is pointing at. Imagine trying to give the time of day in cartesian coordinates…

The spherical coordinates operate just like polar coordinates, but in three dimensions. Take polar coordinates and add another angle. Now we have the distance from the origin, the angle that tells us in which direction of the compass we are pointing, and an angle that tells how far up or down the point is. Now an illustration: a poi is essentially a ball on a chain. A fire poi is when the ball is replaced by a wick and set on fire. Poi enthusiasts spin a pair of poi, holding the end of each chain in one hand. Think again… Where is the ball in space, if we place the origin of our coordinate system at the hands of the poi spinner? (Assuming that the spinner is holding his/her hands together, which is often the case.) Look at this photo by Bucky O’Hare for an illustration. It would be a nightmare to describe the complex pattern drawn by the fire poi in terms of up/down, right/left, forward/backward, but it is actually very easy in spherical coordinates. The distance from the origin is given by the length of the chain and then we just need to know where the poi is around the spinner and how high it is. There! Long live spherical coordinates!

In fairness I have to point out that “we just need to know where the poi is around the spinner and how high it is” is often not simple *if you are the poi spinner*. Especially if you are a beginner spinner, because then you tend to confuse your left and right hand and often define which hand is left or right by whether it currently is on the left or right… And also, if the spinner moves his arms, it may be easier to use a mix of spherical and cartesian coordinates: the spherical coordinates would still describe the movement of the poi with respect to the hand of the spinner and the correction for the movement of the hand itself would be added *afterwards* in cartesian coordinates.

Good point. Maybe we should introduce students to polar and spherical coordinate systems before cartesian coordinates. The former systems would certainly be more familiar if we did this, and eventually they would become the dominant systems. The hard part is getting such a move in the educational system started because the folks who would be teaching this are also generally unfamiliar enough to teach it. We need a similar move to switch from the British measurement system in the USA and UK to metric, which I ponder over along with how to switch to a more environment-friendly philosophy and instill an environmental ethic that we lack altogether.

Researching what influenced Descartes to develop his type of coordinate system might be interesting. He had a crazy idea with his philosophy of dualism, which seems to have led us to the dualistic philosophy we have today regarding the environment–humans apart from nature rather than a part of nature.

Pete said this on 4 September, 2008 at 4:21 am |

Hi Veronika.

I am a pilot, and we often mix cartesian and polar coordinates, using the former to describe where things are in relation to the larger, inertial framework, and the later to describe where things are in relation to either the aircraft or a fixed point. I should clarify by stating that, although latitude and longitude are circular coordinates, the planet is sufficiently large that when seen from the point of view of a prop-driven, single engine aircraft (or, more precisely, in the mind of the pilot), they are essentially orthogonal unless one flies a long distance or approaches the poles.

Lune said this on 15 September, 2008 at 7:19 pm |

Hi Luna,

Thanks for sharing that! I recall reading that pilots use polar coordinates but I had no idea that you would mix the two systems. I am looking forward to telling the students about it next term. It is the best real-life example of mixing the coordinate systems I have come across so far and I am sure they will find it interesting too. (They seemed slightly bewildered when I took out my pair of poi and started to spin them to illustrate my point last time. 😉 )

veronikab said this on 22 September, 2008 at 8:10 am |