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.