Spherical Coordinate System
This article is a very basic primer on the notion of Spherical Coordinate System, and how it is linked to the Cartesian Coordinate System
Suppose you are staring at a clock (I used to do so in every History Class 😜). When it's 3 o’clock, we say that the hands of the clock are a right angle apart, and the same goes with 9 o’clock. We do not measure the distance between the tips of the hands, as doing that would be just plain absurd 😆. Similarly, in many other instances involving circles and spheres, measuring angles is a very intuitive way of quantifying relative position.
In order to assign the position of a point in space, coordinate systems were introduced. The world we live in 3-dimensional, so any point in space can be represented in terms of three independent quantities, the most intuitive (generally) being height, width, and depth. This notion is the root of the Cartesian Coordinate System. Here, an origin is considered, relative to which the position of all points in space would be measured, and three axes (usually denoted by X, Y, and Z)are defined, which are nothing but directed reference lines passing through the origin, which are at right angles to each other. Any point can be represented in the following way — imagine an insect (say the aesthetically beautiful ladybug 😍) is at the origin and can travel only parallel to the three axes (since we are imagining, anything can happen!). So, the set of distances it would travel parallel to each of the three axes (along with their directions considered) would define the three cartesian coordinates of the point.
However, as already stated in the example of the clock, spherical coordinates are more intuitive than cartesian coordinates in cases involving circular and spherical symmetries. Let’s view the three-dimensional world through the eyes of spherical coordinates!
Consider a sphere of radius r with a center O. Let us consider a point P on the sphere. The angle made by OP with the Z-axis is denoted by θ. Also, let the projection of OP on the XY Plane be OQ. The angle between OQ and the X-axis be φ. The set (r, θ, φ) uniquely identifies the point P and distinguishes it from any other point in the three-dimensional space. In other words, point P can be uniquely represented by the spherical coordinates (r, θ, φ).
Connecting Spherical and Cartesian Coordinate Systems
Now that we are familiar with the very basics of the cartesian and the Spherical Coordinate Systems, let us use some simple trigonometry to convert the representation of a point in one coordinate system to the other. The following figure shows the relationship between the two coordinate systems (X, Y, Z) and (r, θ, φ).
