Complex Analysis in Stereographic Projection

Today we are going to go over an area within complex analysis known as Stereographic Projection.

​According to an online dictionary stereographic projection is “a one-to-one correspondence between the points on a sphere and the extended complex plane where the north pole on the sphere corresponds to the point at infinity of the plane” (Dictionary). ​

To begin, complex analysis is used in just about everything we do in life. Wolfram Alpha says complex analysis is “the study of complex numbers together with their derivatives, manipulation, and other properties” (Weisstein). The site goes on to say “Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems”. 

Stereographic projection is used in many areas of the physical life. One area is physics where they use the metric tensor. A metric tensor “is a function which tells how to compute the distance between any two points in a given space” (Stover).

In this way, “the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be ‘arithmetized’ by way of introducing a sort of generalized coordinate system” (Borisenko and Tarapov 1979).

It is said in this problem done by Hobson, that to find “the metric tensor in a new coordinate system, consider the stereographic projection of a sphere of radius r” (Hobson 2006). It is also said that “the projection is done by drawing lines from the south pole through every point on the sphere and extending each line until it intersects the plane tangent to the north pole” (Hobson 2006).

This shows that within physics you can see that complex analysis is used from using stereographic projection on metric tensor.

Additionally, area is within engineering. On this site called the civil engineering notes we can see many cases of examples of how it is used exactly.

It says that “stereographic projection is one of the convenient methods of projecting the linear and planar features” (Fosson, “Structural Geology”). Along with this information it goes on to say that “it allows the three-dimensional orientation data to be represented and analyzed in two dimensions”.

​This site actually gives all areas it is used in from landslide hazard failures, earthquake studies, structural geological analysis, and even mining industry using fossil fuels.

We can see that the application of complex analysis goes very far and wide. This application known as stereographical projection is just one of the many fields within complex analysis that ventures into the worlds of physics and engineering.

​Since these fields are so applied this gives way for us to improve our life as a whole using this type of projection could indeed make even more contributions to the world of fields out there and to all the unknown we have yet to discover.