Mobius transformations

Introduction:

Mobius transformations are a concept in mathematics that have many applications outside of pure mathematics as well. 

What is Mobius Transformation?

A Mobius transformation or Mobius inversion is a geometric transformation that takes a line segment and swivels it through 180 degrees and then connects the resulting two endpoints together. Imagine an equilateral triangle with a line segment that is perpendicular to one side of the triangle, this line segment is our starting point. Our goal is to invert the triangle after transforming our line segment into another part of the triangle. The process to accomplish this goal is as follows:

We start by rotating the line segment about its midpoint through 180 degrees, we now have three segments connected by two points.

The next step is the most complicated one and it involves the inversion. In this step we connect the midpoints of the two segments that result from our first rotation.

When we connect these two midpoints we get a figure that looks like an infinity symbol, or as it is called a Mobius strip.

The image below shows this process visually.

Every point on this Mobius strip can be thought of as being a well defined point in space, some points are exceptional however, for example points A and B on our graph would be considered exceptional points since no line segment can be drawn to point A and B without crossing over itself (self intersection) or passing through another point on the curve.

Mobius Group:

The Mobius transformations are special because if we do one and then reverse the operation we get the same result as if we did it in the opposite order. Also, doing a sequence of these transformations will result in a series of equivalent transformations. 

The group of Mobius transformations is denoted by M  and has 8 elements that can be combined in pairs to produce an equivalent transformation; this is important to note because a Mobius transformed figure can have up to 6 points that are considered exceptional points.

How is Mobius Transformation related to other concepts in Mathematics?

If we look at our Mobius strip as a piece of paper, it would have the shape of a normal bezier curve which is another concept in mathematics. Bezier curves are artifacts that arise from Mobius transformations in 3D. There is one important difference between the Mobius transformation and Bezier curve however, Beziers only deform shapes on 2D surfaces, whereas the Mobius strips can deform 3D shapes such as spheres. The concept behind this fact is that points on an object can be thought of as being localized on an extended surface. In reality, each point is defined by two perpendicular line segments and two perpendicular planes intersecting between these line segments at right angles.

Significance of Mobius Transformation:

Mobius transformations are important for many reasons, the one that is probably most significant is that they are a link between Euclidean space and 3-manifolds. A Mobius strip can be deformed into a 3-manifold without distorting its surface area or volume.

Mobius transformations are also important in Electromagnetism because they give rise to the concept of electromagnetic field tensors and flux which only arises from an object being twisted in a Mobius fashion.

If two 2D surfaces, such as the inside and outside of a sphere, are twisted in such a way that they no longer touch each other when inverted (in Mobius fashion), they are said to be Mobius invariant.

Mobius transformations are also used to determine the best placement of detectors on a spacecraft to minimize self-intersection and maximize overlap between detection zones.

Conclusion:

The Mobius transformations are an essential concept in Mathematics, they are the link between Euclidean space and 3-manifolds. They give rise to the concept of electromagnetic field tensors and flux which only arise from an object being twisted in a Mobius fashion.

Mobius transformations are also important in Electromagnetism because they give rise to the concept of electromagnetic field tensors and flux which only arises from an object being twisted in a Mobius fashion. They also have many other applications, one of these is their use as a method to determine the best placement of detectors on a spacecraft to minimize self-intersection and maximize overlap between detection zones.