In this episode, we look through the geometric heart of complex analysis by exploring how the transformation
reshapes familiar curves in the complex plane.
By solving concrete examples of how this function acts on circles, lines, and
parabolas, we gain powerful insight into the essence of bilinear (Möbius)
transformations. This hands-on approach not only deepens our understanding of
the underlying geometry but also reveals the elegant unity between algebraic
formulas and geometric intuition.
If you find these ideas intriguing,
this episode is an excellent reason to consider picking up my book, where these
concepts are developed in even greater depth.
In “The Riemann Hypothesis Revealed,”
(my book) I explored in detail how the transformation
inverts points with
respect to the unit circle, providing valuable insight into geometric
properties of the complex plane. Now, let’s consider the function
By working through concrete examples of how this function transforms different curves in the complex plane, we’ll gain deeper insight into the fundamental nature of bilinear (Möbius) transformations and their geometric effects.
For example, we can determine the images of:
- Circles such
as
and 
- Families of
parallel lines like
- Lines of the
form
- Lines passing
through a specific point
- The parabola
Studying these mappings not only
deepens our understanding of complex analysis but also demonstrates the power
of geometric transformations in revealing new perspectives on familiar
mathematical objects.
Let 
. Then
so
Equivalently,
We’ll use this to map each curve in
the 
-plane to its image in the 
-plane.
Circles
In -variables:
In terms of 
:
So
So the image is the vertical line
These are circles through the origin
in the -plane; reciprocation sends them to
lines not through the origin.
Circles
Similarly:
So the image is the horizontal line
Pencil of
parallel lines
Equation:
Substitute:
Rewriting:
This is a circle:
Complete the square:
Thus, the image is a family of
circles through the origin, with centers on the line 
, radius 
. (Check: 
satisfies the equation.)
A pencil of parallel lines not
through the origin maps to a pencil of circles through the origin.
Pencil of lines
These are lines through the origin.
Equation:
So the image is again a pencil of
lines through the origin, with slope 
. Geometrically: lines through the
origin map to lines through the origin; the angle is reflected (because 
includes conjugation).
Pencil of lines
passing through a given point
Take all lines through 
. There are two types:
- The unique line through both
and : this
maps to a line through the origin (as above). - All other lines through but not through : each
such line maps to a circle through and
.
So the image of the pencil is a pencil
of circles through and 
(plus one line through the origin
corresponding to the line through and ).
This is the standard inversion
duality: lines not through the origin ↔ circles through the origin.
The parabola
Equation:
Substitute:
So the image is the curve
i.e.
This is not a conic; it’s a more
complicated algebraic curve.
Since , 
and
So we get the parametric view:
which satisfies the implicit relation
above.
The function (say 
)
is at the center of what we call
Bilinear transformations.
A composition of two translations like
a homothety like
and , leads to the well-known Möbius transformation:
where 
are complex numbers that
satisfy 
.
If 
,
is a constant (unless 
,
when it is undefined). Indeed, in the case of 
we have:
Translations,
reciprocations and homotheties are also called Möbius transformations. Actually, the set of all Möbius
transformations with the operation of composition (or product), forms a group.
For each element in the group there
corresponds an element 
called the inverse of , such that 
.
The inverse of
is 
.
Reading my book “TheRiemann Hypothesis Revealed”, you will realize that everything about Möbius
transformations
stems from the classical roots of Euclidean geometry—specifically, the
inversion of points with respect to a circle. This remarkable geometric
insight, to which I dedicate an entire chapter, leads directly to the deduction
of Möbius transformations in a way that, I believe, will truly blow your mind.
Inside you will find
the following result:
The Möbius
transformation that maps the unit circle onto itself is of the form:
The parameter 
influences how points
inside or outside the circle are mapped. If lies inside the circle
meaning 
, then since 
, all points with 
will be mapped to the
interior of the image circle. The situation reverses when 
.
The key insight lies in
how this result is established. A Möbius transformation is uniquely determined by its action on any
three given points in the extended complex plane 
.
Suppose we choose a point 
, 
, and consider the Möbius transformation 
such that .
By leveraging the
property of inversion (see “The Riemann Hypothesis Revealed”), we deduce that must map 
to 
, meaning that 
. The third point is chosen to be 
which satisfies 
.
This selection of three points fully
determines the transformation, illustrating the elegant structure and
flexibility of Möbius mappings.
Epilogue
The simple act of inversion—rooted in
classical Euclidean geometry—serves as the foundation for the rich theory of
Möbius transformations. Each mapping uncovers new perspectives on the complex
plane, illustrating how geometric transformations can illuminate the structure
of mathematical objects in surprising ways. As we continue to explore these
connections, we discover that the language of complex analysis is not just a
collection of formulas, but a vibrant bridge between geometry, algebra, and the
deeper mysteries of mathematics.
Comments
Post a Comment