What are conformal mappings?
Conformal mapping is a concept mostly used in physical applications involving fluid movement, heat conduction, aerodynamics, and electrostatic potential, but as a mathematical concept it can also be applied to any field where surface deformations are present.
Conformal mappings ─or conformal transformations, as they are also called─ are those functions that preserve angle values (or local shapes, with few exceptions at some isolated points) when mapping a domain to a range. The presence of angles implies that conformal mappings are present in 2D shapes in planar shapes.
A little review of functions and transformations
Let's review the basic concepts of what are mappings, but for a formal and wider introduction of the concept of mappings, see the Chapter 5, section 5.2.1 of Foundations of Transcomplex Numbers available for free download.
Definition. A map, also called a function or transformation, and symbolized by small letters such as f, g, etc., is a unique rule from a set A to a set B that assign each element a of A to exactly one element b of B. This is written as
or
and in simpler notation
The element b of B is called the mate (also called the image) under f of the element a of A. The subset of B of all mates of a map f is called the range of f. The subset of A of all the elements of A used by the rule f is called the domain of f.
A function is said to be one-to-one, denoted by (1 − 1), if for every two distinct elements of its domain, the function assigns two distinct elements as mates. In symbols, f is (1 − 1) if and only if
What this last formulas state is that under one-to-one mappings, different elements of a domain are mapped to different elements of the range and vice versa.
A mapping can be (1-1) even when not all the elements the domain are taken into account. What is important is that different elements of a set A,─the domain─, are not mapped to a same element of of the set B, ─the range.
From the standpoint of view of conformal mappings, the range and domain can be either ordered pairs of real numbers, or complex numbers (complex numbers can also be treated as ordered pairs of real and imaginary numbers).
The Golden EBook of Graphs of Mathematical Functions is full of examples of conformal mappings, some of them reproduced below. Chapter 5 of The Golden EBook ... is devoted to some conformal transformations that are usually found in almost every paper or textbook about complex analysis.
An example follows:
The complex sine transformation
The sine function is the transformation that produces the following all familiar sinusoidal curve y = sin( x ) shown below:
This graph takes all real numbers as domain ─here is the X-axis─ and produces real numbers also ─here the Y-axis.
This function is periodic because it repeats itself at a period of 2π. That is sin( x ) = sin( x ± 2πk ) for k = 0, ±1, ±2, ... When k = 0, the sine function takes the basic period; that is, the sin( x ) function is defined for x ≥ 0 and x < 2π; this is called the primary period. Within this range the function is also (1-1).
But the sine function can also de defined for complex numbers. In this case, the domain and the range are also complex.
The complex sine function is written as W = sin(U), where U is a domain of complex numbers x + iz.
Thus,
sin (z) = sin (x) cosh z + icos (x) sinh (z).
This formula is a consequence of the complex identity that states that
Therefore, the two parametric equations to plot the complex sine function are:
u = sin (z) = sin (x) cosh z and v = cos (x) sinh (z).
Similarly to its real-domain counterpart, the complex sine function is also periodic.
The resulting surface for the complex sine transformation is:
This is a composite figure that shows the domain and the range at the same time. In this isometric view, the orange rectangle at the center is the a square domain of complex numbers. In fact, this rectangle includes all the complex numbers x + iz, for -2 ≤ x ≤ 2 and -2i ≤ z ≤ 2i.
Since the rectangle is a smooth continuous region, the range is also a smooth continuous transformation of this rectangle. However, the magic of this mapping is that a flat region is transformed into a warped surface that maintains the angle-preserving feature we mentioned at the beginning.
To better appreciate how this map works, and how angles are preserved under the complex sine transformation, let us take a picture of a chessboard like the one shown below, and use it as the domain of the function.
This chessboard is full of right angles, so the resulting surface must also keep the right angles, no matter the contortions and warps it has. The board is labeled at left with numbers, and at the bottom with letters. For easier identification, the other two board borders are colored.
The transformed chessboard is shown below:
The next figure shows how a portion of the complex domain (a subset of the domain) is mapped. This figure uses more colors such that the red means "high" in the Y-axis (above), and the blue color means "low" (below).
See that a long strip parallel to the X-axis is mapped into a curved strip around the X-axis; making a sort of helicoid.
That is, a simple planar strip of the negative side of the imaginary Z-axis. For this picture, the complex values are: -4 ≤ x ≤ 4 and -2i ≤ z ≤ 2i, with 4 subdivisions per unit on each axis.
One last point is worth mentioning: every right angle (as any other angle) of the domain is preserved under the complex sine transformation, but the angles are preserved and mapped exactly on the surface, the same as the angles are on the planar region. This is mentioned in case some objection arises concerning that the angles doesn't seem exactly right angles in the figures.
If as domain we use a color picture, the result can be impressive, as this the following two pictures show:
Note, however, that the mapping is the same in the three figures above. All the figures obey to the same conformal transformation.
More about complex surfaces.
The figures were drawn with 4DLab-The Program , also available for free download.
For more graphics and mathematical discussions about other conformal transformations see Transcomplex surfaces in this website. For other sites with interesting conformal maps see Some quick links for comparison where you can compare the graphing of complex surfaces with the graphing of transcomplex surfaces.
How M. C. Escher mapped a plane into itself.
We cannot end this little journey into some of the transformation of the plane, without mentioning the artistic effort of Mauritus Cornelis Escher (1898-1972) in trying to capture the essence of the conformality.
Of special interest is the article by Sara Robinson titled: M. C. Escher: More Mathematics Than Meets the Eye . Escher is famous for his intricate and elaborated works, and there is plenty about him elsewhere, but this article is specially dedicated about the presence of conformal mappings in one of his lithographs: Print Gallery (1956) shown below at left. In this lithograph Escher tried to map everything within the borders of the picture into the center of the picture again, but instead of this there is a circular white patch that contains Escher’s monogram and signature. It seems that it was impossible for him to accomplish this task.
Possibly, Escher tried to make an more elaborate and intricate mapping based on the image that appeared "on the tins and boxes of Droste cocoa powder, one of the main Dutch brands" (1904). The continuous repetition of an image onto itself later became to be known as the Droste effect .
Another analysis of the same picture of Escher is Artful Mathematics: The Heritage of M. C. Escher .
Another link worth visiting is Escher and the Droste effect from a webpage from the mathematics department at the University of Leiden. Don't miss the video clips they made where you can clearly see that the blind spot that Escher left at the center of his Print Gallery is the place where the whole lithograph begins to repeat itself .
E. Pérez
jul-10
