I have been spending some time familiarizing myself with the basics of conformal mapping, and found myself somewhat stumped with the limitations of some of the methods I have encountered. Möbius transformations or Schwarz-Christoffel maps, for example, have very strict requirements on where and how they can map the unit disk.
My intuition, however, is that conformal maps should be a lot more general, and a lot more flexible than they are. Consider for example the shape to on the right side in the figure below. Imagine that the three blue closed lines correspond to topographic contour lines of a monotonous hill. Since the hill is monotonous, you can walk to the lowest contour and/or the summit from any point within the hill by walking perpendicular (summit paths: orange lines) up or down the contours (blue closed lines). Since the summit paths and topographic contours are always orthogonal to each other and exist at any point within the hill, the properties of a conformal map (as I understand them) are preserved. So shouldn't there be a conformal mapping from the unit disk (left) to this hill?
Is there something I have misunderstood?
Best Answer
I know of three methods for making a conformal mapping that is not as simple as Moebius transforms and (simple) applications of Schwarz-Christoffel. One reason that the examples you are seeing are "simple" is that the space of shapes (your blue curves) is infinite dimensional and does not have a nice structure (for instance, is not a vector space). Being infinite dimensional, one is inclined to use an infinite amount of data to specify an arbitrary shape -- every shape gives an infinite set of coefficients. However, not every infinite set of coefficients describes a shape -- it is easy for such a representation scheme to yield many objects that fail to be connected, which is a problem for this application.
The "nice" thing about Moebius transforms and Schwarz-Christoffel is that using either only requires a small (finite) amount of information. But this means that these methods cannot give maps that are too complicated. Although, one can increase the complexity via Schwarz-Christoffel by subdividing one's piecewise linear approximations more and more finely.
Luteberget has an overview of the three methods I list below.
(1) Using complicated Schwarz-Christoffel. See the work of Driscoll and Trefethen, for example, https://pdfs.semanticscholar.org/ec28/b851707a35630faf58fdb5690f31cc814b15.pdf , references thereto, and their subsequent work, e.g., https://arxiv.org/abs/1911.03696 .
(2) Use Stephenson's circle packing method, for example, http://www.cs.jhu.edu/~misha/Fall09/Stephenson97.pdf , and references thereto.
(3) Use Marshall's "ZIPPER" algorithm. Examples are visible here: http://sites.math.washington.edu/~marshall/zipper.html . More recent work on ZIPPER: https://arxiv.org/abs/math/0605532 .