- #1
MaxJasper
- 1
- 0
Conformal Mapping for Transforming Regions: Finding a Function
- Thread starter MaxJasper
- Start date
-
- Tags
- Conformal mapping Mapping
In summary, the conversation discussed finding a conformal mapping transform function to map a specific region in the z-plane to the interior of a unit circle in the w-plane. The most general method is a Schwarz-Christoffel mapping, but it is not always solvable analytically. An alternative approach suggested is to split the domain in two and use a mapping from the unit disk to the unit square. However, this method also requires a Schwarz-Christoffel mapping and can be challenging.
- #2
meldraft
- 281
- 2
The most general way to calculate a conformal mapping is a Schwarz-Christoffel mapping. This integral however can't be solved analytically in most cases.
You can split the domain in two along the imaginary axis and try to map the semi circle into one half, and then mirror it to get the other half. I know that you can get that half-shape through ζ=z^2 if you original domain is a square. You can see it here (p. 246):
http://www.math.umn.edu/~olver/pd_/cm.pdf
All you need then is a mapping from unit disk to unit square. One way you could (maybe) do this is by starting with the half-disk, tranform it to a half-plane, and then fold it to a square using a Schwarz-Christoffel mapping. I think that this case can be solved analytically. You then mirror your domain along the imaginary axis and you're done. This is easier said than done of course, but that is the nature of conformal mappings
You can split the domain in two along the imaginary axis and try to map the semi circle into one half, and then mirror it to get the other half. I know that you can get that half-shape through ζ=z^2 if you original domain is a square. You can see it here (p. 246):
http://www.math.umn.edu/~olver/pd_/cm.pdf
All you need then is a mapping from unit disk to unit square. One way you could (maybe) do this is by starting with the half-disk, tranform it to a half-plane, and then fold it to a square using a Schwarz-Christoffel mapping. I think that this case can be solved analytically. You then mirror your domain along the imaginary axis and you're done. This is easier said than done of course, but that is the nature of conformal mappings
Last edited by a moderator:
Related to Conformal Mapping for Transforming Regions: Finding a Function
What is conformal mapping?
Conformal mapping is a mathematical technique used in complex analysis to map one region in the complex plane to another while preserving angles. In other words, it is a way to transform a complex function into a different shape without changing the way it behaves locally.
How is conformal mapping used in science?
Conformal mapping is used in various fields of science such as physics, engineering, and geology. It is often used to map physical systems, such as fluid flow or electromagnetic fields, onto simpler shapes to make calculations and predictions easier. It is also used in the study of fractals and in the design of computer graphics.
What are some real-life applications of conformal mapping?
Conformal mapping has numerous practical applications, including in the design of aircraft wings and in the analysis of electrical circuits. It is also used in cartography to accurately map the curved surface of the Earth onto a flat map, and in weather forecasting to map atmospheric conditions.
What are some limitations of conformal mapping?
One limitation of conformal mapping is that it only works in two dimensions. It also cannot be used to map regions with singularities, such as poles or branch points. Additionally, conformal mapping may not be able to accurately preserve distances, areas, or shapes in some situations.
How is conformal mapping related to other mathematical concepts?
Conformal mapping is closely related to other mathematical concepts such as analytic functions, complex integration, and the Cauchy-Riemann equations. It is also connected to the concept of conformal symmetry, which describes the properties of systems that are invariant under conformal transformations.
Similar threads
-
Topology and Analysis
-
Topology and Analysis
-
Topology and Analysis
-
Topology and Analysis
-
Topology and Analysis
-
Topology and Analysis
-
Topology and Analysis
-
Calculus and Beyond Homework Help
-
Topology and Analysis
-
Advanced Physics Homework Help