# Conformal mapping from polygon with circle segments

• Kurret
In summary, the conversation is about finding a conformal map from a polygon to the upper half plane using circle segments instead of lines. The Schwarz-Christoffel mapping is the closest option, but the speaker is seeking tips or references for further exploration.
Kurret
I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping.

Anyone has any tips?

I don't know about this subject, but I notice that you can find numerous hits by searching for the topic of conformal mapping and "circular arc polygons". (For example: http://en.wikipedia.org/wiki/Schwarzian_derivative mentions circular arc polygons.)

If you find a reference that seems to do what you want and have a specific question about it, there are probably forum members who can explain it.

## 1. What is conformal mapping?

Conformal mapping is a mathematical technique used to transform a complex shape onto another shape in a way that preserves angles between intersecting curves. This means that the map preserves the shape and size of small regions in the original shape.

## 2. How is conformal mapping used in transforming polygons with circle segments?

Conformal mapping can be used to transform a polygon with circle segments onto a different shape, such as a circle or a rectangle, while preserving the angles between the curves. This is useful in various applications, such as in engineering and physics, to simplify complex geometric problems.

## 3. What are the benefits of using conformal mapping?

One of the main benefits of using conformal mapping is that it allows for complex shapes to be transformed into simpler shapes, making them easier to analyze and understand. Additionally, conformal mapping can preserve important properties, such as angles and distances, which can be useful in various applications.

## 4. Are there any limitations to conformal mapping?

While conformal mapping can be a powerful tool, it does have some limitations. One limitation is that it can only be applied to certain types of shapes, such as polygons with circular segments. Additionally, conformal mapping may not preserve all properties of the original shape, such as area, which may be distorted in the transformed shape.

## 5. What are some real-world applications of conformal mapping?

Conformal mapping has many practical applications in fields such as engineering, physics, and mathematics. It is commonly used in the design and analysis of complex structures, such as buildings and bridges. In physics, conformal mapping is used to study the behavior of electric and magnetic fields around objects with complicated shapes. It also has applications in computer graphics and image processing, where it is used to transform and distort images in a controlled manner.

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