Conformal mapping. From an ellipse to a rectangle

In summary, the conversation discusses the possibility of transforming an ellipse into a rectangle and the use of conformal mappings. The possibility of such a transformation is explored, with the understanding that conformal mappings preserve angles except at four points. The conversation also mentions a book by Kreyszig that discusses the transformation from a rectangle to an ellipse, but notes that it is not conformal somewhere. A potential solution is suggested using Schwartz-Christoffel.
  • #1
traianus
80
0
Is it possible to transform an ellipse

x^2/a^2 + y^2/b^2 = 1 ("a" minor or major semiaxis)

Into a rectangle?
If so, how can I do it? I am not very familiar so please explain all the details. I know the transformation from a circle to an airfoil, but not this one.
 
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  • #2
Conformally? I don't think so. Conformal mappings preserve angles.
 
  • #3
I know that there is a transformation from a rectangle to an ellipse (book advanced enginnering mathematics by Kreyszig) but it is not conformal somewhere
 
  • #4
Like I just said, conformal mappings preserve angles. There are at least four places where a rect-angle ('rect' means 'right', 'angle' means 'angle') cannot be mapped conformally to an ellipse. Are there any right angles on the boundary of an ellipse?
 
  • #5
AKG, what you are saying is obvious and I knew it. But, except these four points, is there a transformation?
 
  • #6
Are you really looking to turn an ellipse into a rectangle, or are you more interested in their interiors?
 
  • #7
I am trying to transform the ellipse into a rectangle
 
  • #8
traianus said:
I know that there is a transformation from a rectangle to an ellipse (book advanced enginnering mathematics by Kreyszig) but it is not conformal somewhere

Hi, I found your quote while doing a search for transforming a rectangle to an ellipse. I found the book you mentioned and could not find the information I am looking for.

Can anyone assist me in finding out how to transform a rectangle into an ellipse?

Thanks
 
  • #9
What you can do is map the ellipse to the real axis and map the real axis to a regular 4-sided polygon using Schwartz-Christoffel.
 

1. What is conformal mapping?

Conformal mapping is a mathematical technique used to transform a complex shape onto another shape while preserving angles. It is often used in physics, engineering, and other sciences to simplify complex systems and make calculations easier.

2. How does conformal mapping work?

Conformal mapping works by using complex functions to map points from one shape onto another. These functions are designed to preserve angles, so that the resulting shape is a scaled and rotated version of the original shape. The equations used in conformal mapping can be quite complex, but they are based on principles of complex analysis.

3. What is the importance of conformal mapping in science?

Conformal mapping is important because it allows scientists to simplify complicated systems and make them easier to analyze. It is also useful in visualization, as it can be used to create accurate maps or diagrams of complex shapes. In addition, conformal mapping has applications in fields such as fluid dynamics, electromagnetism, and quantum mechanics.

4. Can any shape be conformally mapped onto another shape?

No, not all shapes can be conformally mapped onto another shape. The shape must be "simply connected," meaning it has no holes or gaps. Additionally, the transformation must be one-to-one and onto, meaning each point in the original shape is mapped to a unique point in the new shape.

5. How can conformal mapping be used to transform an ellipse into a rectangle?

To transform an ellipse into a rectangle, a special type of conformal mapping known as a "Schwarz-Christoffel mapping" can be used. This mapping uses a complex function to transform the points on the ellipse onto the corners of a rectangle. The resulting rectangle will have the same aspect ratio as the original ellipse, but with straight sides instead of curved ones.

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