Mobius Transformations, quick question concepts.

In summary, the condition ad-bc≠0 is necessary for a mobius transformation to be invertible. If ad-bc=0, the resulting mapping would be a constant, making it impossible to invert. This is why it is considered undefined and not simply an identity map.
  • #1
binbagsss
1,278
11
So a mobius transformation is defined as [itex]\frac{az+b}{cz+d}[/itex]=f(z).
Where ad-bc≠0.


My question is just deriving this condition ad-bc≠0.

I understand that the condition describes the case were the mapping leaves all points unchanged. This is described as undefined in some textbooks...(why is this, isn't it then just an identity map?)

But not by setting z=f(z).

I can't see why this condition would not equally be desribed by z=f(z)?

Many Thanks for any assistance.
 
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  • #2
If ad-bc=0 then f(z) = b/d which is a constant.
Therefore, the equation y=f(z) can be inverted with respect to z only if ad-bc is not zero.
 
  • #3
But I can't see why this condition would not equally be desribed by z=f(z)?
 
  • #4
binbagsss said:
But I can't see why this condition would not equally be desribed by z=f(z)?

I guess I don't see why anyone would describe that mapping where ad-bc=0 as 'leaving all points unchanged'. It doesn't. It maps everything into a constant.
 

FAQ: Mobius Transformations, quick question concepts.

1. What are Mobius Transformations?

Mobius Transformations are a type of mathematical transformation that involves mapping points on a plane to other points on the same plane. They are also known as conformal mappings or fractional linear transformations.

2. How do Mobius Transformations work?

Mobius Transformations are based on the concept of complex numbers and can be represented by a mathematical formula. This formula involves the use of a complex variable, which is a number with both real and imaginary parts. The transformation is achieved by multiplying the complex variable by a constant, adding another constant, and then dividing by a different constant.

3. What are some real-world applications of Mobius Transformations?

Mobius Transformations have a wide range of applications in various fields such as physics, engineering, computer graphics, and robotics. They are used to model and analyze complex systems, create visual effects in movies and video games, and design robotic movements.

4. Can you provide an example of a Mobius Transformation?

One example of a Mobius Transformation is the transformation that maps the unit circle in the complex plane to itself. This can be represented by the formula f(z) = (az + b) / (cz + d), where a, b, c, and d are constants. This transformation preserves the shape of the circle and can be used to create interesting patterns and designs.

5. Are there any limitations or restrictions to Mobius Transformations?

Yes, there are some limitations and restrictions to Mobius Transformations. For example, the constants used in the formula must be complex numbers and cannot be zero. Also, some transformations may result in singularities or points where the transformation is not defined. Additionally, Mobius Transformations are only applicable to points on a plane and cannot be extended to higher dimensions.

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