Mobius Transformations and Geometric Interpretations

In summary: What's the general form of a circle?In summary, the conversation discusses how any transformation can be split into a Translation, Dilation-Rotation, Inversion, and Translation. It also explores how this decomposition can be used to show that any straight line or circle is transformed into another straight line or circle when applying a mobius transformation. The conversation also touches on the general form of a circle and how inversions can be used to transform a straight line into a half circle or an entire disk.
  • #1
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Homework Statement


a) Show that you can split any transformation into a Translation, Dilation-Rotation, Inversion, and Translation.

b) Show using part a) that any straight line or circle is send to a straight line or circle when applying the mobius transformation.



Homework Equations



Mobius transformation is of the form:
(az+b)/(cz+d)
WHERE:
ad-bc != 0



The Attempt at a Solution


I believe I finished part a (mostly) though I'm a little unsure how to do the simple case.

First if c!=0 then we can split up the equation (az+b)/(cz+d)

into:

(a/c) - (ad-bc)/c^2 * 1/(z+d/c)
Thus
T_4(z) = z + (a/c)
T_3(z) = z * -(ad-bc)/c^2
T_2(z) = 1/z
T_1(z) = z + (d/c)

Where (az+b)/(cz+d) = T_4(T_3(T_2(T_1(z))))

If c = 0 then d!=0 and the mobius transformation takes the form:
(az+b)/d = (a/d)z + (b/d) which is just a rotation-dilation and a translation. Though I have no clue how to show this as 4 different functions (one of which being an inversion). I'm thinking that's its ok for me to leave my answer for part a as is, though if anyone has any hints on finding 4 functions for this then by all means don't hesitate to tell me to keep looking.

b) This is where I'm having trouble.

Translations don't change the geometric structure so I'm not going to discuss them.

Inversions send z to z_bar / |z|^2 ~ Ie: Flips z over the real axis and then scales them to 1/|z|.

I can see how this may send a straight line to a half circle.. though I don't see how it could possibly encompass an entire disk.

Rotation-Dilation rotates z by some theta and then scales it by some radius.

Basically I'm a little lost on how to start on these, any hints you can provide would be appreciated.
 
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  • #2
Your part a) is fine. When c=0, you can't decompose the mobius transformation in the way they suggest. For b) what part don't you get? You seem to have the right idea for translations and rotation-dilations. It's just inversions, right?
I can see how this may send a straight line to a half circle.. though I don't see how it could possibly encompass an entire disk.
Why?
 

1. What are Mobius transformations?

Mobius transformations, also known as linear fractional transformations, are complex functions that map the complex plane to itself. They are expressed as ratios of two linear functions, and can be used to transform geometric shapes in the complex plane.

2. What are some common geometric interpretations of Mobius transformations?

Some common geometric interpretations of Mobius transformations include rotations, translations, dilations, and inversions. These transformations can be used to manipulate shapes such as circles, lines, and parabolas.

3. How do Mobius transformations preserve angles?

Mobius transformations preserve angles because they are conformal, meaning they preserve the local angle at each point in the complex plane. This property makes them useful for preserving the overall shape of a geometric figure while transforming it.

4. Can Mobius transformations be applied to three-dimensional objects?

No, Mobius transformations can only be applied to objects in the complex plane. They are not applicable to three-dimensional objects because they are defined using complex numbers, which do not have a natural extension to three dimensions.

5. What are some real-world applications of Mobius transformations?

Mobius transformations have applications in various fields such as computer graphics, physics, and engineering. They are commonly used in computer algorithms for creating 3D graphics, and also have applications in modeling fluid flow and electrical circuits.

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