Linear fractional transformation

In summary, the conversation discusses the process of determining a linear fractional transformation for a circle that maps onto a line or vice versa. The suggested method is to pick three distinct points on the circle and map them to three distinct points on the line. The conversation also touches on the concept of cross ratio and the importance of fixing orientation when choosing points. The conversation also briefly mentions the possibility of using chat programs to further discuss the topic. However, it is clarified that there is no linear fractional function that can directly map a line onto a circle.
  • #1
sweetvirgogirl
116
0
sooo ...
i am kind of clueless about how to determine a linear fractional transformation for a circle that maps on to a line or vice versa ...


like i do *kinda* get how to map a circle on to a circle ... or a line on to a line ...
 
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  • #2
Well, a line is just a circle that passes through the point at infinity! So...
 
  • #3
Hurkyl said:
Well, a line is just a circle that passes through the point at infinity! So...
the way prof showed it in class ... he said pick three points on the circle and send it to three points on the line (or vice versa if we are trying to map a line to a cirle) ... but i am not sure how do i know what those three points are going to map to ...

on a circle |z| = 1, we could send i to infinity and -i to infinity and then the third point? maybe i am taking a wrong approach ... or maybe the prof didnt mean to tell us to take this approach and i misunderstood him
 
  • #4
sweetvirgogirl said:
on a circle |z| = 1, we could send i to infinity and -i to infinity and then the third point? maybe i am taking a wrong approach ... or maybe the prof didnt mean to tell us to take this approach and i misunderstood him

A linear fractional transformation sends only one point to infinity, you can't send two there. You have to take 3 distinct points on your circle and map them to 3 distinct points on your line.
 
  • #5
shmoe said:
A linear fractional transformation sends only one point to infinity, you can't send two there. You have to take 3 distinct points on your circle and map them to 3 distinct points on your line.
oops typo ... i meant we could send i to infinity and -i to - infinity and then the third point?

(-infinity and infinity are different, right?)
 
  • #6
No, infinity and -infinity are the same.
 
  • #7
AKG said:
No, infinity and -infinity are the same.
woopsie ... let's restart then ...

I send i to infinity, -i to zero ... what about the third point?
it has to do something with fixing orientation, right? how does that work?

(like i know the third point can not be totally arbitrary like the first two points were, right?)
 
  • #8
The third point just has to be on the 'target' line. If you're going for the real axis, you could take the third point to be 1. Have you seen the cross ratio?

Different choices of the third point on the real axis will get different transformations, but will still send your circle to the real axis.
 
  • #9
shmoe said:
The third point just has to be on the 'target' line. If you're going for the real axis, you could take the third point to be 1. Have you seen the cross ratio?

Different choices of the third point on the real axis will get different transformations, but will still send your circle to the real axis.
no we haven't covered cross ratio in class yet.

so i *sorta* get how to send a circle to a line ... ...
the circle |z| = 1 and the line Re((1+i)w) = 0 ... so i sent 1 to infinity, -1 to 0 and i = -1 and i came up with i*(z+1)/(z-1) ...
the book has the answer as (1-i)*(z+1)/(z-1)
i want to make sure my answer is right ...

also... to send a line to a circle ... i pick three arbitrary distinct points on the line and send them to three distinct points on the circle?

also ... if you have free time and you don't mind talking to a stranger over yahoo/aim ... please let me know ... thanks! :)
 
  • #10
sweetvirgogirl said:
so i *sorta* get how to send a circle to a line ... ...
the circle |z| = 1 and the line Re((1+i)w) = 0 ... so i sent 1 to infinity, -1 to 0 and i = -1 and i came up with i*(z+1)/(z-1) ...
the book has the answer as (1-i)*(z+1)/(z-1)
i want to make sure my answer is right ...

-1 is not on that line.

sweetvirgogirl said:
also... to send a line to a circle ... i pick three arbitrary distinct points on the line and send them to three distinct points on the circle?

Yes.

sweetvirgogirl said:
also ... if you have free time and you don't mind talking to a stranger over yahoo/aim ... please let me know ... thanks! :)

I don't have any chat programs, sorry.
 
  • #11
shmoe said:
-1 is not on that line.
how exactly do you determine that? coz the way i saw it ... -1 was on the line ... maybe i need some rest
 
  • #12
If -1 were on that line, then -1 would be one of the w such that Re((1+i)w) = 0, i.e. we would get:

Re((1+i)(-1)) = 0
Re(-1-i) = 0
-1 = 0
 
  • #13
There is no linear fractional function that maps a line on to a circle, or a circle on to a line, (unless the circle is a point), since linear fractional function preserves convexity. i.e., image of any convex set is convex , and the inverse image of any convex set is convex. See Stephen Boyd, Lieven Vandenberghe: Convex Optimization, Page 39
 
Last edited:

1. What is a linear fractional transformation?

A linear fractional transformation, also known as a Möbius transformation, is a mathematical function that maps complex numbers to other complex numbers. It is a type of rational function that can be expressed as a ratio of two linear polynomials.

2. How is a linear fractional transformation represented?

A linear fractional transformation can be represented as f(z) = (az + b)/(cz + d), where a, b, c, d are complex numbers and z is the input complex number. This representation is also known as the standard form of a linear fractional transformation.

3. What are the properties of a linear fractional transformation?

The main properties of a linear fractional transformation are: linearity, meaning that the transformation preserves straight lines; conformality, meaning that angles are preserved; and orientation-preserving, meaning that the transformation preserves the counterclockwise orientation of curves.

4. What are some applications of linear fractional transformations?

Linear fractional transformations have various applications in mathematics, physics, and engineering. They can be used to solve problems in complex analysis, conformal mapping, and differential equations. In physics, they are used to model physical systems and in engineering, they are used in control systems and signal processing.

5. Can any complex function be represented as a linear fractional transformation?

No, not all complex functions can be represented as linear fractional transformations. Only functions that are bijective, meaning they have a one-to-one correspondence between input and output, can be represented in this form. Additionally, functions that are not analytic, meaning they have discontinuities or singularities, cannot be represented as linear fractional transformations.

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