Questions tagged [lft]

For questions about linear fractional transformations.

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Converting a Linear Fractional Transformation as a Hyperbola

I'm trying to show algebraically that a Linear Fractional Transformation of the form $$f(x)=\frac{(ax+b)}{(cx+d)}$$ can be written as hyperbolas of the form $$(x-h)(y-k)=m$$ I started by expanding ...
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1answer
68 views

how to draw LFT?

How to draw this dark area to $w=\frac{z+1}{z-1}$ so difficult to draw LFT...
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2answers
191 views

Reference request: Where is this trigonometric identity found?

[Note that this is a reference request; I already know a couple of routine ways to prove the identity.] In April I posted this answer. Then yesterday I had occasion to conjecture that in general $$ \...
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0answers
44 views

Linear fractional transformations in several variables?

Is there a general theory of functions of several variables that are linear fractional transformations in each variable separately, and treating their application to geometry of circles, spheres, tori,...
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1answer
2k views

Condition for points to be concyclic

I want to prove that if $\dfrac{z_1-z_4}{z_1-z_2} \times \dfrac{z_2-z_3}{z_4-z_3}$ is real, then the four complex numbers are concyclic. Now I'm aware that this can be done by drawing them up ...
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0answers
116 views

Reference request: certain special LFTs

$\newcommand{\cs}{\operatorname{cs}}$ My actual question is at the bottom. Let $$ a\diamond b = \frac{a+b}{1+ab} $$ and $$ a\dagger=\dfrac{1-a}{1+a}. $$ Then $$ a\dagger\dagger = a $$ and \begin{...
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1answer
125 views

Showing reflections are hyperbolic isometries in $\mathbb{D}$.

I am interested in showing that isometries in $\mathbb{D}$ are either conformal self-maps in $\mathbb{D}$ or they are compositions of conformal self-maps with $z\mapsto \bar{z}$. It is given that ...
3
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1answer
451 views

The fixed points of analytic self-maps of $\mathbb{D}$

So far, I have assumed that $z_1$ is a fixed point of an analytic self map of $\mathbb{D}$. Then, I summoned the conformal self map of $\mathbb{D}$, $\phi$ to take $z_1\to 0$. It follows from Schwarz ...
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3answers
79 views

Is there a name for this particular linear fractional transformation?

Is there a conventional name for this function? $$ \begin{align} g(t) & = \frac{1+it}{1-it} \\[15pt] & = \frac{1-t^2}{1+t^2} + i\frac{2t}{1+t^2}. \end{align} $$ This function comes up from ...
6
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1answer
326 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
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0answers
209 views

Möbius tranformation

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if $c\...
3
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0answers
114 views

Characterizations of a linear fractional transformation

Consider the function $$ g(t) = \frac{1+it}{1-it} = \frac{1-t^2}{1+t^2} + i \frac{2t}{1+t^2}. $$ (The second equality holds except when $t=i$.) It seems to be widely known that this function is the ...
2
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1answer
282 views

Tangent half-angles and linear fractional transformations

Suppose $z=x+iy$, $x$ and $y$ are real, and $|z|=x^2+y^2=1$ so that $z=e^{i\alpha}$ for some real $\alpha$. Then for some real $\gamma$, $$ \begin{align} e^{i\gamma} = f(e^{i\alpha}) = f(z) & \...
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1answer
328 views

Linear Fractional Transformation help

I am given this problem from a past test that I am trying to figure out, I have tried finding the conjugate and going about it. But i am not getting the right transformation. Please help out. Show ...
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2answers
1k views

circles and linear fractional transformations

I'm realizing how little (in some respects) I know about circles. Here's something that emerged out of something I was fiddling with. My question is whether this is "well known" in the way that $...
4
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2answers
451 views

is this a linear fractional transformation(LFT)?

Now, suppose the transformation(in 2d) I am working with has two separate functions for $x$ and $y$. That is, the transformation for $x$ is of the form $$ x'=\frac{x}{x+y} $$ and the transformation ...