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Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

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58 views

Triangle mapping to triangle geometric construction

What Moebius transformation $$ w=R \, \frac {z-p}{z-q}$$ sends a given equilateral triangle $abc$ to another one $ABC$ in the plane ? Since their circum=circles can be inter-mapped through ...
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118 views

Solving the functional equation $\tau \left(\frac{-1}{z}\right) = - \tau(z)$

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. Find $\tau: \mathbb{H} \to \mathbb{C} $, holomorphic and non-constant, satisfying $\tau \left( \frac{-1}{z} \right) = - \tau(z)$. There ...
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1answer
358 views

Only three parameters to define a Möbius transformation

The Möbius transformation $$ w=f(z)=\frac{az+b}{cz+d}, $$ with $a,b,c,d$ complex constants is uniquely determined by three complex parameters. This is clear because we can divide $f(z)$ by, say $...
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1answer
173 views

finding a möbius transformation for power series

Let f be the function with domain $D = {{z \in C : |z| < 8}}$ given by f(z)= $\sum (i^nz^n)/(8^n) $ find a mobius transformation g such that f(z)=g(z) for all $z\in D$. Calculate f(i) and f'(0) I ...
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100 views

Find the Mobius transformation which maps these ordered points

So I have these four transformations given by: $$-1+i \mapsto -1 $$ $$0 \mapsto -i $$ $$1-i \mapsto 1$$ And I need to combine them into a single transformation. But every way I tried seems to get ...
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54 views

What exactly is an Appollonian Gasket?

I've been trying to learn about these fractals and haven't found many good sources, if the explanation I'm looking for is long-winded sources on the subject would also be appreciated. The first thing ...
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1answer
1k views

Finding the Fixed points of a Möbius Transformation

Is there a method or algorithm that can be used to find the Fixed points of a Möbius transformation on $\mathbb{C}$?
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1answer
227 views

function $f$ is conjugated by a Möbius transformation, fixed points

When a function $f$ is conjugated by a Möbius transformation $h$, i.e. $g \circ h = h \circ f$ any fixed point of $f$ is mapped to a fixed point of $g$ and the multiplier is equal. I understand why ...
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2answers
507 views

Inverting a Linear Fractional Transformation

I have a particular Linear Fractional Transformation from $\mathbb{C}$ to $\mathbb{C}$, I am using to solve Laplace's equation, and I was hoping to find the inverse of this transformation. Is there a ...
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2answers
271 views

How to combine these transformations into a single Mobius transformation [closed]

So I have these four transformations given by: $$z \mapsto z - \frac i 2$$ $$z \mapsto -z $$ $$z \mapsto \frac{z-i}{z+i}$$ $$z \mapsto z+1 $$ And I need to combine them into a single ...
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2answers
347 views

Transitivity of mobius groups

I am trying to understand the transitivity of m\"obius groups. I know that mob$(\mathbb{\bar{C}})$ acts transitively on the set of triples of distinct points. Does that mean that it also acts ...
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Given a geodesic $I$ and a point $p$ not on $I$, there is a point $q$ on $I$ such that the geodesic joining $p$ and $q$ meets $I$ orthogonally.

The theorem I want to prove is this Given a geodesic $I$ in hyperbolic space and a point $p$ not on $I$, there is a unique point $q$ on $I$ such that the geodesic joining $p$ and $q$ meets $I$ ...
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1answer
387 views

How to show that Mobius transformation is surjective on the unit disk?

For $z\in \hat{\mathbb{C}}$, let: $$T(z)=\frac{az+b}{cz+d}~~~~~~~~~~~~~~~~ad-bc=1$$ be the mobius transformation on the Riemann sphere. 1) Show that $T:\hat{\mathbb{C}}\longrightarrow \hat{\mathbb{...
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Simplifying the “cat's eye” function $f(z) = \frac {(e^x + 1)z + (e^x - 1)} {(e^x - 1)z + (e^x + 1)}$.

So I'm trying to find a solution of this function for $z = e^{i\theta}$ and for $x \in \Re$ and $\theta \in \Re$ $$f(z) = \frac {(e^x + 1)z + (e^x - 1)} {(e^x - 1)z + (e^x + 1)}$$ First of all, ...
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1answer
869 views

Finding the image of this line under 1/z

Let $T(z)=1/z$. Find the image of $y=2x+1$ under $T$. I assume x and y are the real and imaginary part of z. Basically what I need was let $w=1/(x+(2x+1)i)$ then multiplied by the complex conjugate ...
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3answers
2k views

Why does the inverse of a mobius transform not get divided by the determinant?

for a mobius transform $\frac{az+b}{cz+d}$ it's inverse is given by $\frac{dw-b}{-cw+a}$ since this can be put into a matrix form why is there not a requirement for the inverse to be divided by the ...
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Geometric imports

In a theorem of mobius transformation, we have to prove that every mobius transformation is the resultant of mobius transformations with simple geometric imports. What is the meaning of simple ...
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1answer
128 views

Complex mapping of an annulus problem

I am having a hard time understanding how to set up this map. $f$ maps the annulus holomorphically onto itself but permutes the inner and outer boundaries $|z| = a$ and $|z| = b$, $0 < a < b$. ...
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87 views

Conformal map from a trapezoid like subset of a unit disk to the upper half plane.

I'm having some trouble with finding a conformal map for this problem. Find a holomorphic one-to-one map from the open set bounded by the unit semicircle $|z| = 1$, ${\bf Im}\, z > 0$ and the ...
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1answer
58 views

Möbius transformation, unit disk to real axis [closed]

$$\begin{align}i &\to 1 \\1 &\to \infty\\-1& \to 0\end{align}$$ Unusually use the cross product but I can't do this with the infinity. I know it's the unit disk to the real axis.
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1answer
65 views

The images of disjoint regions separated by a circle or line are disjoint under a linear fractional transformation

It is known that linear fractional transformations (LFTs) take lines and circles to either a line or a circle. $$ T: \{ \text{ Line or Circle } \} \mapsto \{ \text{ Line or Circle } \}$$ I would ...
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1answer
94 views

Fundamental domain for (2,3,3) triangle group

I have 2 mobius transformations that generate the tetrahedral symmetries (all 12 of them). This corresponds to the A4 subgroup of S4, I believe. Expressed as matrices, they are: U = [.5(1-i), .5(1-i)...
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1answer
113 views

Mapping of a horizontal line $y = c$ by $w = \frac{1}{z}$ onto which region?

I was trying to solve this problem , so $y = c$. Suppose for ease let us take $c > 0$. so $w = \frac{1}{z} = \frac{1}{x + iy} = \frac{x - iy}{x^2 + y^2} = \frac{x - ic}{x^2 + c^2} = \frac{x}{x^2 ...
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115 views

Automorphism groups of domains of the complex plane

What is the automorphism group for $\mathbb{C}\setminus \{0\}$? Show that it has two connected components, and that the connected component containing the identity is transitive. What is the ...
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2answers
744 views

Möbius transformation sending line and circle to concentric circles

I want to find some Möbius transformation sending the line $l$ defined by $Re(z) = 5$ and the circle $\vert z\vert = 4$ to concentric circles in the complex plane. I know that Möbius transformations ...
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2k views

Complex modulus of $\left|\frac{-3z+2i}{2iz+1}\right|$ given that $\left|z\right|=\frac{1}{\sqrt3}$

The following question was on my first year algebra exam way back in 1989. If $\left|z\right|=\frac{1}{\sqrt3}$, then find $\left|\frac{-3z+2i}{2iz+1}\right|$. I couldn't figure it out then, and ...
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1answer
447 views

Find a fractional linear transformation $f$ that maps $\{z:\ Im(z)>0\}$ onto $\{z: |z|<1\}$ and such that $f(i)=0$ and $Re (f'(i))=0.$

Find a fractional linear transformation $f$ that maps $\{z:\ Im(z)>0\}$ onto $\{z: |z|<1\}$ and such that $f(i)=0$ and $Re (f'(i))=0.$ I consider $x$-axis maps to the unit circle. First, let $f(...
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79 views

Möbius squared, a better illustration or not?

Making a Möbius sling from a slip of paper makes very different effects on the direction that gets the the short ends (North - South) and direction that get the long sides (East – West). The North – ...
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2answers
312 views

Is'nt the paper Möbius strip a 3d object that is very different from the 2d Möbius band?

The reason why the paper Möbius strip is twice as long as the paper strip used, is, as far as I can underständ it, that paper has two more sides, apart from the two dimensional. A common paperslip cut ...
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63 views

mobius transformation 2

I need a Möbius transformation from the set of $z$s contained in the unit circle centered at the origin with real part greater than $0$ to the infinite sector bounded by the lines making $45$ degrees ...
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1answer
185 views

Proving a Mobius function forms a group under compositon

So I'm trying to prove a mobius function forms a group under composition. I know this has been asked (3 years ago) here: Proving the set $\mathcal H$ of Möbius transformations is a group under ...
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1answer
130 views

Moebius circle to circle operators

To map $$ (x-h)^2+(y-k)^2 = r^2 $$ to $$ (x-H)^2+(y-K)^2 = R^2 $$ in the complex plane is there a easy/quick way to determine $ a,b,c,d $ in terms of $ A,B,C,D $ the Moebius transformation $ w =...
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What should I learn to be able to study Mobius transforms?

I'm currently a student who is very interested in the Mobius transform. However, I tried reading some papers on the topics, and with my current knowledge of maths, I am not able to completely ...
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1answer
437 views

Fractals and Kleinian groups - Rendering the limit set

I am talking about this: I recently read the book "The fractal geometry of nature" by Benoit B. Mandelbrot. There was one particular fractal I found very beautiful: A limit set of some group of ... ...
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2answers
2k views

Find the image of the Möbius transformation

My question is: Find the image of the area $C=\{z\in \mathbb{C}:|z+3|\geq 3\}$ of the Möbius transformation $w=f(z)=\frac{z}{z+6}$. I have drawn the image in the z-plane and then taken three points $...
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1answer
361 views

Möbius transformation, determine the image area for the w-plane

Can someone help me with this question. Let $D\{ z\in \mathbb{C}: Re(z)<0,|z-1|<2\}$ and look at the Möbius transformation $f(z)=\frac{z-i}{z+1}$. Determine the image area $D'=\{f(z):z\...
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1answer
297 views

The riemannian metric on the unit disk of complex form.

Claim: Suppose that $f$ is a conformal map from the unit disk to the unit disk, $f(z_1) = w_1$ , $f(z_2) = w_2$, then we it can be derived that $|\frac{z_1-z_2}{1-\bar z_1 z_2}| = |\frac{w_1-w_2}{1-\...
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1answer
121 views

A dilation is a Mobius transformations

I want to show that for each $a \in \mathbb{R}_{>0}$ the dilation $\vec{p} \to a\vec{p}$ is a Mobius transformation. I'm told that this can be done by composing the inversions $I_{0,r_1}$ and $I_{...
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1answer
412 views

How can I show the unit circle in z-plane maps to a line?

For example, it is known that if |c| = |d|, then the linear fractional transformation: $w(z)=\frac{az+b}{cz+d}, ad-bc \neq 0$ should map the unit circle ($z=e^{i \theta})$ to a line. How would I ...
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1answer
248 views

Mobius transformation that maps interior of a circle to a half plane bijectively

Construct a Mobius transformation that maps the interior of the circle $\{z \in \mathbb{C}:|z-3|=2\}$ bijectively onto the half plane $\{z \in \mathbb{C}:Re(z)<-1 \}$. I drew the two graphs but I ...
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1answer
188 views

Working with Mobius Transformation

I want to find the transformation that maps $z_1 = 0,\,z_2 = 1,\,z_3 = \infty$ to $w_1 = 1,\,w_2 = 1+i,\,w_3 = i$. Under this mapping what is the image of the line $\text{Im}z = \text{Re}z$, the real ...
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84 views

How is this picture regarding Möbius transformation formed?

The following picture is from the Wikipedia article on Möbius transformation. It is explained under the picture that Pre-images of the unit circle are circles of Apollonius with distance ratio $c/...
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1answer
154 views

Image of a circle under a map

While trying to show (explicitly) that the image of the circle $|z-\frac{1}{2}|=\frac{1}{2}$ under the map $w=\frac{1}{z}$ is the line $u=1$, a few questions arose. (Here I put $w=u+iv$.) Here is ...
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27 views

If two Möbius maps are equivalent, then they relate by constant

Let $\mu_1=\frac{a_1z+b_1}{c_1z+d_1}$ and $\mu_2=\frac{a_2z+b_2}{c_2z+d_2}$. Then, if $\mu_1=\mu_2$, there exists $0\ne\kappa\in\mathbb{C}$ such that $a_2=\kappa a_1$, $b_2=\kappa b_1$, $c_2=\kappa ...
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211 views

Why is Möbius map not holomorphic on $\mathbb{C}_\infty$?

Let $$\mu(z)=\frac{az+b}{cz+d}$$ be a Möbius map. Then one can show that $\mu$ is holomorphic on $\mathbb{C}\backslash\{\frac{-d}{c}\}$ since $\mu$ can be decomposed into four holomorphic functions. ...
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1answer
100 views

Determining where a Möbius map takes annuli

I'm somewhat puzzled as to how to determine where a certain Möbius map takes regions. For example, consider the annuli $$A_{1,2}:=\{z\in\mathbb{C}:1<|z|<2\}$$ $$A_{\frac{1}{2},1}:=\{z\in\mathbb{...
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61 views

Möbius maps preserve cross-ratios [duplicate]

My question is different from this question since my question asks to clarify the proof, while the other question asks to check if the proof is correct. Let $z_1, z_2, z_3\in\mathbb{C}_\infty$ be ...
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1answer
248 views

Find Möbius transformation fixing $i$,mapping im-axis to im-axis and $\vert z-i\vert < 2$ to the upper half plane,

I want to find a Möbius transformation $T$ such that T fixes $i$, Sends the region $\vert z-i\vert < 2$ to the upper half plane, Maps the imaginary axis onto itself I'm really stuck on this even ...
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1answer
138 views

Möbius map on open unit disk which goes to open unit disk

Consider the Möbius map $$m(z) = e^{i\theta}\frac{z-\alpha}{\bar{\alpha}z-1},$$ where $\theta$ is a real number. Let $z$ be in the open unit disc $D=\{z\in \mathbb{C}:\left|z\right|<1\}$. What I ...
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1answer
174 views

$GL(2, \mathbb{C})$ and $SL(2, \mathbb{C})$ isomorphic to Möbius group?

I'm wondering if homomorphisms $\Phi:GL(2,\mathbb{C})\to \mathcal{M}$ and $\Theta:SL(2,\mathbb{C})\to \mathcal{M}$ are isomorphisms. One can prove that $\Theta$ is sujective, but then $\Phi$ must also ...