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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8
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3answers
164 views

Is a rational function which maps all circles/lines to circles/lines a Möbius transformation?

It is well-known that Möbius transformations map circles and lines to circles and lines. (Here and in the following, “line” means a line in the extended complex plane $\hat{\Bbb C}$, including the ...
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1answer
45 views

If $f$ is holomorphic in $D = \Bbb D\cap\{\Re(z)>0\}$, $f(a)=a$ for $a\in D$ and $f(D)\subset D$, how to show that $|f'(a)|\le 1$?

Consider $D = \{z\in\Bbb C : |z|<1,\;\Re(z)>0\}$. Take $a\in D$ and consider $f$ a holomorphic function in $D$ such that $f(a)=a$ and $f(D)\subset D$. How can we prove that $|f'(a)|\le 1$? I've ...
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1answer
70 views

Finding a Möbius transformation sending a set onto the open unit disk.

Let $g : G \longrightarrow \mathbb C$ be an analytic function on a region $G.$ Let $a \in G.$ Then for any $r \gt 0$ there exists a Möbius transformation $T_r$ such that $T_r \left (\mathbb C \...
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2answers
44 views

How to show directly that $T(G \setminus \{a\})$ is open?

Let $G \subseteq \mathbb C$ be open and $a \in G.$ Let $T$ be a Möbius transformation defined by $z \mapsto (z-a)^{-1}.$ Then show that $T(G \setminus \{a\})$ is open. I can able to conclude that by ...
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0answers
61 views

Is this a valid conformal map for my problem?

I need to find a conformal map $$f\colon \mathbb{D} \to \{z\;|\;|z|<1,\,z\notin \mathbb{R}_{\geq 0}\}.$$ This is an excercise from one of my books and their solution is completely different from ...
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1answer
42 views

How do I choose the three pairs of points for a moebius transform?

I somehow don't really understand how to choose the points for a moebius transform. I know that a moebius transform maps circles and lines to circles and lines and that it is a conformal(biholomorphic)...
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0answers
23 views

Möbius transformation of a 3D surface mesh

Given a mesh M with vertices V, edges E and faces F, I want to apply the Möbius transformation on it to generate new positions of the vertices V'. Can someone tell me the mathematical formulation/...
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0answers
29 views

Images and radii of circles under Möbius transormation

I've been trying to follow the working in this paper on Steiner Chains for a project. https://ima.org.uk/8218/exploring-steiner-chains-mobius-transformations/ Could someone help me understand how ...
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0answers
47 views

Isometry groups of $\mathbb H^n(R)$ (hyperbolic spaces)

I know that the group of isometries on $\mathbb H^n(R)$ is $O^+(n,1)$ (the orthochronous lorentz group) which i have proved using the Hyperboloid model. I have also been able to show for n=2 case ...
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29 views

How to plot points more evenly when transforming a straight line using the Möbius transformation $f(z)=\frac{1}{z}$?

I learned a bit about Mobius transformations. I am having the following problem: If I take the following points (in a straight line): $$1-1000i,1-999i,\dots,1+999i,1+1000i$$ And graph them after ...
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1answer
45 views

Mobius transformation between those two domains

I am trying to understand why there is no such mobius transformation from $A$ to $B$ . $$B= D_1(0) / 0 , A=\{z \mid \operatorname{Im}(z)+\operatorname{Re}(z)>0\}$$ I do not understand how do I ...
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0answers
28 views

Why there is not a mobius transformation between those two domains? What is the general approach should I use?

I am learning about conformal mapping, and I found it hard to answer question from the kind "determine if there exist mobius transformation between $A$ and $B$" for example: $$A=\left\{z \in ...
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2answers
69 views

Prove that a certain Möbius transformation looks like $f(z)=(az+b)/(\bar{a}z+\bar{b})$.

I have some trouble proving this: A Möbius transformation $f$ is such that $f(\mathbb{R}\cup\{\infty\})=S^{1}$ iff it looks like $$ f(z)=\frac{az+b}{\bar{a}z+\bar{b}} $$ with $a$ and $b$ in $\mathbb{C}...
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0answers
24 views

Does every one-one and onto analytic map on the unit disk send boundary to boundary?

Let $D$ be the unit disk and $f \in \text {Aut} (D).$ Then can we always say that $f(\partial D) \subseteq \partial D\ $? What I know is that if $f \in \text {Aut} (D)$ with $f(a) = 0$ for some $a \in ...
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0answers
31 views

Analytic Function on a Half Plane Bounded at the Boundary Must Be Bounded

I want to verify an argument that supposedly shows that any analytic function in, let's say, the right half plane that is bounded in the boundary must be itself bounded by the same bound. That is if ...
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0answers
28 views

The composition of inversions of two non-intersecting circles is hyperbolic

I am solving a question from Gareth A. Jones book Complex Functions. The question is: (i) Show that the circle \begin{equation}\label{1} a z \bar{z}+\bar{b} \bar{z}+b z+c=0 \quad(a, c \in \mathbb{...
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0answers
30 views

Show Möbius-Transformation with certain properties

I've got to show that such transformation $f$ with mapping points $f(0) = -1$, $f(i) = 0$, $f(+\infty) = 1$ has properties $f(D_1) = D_2$ where $D_1 = \{z \in \mathbb{C} \vert \quad \text{Im$(z)$}\geq ...
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2answers
44 views

Explicit formulas of how to extend $PSL_2(\mathbb{C})$ action of Möbius group on upper-half space model of $\mathbb{H}^3$?

EDIT: Solved, it was an algebra mistake. I posted the correct answer below. I'm interested in explicit formulas for the action of the isometries of the Mobius group on the upper-half space model of $\...
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2answers
43 views

Why is the complex conjugate function (on the upper half plane) not a Möbius transformation?

I want to show that the map $z\mapsto -\overline z$ is not a Möbius transformation. I've seen the answer to do something with analytic maps, which we haven't learned, so is there a more basic proof to ...
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0answers
38 views

Visualizing Projections on Riemann Sphere

I am studying Visual Complex Analysis by Tristan Needham. I was using pen and paper as well as some 'python' code to visualize the 2D mappings like $z \rightarrow e^z$ but now I came across a question ...
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32 views

Will $S(\infty)$ lie on the same circle as that of $S(\mathbb R)$ for every Möbius transformation $S\ $?

Let $S : \mathbb C_{\infty} \longrightarrow \mathbb C_{\infty}$ be a Möbius transformation. Does $S$ necessarily send $\infty$ to $\infty\ $? I am actually following Conway's book on Complex Analysis....
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0answers
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A question on the conclusion of the proof of "$Sl(2, \Bbb R)$ is transitive on the upper half plane".

$Sl(2, \Bbb R)$ is transitive on the upper half-plane. Here I have a question on the conclusion. I have proved that for any $w \in \Bbb H$ there exists $z \in \Bbb H$ s.t $A⋅z=w$ for some $A \in Sl(2, ...
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0answers
25 views

Mobius transformation mapping the upper half plane to itself different approach

I am trying to use the fact that every bi- holomorphic function from the unit disk to itself can be writen as $$ \varphi(z)=\frac{\lambda(z-a)}{1-\bar{a} z} $$ for every $a,\lambda \in D_1(0)$ to show ...
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1answer
34 views

General form of Matrix in SU(1,1),PSU(1,1) and Unit Disk Automorphisms

I have the definition of $SU(1,1)$ as $SU(1,1)=\{M\in M_{2}(\mathbb{C}) ; M^*JM=J, detM=1\}$ where $J=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, and $PSU(1,1)$ as the image of $SU(1,1)$ ...
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0answers
21 views

Logic about image of a Bilinear transformation.

I have to find image of $\{z\mid Re(z)\leq 0\}$ under a complex bilinear transformation which maps points $0,-i,\infty$ to $10,5-i5,5+i5$ respectively. I am thinking like, as all points $0,-i,\infty$ ...
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0answers
50 views

Conformal map from $\{z \in \mathbb{C}: \operatorname{Re}(z)>0, \operatorname{Re} (z^2)>1 \}$ to unit disk?

How to find the conformal map from $\{z \in \mathbb{C}: \operatorname{Re} (z)>0, \operatorname{Re}(z^2)>1 \}$ to unit disk? I want to first use $z^2$ to map it into $\{z \in \mathbb{C}: \...
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1answer
77 views

Properties of the Möbius transformation

I am given four points on a straight line, which are mapped to four points on a circle. I confirmed that the latter four points are concyclic by calculating their cross ratio, which is real. Does a ...
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0answers
32 views

Does series $\sum^\infty_{n=0}(\frac z{1+z})^n$ converge uniformly on compact subsets of $-\frac{1}{2}+\mathbb{E}$?

Let $\mathbb{E}$ denote the open right half-plane, that is $\mathbb{E}=-i\mathbb{H}.$ There are at least two answers on MSE showing ordinary convergence in $-\frac{1}{2}+\mathbb{E}=\{z:\Re z>-\frac{...
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1answer
61 views

Prove conjecture about circle pairing centers

I'm desperately trying to prove a statement which I need for some other results in my bachelor final project (I already spent several hours on this and it's driving me crazy). I would be very happy if ...
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2answers
49 views

Does a Möbius transformation map the diameter points of a circle to diameter points?

The Möbius transformation given by $(az+b)/(cz+d)$ maps a circle to circle or a line. Say we are given the unit circle centered at the origin and that the pole $-d/c$ does not touch the boundary of ...
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0answers
21 views

Complex function mapping second sphere tangent to unit sphere

Not sure about the question, please help correct it to convey properly or I could even delete if not meaningful. The common intersecting plane is through the north pole of top unit sphere. By Möbius ...
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1answer
59 views

Is there a conformal map that preserves the Poincaré disc and maps two points $z_1, z_2$ to different points $z_1', z_2'$?

Or does such a map not even exist? I think the wanted map has to merely translate/rotate/maybe reflect the underlying lattice. I would like to tessellate the hyperbolic plane. My idea is to draw the ...
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0answers
33 views

Iterating Mobius Transformation gives Identity

On AoPS, I found the problem: Find all $f=\frac{ax+b}{cx+d}$ with $a,b,c,d\in\mathbb R$ which when iterated $n$ times gives the identity (group theoretic order $n$). With some motivation to solve ...
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0answers
29 views

Fixed points of unit disk automorphism

Given mobvius transformations of the form $$f(z) = \eta\frac{a-z}{1-\bar a z}$$ with $\vert \eta \vert = 1$ and $a\in\mathbb{D}$ ($\mathbb{D}= $ open unit disk). I have to proof that if this ...
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1answer
49 views

Looking for a Conformal Map

Given the image, I would like to conformally map the plane minus the cut of this contour to a plane with a cut along the positive real axis say. As pointed out by @saulspatz Möbius transforms cannot ...
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1answer
59 views

Proving that the Möbius transformation is the identity

Given the Möbius-Transformation $T: \mathbb{D} \rightarrow \mathbb{D}, T(z)=\eta\frac{a-z}{1-\overline az} $ for some fixed $a \in \mathbb{D}, \eta \in \partial \mathbb{D} $ I am trying to show that ...
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0answers
18 views

Bibliography about Möbius transformations

I'm looking for some bibliography about möbius transformations and Möbius transformations on quaternions. I've been looking for some books, but I haven't been able to find them,so, if anyone could ...
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1answer
46 views

Conceptual understanding of the global triviality of $SL(2,\Bbb{R})$ with typical fiber $S^1$

In Saunders' book on jet bundles, the following example is given: Let $H=\{z\in\Bbb{C}\,:\,\text{im}(z)>0\}$ be the upper half plane regarded as a 2-dimensional real manifold, and consider the ...
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0answers
20 views

Mapping Im(z)>0 to a shifted circle - Möbius transformation

I want to map $Im(z)>0$ to $|z-i|<1$ using a Möbius transformation. I have a simple question regarding that. I know I can choose 3 points on the domain boundary and map them to the circle, but ...
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2answers
36 views

How can you write a Möbius transformation that maps the outside of a circle to the inside of the cirle?

I am supposed to map the outside of the unit disk in the first quadrant onto the unit disk. I don't understand how you map the outside of a disc ( or part disc) to the inside.
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0answers
38 views

mapping complement of half disk to unit disk

Let $K=\{ Im z \geq 0, |z| \leq 1 \}$ be the closed set of the intersection of the closed disk with upper half plane. I need to find a conformal map that takes complement of $K$, i.e. $\mathbb{C} - K$ ...
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0answers
22 views

conformal map from a disk minus a radial segment to unit disk

I need to find a conformal map from the disk minus a radial segment $\Omega$ to the unit disk $\mathbb{D}$, where $\Omega = \{ z \in \mathbb{C}: |z|<1, z \notin [\frac{1}{2},1) \}$. I have tried ...
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0answers
34 views

conformal map $\{ Re(z) > 0 : |z-1|>1 \}$ onto unit disk

How many conformal mapping we apply to map the exterior of the disk $\{z : |z-1|>1 \}$ in the right half plane to the unit disk?
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1answer
49 views

Möbius transformation of an area between two lines

$\DeclareMathOperator{\Re}{Re}$I wish to find where all the points in $$\left\{ z:\,\,\,-1<\Re\left(z\right)<0\right\} $$ are mapped to under $$f\left(z\right)=\frac{1}{z+1}$$ I found out $\Re\...
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0answers
17 views

writing holomorphic functions on unit disc as product of mobius maps

If $f$ is continuous on the closed unit disc and holomorphic in the open unit disc such that if $|z|=1$ then $|f(z)|=1$. Then we claim there are points $a_1, a_2,\cdots, a_n \in \mathbb{D}$, not ...
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2answers
80 views

Image under a Mobius Transformation

I am studying the Mobius transformation $$f(z) = \frac{z(1+i)}{z - i}$$ I need to find where the transformation maps the real axis, the imaginary axis and the unit circle. I know that Mobius ...
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0answers
30 views

automorphism group of the disc $\mathbb{D}-\{p,q\}$

The automorphism group of the punctured disc $A=\mathbb{D}-\{a\}$ consists of all holomorphic maps $f$ from $A$ onto itself, and since $A$ is bounded, then $f$ should be bounded, and so $a$ is ...
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1answer
53 views

finding an automorphism group of the unit disc

Can I find automorphism group of the domain $\mathbb{D} - a $ ? where $\mathbb{D}$ is the unit disc with center 0 in the complex plane and $a$ is a point in $\mathbb{D}$
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0answers
84 views

Mathematica for computing number and diameter of closed curves in the Riemann Sphere

I am currently learning Python but I don't know (YET!) any language like Mathematica, Matlab, Maple, Octave and so on. I have a Math problem I would like to attack numerically and I need to know which ...
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2answers
73 views

Find the image of the $x$ and $y$ axes under $f(z)=\frac{z+1}{z-1}$

Find the image of the $x$ and $y$ axes under $f(z)=\frac{z+1}{z-1}$ Atemppt Notice that the real axis is given by $x+0i$ for $x\in \mathbb{R}$, then the image are $f(x)=\frac{x+1}{x-1}$ which is a ...

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