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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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How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?

Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
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Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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1answer
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Mobius transforms - inversion sends line/circle -> line/circle

I'm trying to prove that the inversion mapping $f(z) = \frac{1}{z}$ sends circles or lines to circles or lines. Apparently the set $$\{z \in \mathbb{C}: |z-a|^2 = r^2 \}$$ describes either a circle ...
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Prove transformation $z \mapsto \frac{az+b}{cz+d}, z = x+ iy, ad-bc = 1$ is isometry of Hyperbolic plane.

Prove transformation $$f: z \mapsto \frac{az+b}{cz+d},\ z = x+ iy,\ ad-bc = 1$$ is isometry of Hyperbolic plane $$M=\{(x,y)\in \Bbb R^2:y>0\} \text{ with Riemannian metric } g= \frac{1}{y^2}(dx \...
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Find Möbius transformation for half-plane to unit disk $|w|<1$?

Consider the half-plane depicted in the following figure How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found? What are the steps and things to think ...
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Möbius transformation: area outside of two circles mapped onto the interior of a circular ring

In my current homework, there's the following task: "The area outside of these two circular discs $K_1=\{z{\in}\mathbb{C}:|z-\frac{5}{2}|\le\frac{3}{2}\}$, and $K_2=\{z{\in}\mathbb{C}:|z+\frac{5}{2}...
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Stereographic projection combined with rotation is a Möbius transformation

So for my Complex Analysis class, I need to prove the following question: Consider the function that maps a point from $\mathbb{C} \cup \{\infty\}$ to the sphere via inverse stereographic projection, ...
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Finding the image of the unit disc under a Mobius transform

I am working through an example of mapping the unit disc under the following Mobius transform: $$f(z)=\frac{iz+3}{iz-1}=1+\frac{4}{iz-1}$$ This can be written as a composition of elementary Mobius ...
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number theory and geometry question-Möbius transformation [duplicate]

I am trying to justfiy the distinct points z1, z2, z3, z4, w1, w2, w3, w4 ∈ C ∪ {∞} such that there is no Möbius transformation T with T(zi) = wi for all i = 1, 2, 3, 4. I am having a real hard time ...
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Mobius transformations between two sets

I am doing some revision of complex analysis, and am stuck on this question. I am looking for A mobius mapping sending the set {z: |z+1|<$\sqrt{2}$}, |z-1|<$\sqrt2$} onto the sector {z:3pi/4< ...
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Möbius transformations mapping non-unit circle to non-unit circle

I have a problem in which I need to find a möbius transformation which has as one of the criterion to map the circle $|z−2+i| = \sqrt5$ onto the circle $|w+2| = 2$, I dont really understand how to ...
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1answer
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Determine $z\in\mathbb{C}$, $r>0$ so that $0,1,2+i\in \partial B_r(z)$

I'm struggling to see for a method to start this question. It looks like a question related with mobius transforms. We have studied about determining the mobius transform when points from the domain ...
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2answers
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Mobius Transformation viewed as a mapping on $\bar {\mathbb C}$

The question I have on hand is as follows: Suppose that a Mobius Transformation z $\to \frac{az + b}{cz + d}$ (viewed as a mapping on $\bar {\mathbb C}$) maps $\infty \to \infty$. What information ...
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1answer
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Vertical Line through z=0 in complex plane mapped with f(z)=(1+z)/(1-z)

I have the vague notion that the imaginary axis maps to a circle with f(z)=(1+z)/(1-z). $$ \begin{array}{lll} f(\infty) & = & -1\\ f(i) & = & e^{\pi/4}\\ f(-i) & = & e^{-\pi/4}\...
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Conformal property of transformation [closed]

I want to know if a LFT, $F$, is conformal on the hyperbolic plane $\mathbb H^2$ , that is if we have the curves $\Gamma_1$ and $\Gamma_2$ that intersect at a point $P$ making the an angle $X$, then $...
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Find the image under the Mobius transformation $f(z) = 1/z$ of $|z-3i|=1$ [closed]

I am given $|z-3i|=1$ which is a circle with radius $1$ and centre $(0,3i)$ on the complex plane. I want to find the image (to sketch it) under the transformation $1/z$ WITHOUT taking points and ...
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Conformal Mapping Question about Mobius Maps, mapping 1 region to another

I am trying to map the region G, {|z|<1, |z+i|> (2)^0.5} to the infinite vertical strip at x = +/- pi. I have started by using a Mobius Map which sends the two common points of the circles to 0 ...
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Finding the Möbius transformation from the unit disk to the half plane $\{Re(z)\geq3\}$

I want to find the Möbius transformation from the disk $\{|z-1|\leq2\}$ to the half-plane $\{Re(z)\geq3\}$ that moves the point $0$ to $4+4i$. I know that by specifying the values at 3 points, the ...
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If $w:\mathbb{D}\rightarrow\mathbb{D}$ is a Möbius transform and $||f||_{\infty} \leq 1$, why is $||w(f(z))||_{\infty} \leq 1$?

Denote the unit disk by $\mathbb{D}$. Let $w: \mathbb{D} \rightarrow \mathbb{D}$ be a Möbius transformed defined by $w(z) = \frac{z-\lambda}{1-\overline{\lambda}z}$ where $\lambda \in \mathbb{D}$ is ...
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Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$

Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
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Mobius transformation maps $\Bbb R_∞$ onto itself iff we can choose its coefficients to be real

I have seen many solutions which are very intricate and/or long. The solution I had was much shorter which lead me to believe that it was incorrect. It went like this : $T$ be a Mobius transformation ...
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Variant of Schwarz-Pick for Different Bound/Disk

All, I'm looking to prove an alternate version of the Schwarz-Pick Lemma: Let $f:D(0,r) \rightarrow \mathbb{C}$ be holomorphic, and suppose that $|f(z)| \leq U \quad \forall z \in D(0,r)$. Then, $\...
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Is this a simpler proof that all transformations mapping D(1:0) onto D(1;0) are of the form $\;\;e^{i\lambda}\frac{z-\alpha}{\bar{\alpha}z-1}$

The question is from an exercise (2.13) in Introduction to Complex Analysis by H.A. Priestley. Before writing this I did check all the questions that might have the answer, but where proofs were given ...
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Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
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Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$.

Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$. Using the map $f(z)=i (2/\pi)(\log 2) z$ we get the image of $f$ as $\{ z: \log 1/2 &...
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Describing curves of complex valued functions

I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions: 1.$f(z)=exp(z^2)$ 2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$ My thoughts: I can write down what the ...
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1answer
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Does a Möbius transformation having fixed points not on the object return the entire complex plane as image?

This arises from a (very early) exercise in H. A. Priestley's Introduction to complex analysis. Given the transformation $\frac{z-1}{z+1}$ we were to find the invariant (=fixed?) points, which are $\...
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Mobius transformation produces either a circle or a line…

The exercise (from H A Priestley) required a transformation that sent $\:0, 1, {\infty}$ to $1, 1+i, i$. I knew the transformation that sent $z_1, z_2,z_3,$ to $0, 1, {\infty}$ ie $$\frac{(z-z_1)(z_2-...
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A homeomorphism $T$ from extended complex plane to itself preserving cross ratio is a Mobius map.

Cross ratio preserving means $(Ta,Tb,Tb,Td)=(a,b,c,d)$ where $(a,b,c,d)=\dfrac{(a-b)(c-d)}{(a-d)(c-b)}$. If we assume $T$ fixes infinity, can we prove $T$ is affine?
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Mobius Geometry identity

How do you prove that all figures consisting of three distinct points are congruent in Mobius Geometry? I understand it relates to the Fundamental Theorem of Mobius Geometry. The concepts of which ...
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conformal automorphism $f$ of $D$ that interchanges

Let $a$ and $b$ be distinct points in the unit disk $D$. Show that there exists a conformal automorphism $f$ of $D$ that interchanges $a$ and $b$; that is, $f(a) = b$ and $f(b) = a$. Idea: we know ...
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Mobius transform |z|<1 to the right half plane

Find a Mobius transformation mapping the unit disk {|z| < 1} into the right half-plane and taking z = −i to the origin. My workings: $\phi(t) = \frac{az+b}{cz+d}$ We map -i to the origin (0) by ...
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1answer
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Conformal mapping of a domain to $f(z)=z^3$

Let W be the domain ${Im(z) < 0, Re(z) > 0}$. Sketch and describe the image of W under the conformal map $f(z) = z^3$. I have absolutely no idea how to tackle this practice exam question. I ...
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1answer
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Are $\Bbb R^2\backslash\{(x,y) : x\le 0,y=0\}$ and the unit disc homeomorphic? Difference between conformal map an homeomorphism.

The Riemann mapping theorem says that there exsists a (bijective) conformal map $f$ between $\Omega =\Bbb C\backslash \{z\in\Bbb C: Im(z)=0, Re(z)\le0\}$ and the unit disc $D_1$. $f$ is the ...
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1answer
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Finding the inverse of $\phi_{\lambda}^2$ where $\phi_{\lambda}$ is the Mobius transform on $\mathbb{D}$

Let $\mathbb{D}$ denote the open unit disk. Fix $\lambda \in \mathbb{D}$. Define the Mobius transform $\phi_{\lambda}:\mathbb{D}\rightarrow\mathbb{D}$ by $$\phi_{\lambda}(z) = \frac{z-\lambda}{1-\...
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Image of unit disk under inversion - Mobius transformations

I am trying to find the image of $\{z \in \mathbb{C} : |z| < 1\}$ under $f(z) = \frac{1}{z}$ Let $$w = \frac{1}{z} \Rightarrow z = \frac{1}{w} = \frac{1}{u+iv} = \frac{u-iv}{u^2 + v^2}$$ if $w = ...
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What is actually the geometry or analysis behind the fact that $Mob(\hat{\Bbb C})$ is simple?

Let, $Mob(\hat{\Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $\hat{\Bbb C} \to \hat{\Bbb C}$ . I have been able to prove that (i) $Mob(\hat{...
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mobius transformation given points

Let P,Q,R ∈ ˆ C be the points P = − √2 + i√2 , Q = 2i , R = √2 + i√2 . Let M : ˆ C→ ˆ C be the Mobius transformation with M(P) = Q , M(Q) = R The points P,Q,R lie on a common hyperbolic line (you do ...
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What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. ...
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1answer
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Using conformal maps to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$

I'm trying to solve a question which asks me to solve the Dirichlet problem on $U = \{z : \text{Im}z \geq 0 \}$ on the condition that (where we let $z = x+iy$) $u(x,0) = 0 $ when $ |x| >1 $ and $ u(...
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Solving the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ using a conformal map

I'm trying to solve the Dirichlet problem on $ U = \{z: \text{Im} z \geq 0\}$ with the conditions $u(x,0) = 0$ when $x>0$, $u(x,0)=1$ when $x<0$. To do so, I'm supposed to use conformal maps ...
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1answer
177 views

Subgroup of Möbius transformations which are isometries with respect to the standard metric on the Riemann sphere

I'm trying to find which subgroup of Mobius transformations are isometries with respect to the standard metric on the Riemann sphere (the one induced from the Euclidean metric on $\mathbb{R}^3$). The ...
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1answer
122 views

Finding a conformal map from the intersection of two disks to the unit disk.

I'm trying to solve a problem which asks me to find a conformal mapping from $\{z\in \mathbb{C}: |z-i|< \sqrt2$ and $|z+i|<\sqrt2\}$ onto the open unit disk. I'm really new to these and I'm a ...
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2answers
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Möbius transformation/biholomophic funtion

I have to show, that the Möbius transformation $$ T(z) = \frac{z-z_0}{1-\bar{z_0}z}$$ is an biholomorphic function on $ \mathbb{D}$. $ \mathbb{D}:=\{ z \in \mathbb{C}: |z|<1 \}$ and $z_0 \in \...
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1answer
150 views

Finding bilinear transformation which maps $|z|=1$ on to $|w|=1$ [closed]

How can I show that every bilinear transformation which maps $|z|=1$ on to $|w|=1$ must be of the form $$w=K\frac{z-\alpha}{\overline{\alpha}z-1}$$ where $|K|=1$? Please help me. Thanks.
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1answer
41 views

Mobius transformation producing a curved triangle with 3 intersecting circles

Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
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2answers
209 views

Möbius transformation mapping

I would like to understand how to choose the right Mobius transformation. For example, the Mobius transformation that maps upper half plane onto the unit disk is: $z \rightarrow \frac{z-i}{z+i}$ ...
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3answers
155 views

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H$. What kind of points have non-trivial stabilizer? And how many orbits are there?

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H= \{z \in \mathbb{C} | \Im(z) > 0 \}$ via Mobius transformation. $$ \text{ For } \gamma =\begin{bmatrix} a &b \\c&d \end{...
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1answer
35 views

Finding the norm of $w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $\mathbb{R}^n$

I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function $$M_w(z) = w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z \in \mathbb{R}^n$ and $|w| < 1$. (...
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0answers
55 views

Image of square under Mobius Transformation

Am i right that there is no Mobius transformation $$ h(z) $$ that sends a square to rectangle "A" with vertices in $$ 0, 2, i, i+2 $$because if we inscribe a cricle into square then it has 4 points $$ ...