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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no wandering domain theorem for circles

I am trying to understand the proof of the no wandering domain theorem from Beardon's iterations of rational functions and thought a good start would be to omit the quasiconformal structures part and ...
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Mobius transformation mapping three specific points to three specific points

I'm having trouble understanding point (30) on page- 155 of Visual Complex Analysis, the following is given: Let $C$ be the unique circle through the points $q,r,s$ in the z-plane, oriented so that ...
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Find a mobius transfromation that maps $|z|=2$ into $|z+1|=1$

Find a mobius transfromation that maps $|z|=2$ into $|z+1|=1$. mapping $-2$ and $0$ to $0$, $i$ respectively. I started by substituting in the general form $T(z)= \frac{az-b}{cz-d}$. I get $b = 2a ...
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Determine whether the family $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \}$ is normal

Let $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \: \forall z \in \mathbb{D}\}$. I want to determine whether this is a family of normal functions. In ...
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Extending an automorphism of the upper-half plane to the whole Riemann sphere

We know the automorphism group of the upper-half plane $\mathbb{H} \subset \mathbb{C}$ is given by $\Big\{ \frac{az+b}{cz+d} \quad \big|\quad a,b,c,d \in \mathbb{R} \quad \text{and } ad-bc>0 \Big\}$...
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Conceptual Question

Hi guys I am a engineering student(in my 4th semester). I was having trouble with the following question. Kindly tell me how to approach and solve this question. Also please tell me some nice online ...
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Changing Non Convex Optimization Problem into a convex problem

enter image description here I have this optimization problem to solve, where the bold m and h are vectors. Since | x_{k} | >= 1 so its feasible region is outside the unit circle. So my question is ...
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Is there a Möbius transformation mapping an arbitrary disk to the unit disk?

I have the following problem. I'm interested in finding a Möbius transformation mapping an arbitrary disk with radius $R$ centered at $z_0$ to $\Bbb{D}$. I thought that it will be something of the ...
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Rational functions with Mobius symmetries

I'm interested in describing solutions to the equation $ f = f \circ \gamma $, where $ f : \mathbb{C} \rightarrow \mathbb{C} $ is a rational function of a given degree, and $ \gamma : \mathbb{C} \...
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Automorphism group of unit disk acts transitively?

Is there a quick way to see that $\text{Aut}(D)$, the group of conformal automorphisms, acts transitively on the unit disk $D$? I know that one can equivalently consider the projective special linear ...
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What are the infinitesimal generators of the Mobius transformation

I understand that the Mobius transformation $$f(t)=\frac{at+b}{ct+d}$$ is isomorphic to $SL(2)$ for $ad-bc=1$. I also know how to get the infinitesimal generators for the $SL(2)$ group. i.e. the trace-...
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Mechanics of Clifford Algebra valued Möbius Maps

The Möbius maps $$ (ax+b)(cx+d)^{-1} $$ Where $a,b,c,d,x \in \mathbb{C}$ are well-understood, but how do the ones where the variable and coefficients are in Clifford Algebras act? Are there any good ...
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Resources/explanations for conformal geometry with “null cones at infinity”

In the Wikipedia article on conformal geometry https://en.m.wikipedia.org/wiki/Conformal_geometry there’s a section in Mobius geometry that says it’s the study of pseudo Euclidean spaces with either a ...
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Formula for action of $\operatorname{SL}(2,\mathbb{C})$ on hyperbolic 3-space [duplicate]

It's pretty standard in 3-manifold topology and hyperbolic geometry that $\operatorname{PSL}(2,\mathbb{C})$ is the orientation-preserving isometry group of hyperbolic 3-space $\mathbb H^3$. I haven't ...
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Law of cosines in proving invariance of Mobius energy

I'm currently reading Freedman's paper on the Mobius invariance of knot energy, and I'm stuck on a particular equality (2.8). Let $\gamma$ be a curve in $\mathbb{R}^3$ parametrized with respect to arc ...
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Show that the center of the circle of the Mobius transformation $f(z)=i(1-z)/(1+z)$ of $z=k+iy$ is always on the $y$ axis.

How can I show that the center of the circle of the Mobius transformation $f(z)=i(1-z)/(1+z)$ of $z=k+iy$ is always on the $y$ axis when $k$ is a constant? I know that this is a direct result of the ...
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What is the locus of all shape parameters of ideal tetrahedra which share the same volume?

I'm taking a class in hyperbolic knot theory out of Jessica Purcell's book, and I was curious about some volumes and classifications of ideal tetrahedra in $\mathbb{H}^3$, with the upper half $n$-...
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Find the image of the upper half-plane Im(z) > 0 under the transformation $w = \frac{(1-i)z+2}{(1+i)z+2}$

How can I solve this question? Given that the transformation part is very much similar to $f(z) = \frac{az+b}{cz+d}$, the requested transformation is probably a Möbius transformation. I am not ...
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2 votes
1 answer
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Analytic maps from lower half plane onto itself.

I know that the bilinear map $f(z)=\frac{az+b}{cz+d},a,b,c,d\in\mathbb{R},ad−bc>0$, maps the upper half plane onto itself. But I wonder what is the map which maps lower half plane onto itself such ...
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Möbius Transformation that moves zero

Find a Möbius Transformation on complex plane, that moves $|z|<1$ to $|w|>2$ and point (0, 0) to (4, 0). If we don't concern second condition, we can use $w(z) = \frac{2}{z}$. But later we can't ...
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How to construct a Mobius group corresponding to a given fundamental triangle?

Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Mobius transformations: $$z'=z+1$$ $$z'=-\frac{1}{z}$$, and immediately after that, ...
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Why can we assume the other endpoint of this hyperbolic line lies after $x=1$? (Anderson's Hyberbolic Geometry Prop. 3.22)

I'm reading through James Anderson's Hyberbolic Geometry. Proposition 3.22 and the first step of it's proof is as follows Proposition 3.22: Let $l_0$ and $l_1$ be ultraparallel hyperbolic lines in $\...
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Show that any mapping from the extended complex plane to the Riemann Sphere back to the extended complex plane is a Möbius Transformation [duplicate]

Definitions: Extended Complex Plane $\mathbb{C}^\infty = \mathbb{C} \cup \{\infty\}$. Stereographic projection: A mapping from a sphere in $\mathbb{R}^3$ to the extended complex plane. Möbius ...
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Any finite abelian group of Möbius transformations is either isomorphic to $C_2 \times C_2$ or is cyclic.

I want to solve the following question Show that if a non-trivial element of $\mathcal{M}$ has finite order, then it fixes precisely two points in $\mathbb{C}_{\infty}$. Hence show that any finite ...
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1 answer
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when is this inequality feasible?

I have the following inequality that has to hold for all frquencies $w$ $${\left[ {\begin{array}{*{20}{c}} {H\left( {jw} \right)}\\ 1 \end{array}} \right]^*}\left[ {\begin{array}{*{20}{c}} { - 2mn}&...
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1 vote
1 answer
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What is the essence behind representing Möbius transformations as matrices?

I know that Möbius transformations generally maps lines/circles to line/circles using a function $f\left( z \right) = \frac{{az + b}}{{cz + d}}$ defined over $\mathbb C$. However, what I do not ...
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how to obtain this mobius transformation?

The Möbius transformations $f\left( z \right) = \frac{{\frac{1}{2}\left( {{k_u} - {k_l}} \right)z}}{{\frac{1}{2}\left( {{k_u} + {k_l}} \right)z + 1}}$ bijectively maps the circle that is symmetric ...
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-3 votes
1 answer
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what is the mobius transformation that maps a circle with center $z_0$ and radius $R$ to the unit circle? [closed]

I want to find the mobius function $f(z)$ that transforms the circle with center $z_0$ and radius $R$ to the unit circle centered at the origin.
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1 vote
2 answers
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Bounding the derivative of a holomorphic function at the origin.

Let $f : \Bbb{D} \to \Bbb{D}$ be a holomorphic function. If $f(0)=f(1/2) = 0$, then prove that $|f'(0)| \le 1/2$ Here is the previous problem, which I was easily able to solve using the Schwarz lemma ...
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Prove the Mobius transformation is the group of all biholomorphic preserving the unit disk

The wikipedia entry Mobius transformation states the following. The subgroup of all Möbius transformations that map the open disk $D:= \{z :|z| < 1\}$ to itself consists of all transformations of ...
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Conformal map from $\Omega = \{z \in \mathbb{C} | \operatorname{Re}(z)>0\} \setminus [0,1]$ to unit disk

Find a conformal map $f$ from $\Omega = ${$z \in \mathbb{C} | \operatorname{Re}(z)>0$}$ \setminus [0,1]$ to unit disk $\mathbb{D}$. Also make sure that $f(2) = 0$. I drew $\Omega$ (see image below,...
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Making a Möbius transformation such that $f(3)=0$, $f(2)=1$, $f(1)=\infty$

We want to make a Möbius transformation such that $f(3)=0, \ f(2)=1, \ f(1)=\infty$, so we use the cross-ratio: \begin{equation} \frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)...
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1 vote
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Mobius transform and defining an hyperbolic angle

Let $\alpha,\beta:[-1,1]\to\Bbb{C}\simeq\Bbb{R^2}$ be two paths, s.t $\alpha(0)=\beta(0)=\zeta$. Let $A:\hat{\Bbb{C}}\to\hat{\Bbb{C}}$ be a mobius transform s.t $A\zeta\neq\infty, A\in SL_2(\Bbb{C})$. ...
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Möbius transformation with infinity in both the $w$-plane and $z$-plane.

I want to find the Möbius transformation which brings $f(0)=\infty$, $ f(\infty)=0$, and $f(5)=i$. If I use the formula \begin{equation} \frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{(z-z_1)(z_2-z_3)...
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1 vote
1 answer
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Showing that two circles in w-plane tangent each other if there is a Möbius transformation that takes them to two parallel lines

I want to show that two parallel lines in z-plane yield two tanging circles in w-plane. I start with two lines, $L_1=i+x$ and $L_2=3i+x$. Then I want to use the formula: \begin{equation} \frac{(w-...
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2 votes
1 answer
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Find a Möbius transformation such that a semi-circle in the upper half plane is mapped to $i\mathbb{R}_{>0}$

I am giving the semi-circle $\ell = \{z\in\mathbb{H}:|z|^2=4\}$ and I want to create a Möbius transformation $M:\mathbb{H} \to \mathbb{H}$ via $M(z) = \frac{az+b}{cz+d}$, where $ac-bd > 0$ such ...
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Show that the Möbius group (or general linear projective space $PGL(2,\mathbb{C})$) is not simply connected.

How to show that the Möbius group (or the general linear projective space $PGL(2,\mathbb{C})$ is not simply connected? For a definition see: Möbius transformations. For more background see: projective ...
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2 votes
2 answers
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Explicitly representing an isometry as a composition of circle inversions

Let $\mathcal C$ be a circle $|z - z_0| = a$ in the complex plane. Then inversion in $\mathcal C$ is the map $z \mapsto z_0 + \frac{a^2}{\overline{z} - \overline{z_0}}$, which is easily seen to be ...
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Proof that the composition of two inversion with the same center is a homothety

It is asking me for the inversion of two concentric circles, but I just have no idea how to solve this, I've been stuck for a while
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1 vote
1 answer
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prove a property for the stabiliser of 0 by the map apply action of a discrete subgroup of Aut(D)

It’s a note from Donaldson’s Riemann Surface,section 3.2.3. Aut(D) is the holomorphic automorphism group of the unit disk in the complex plane. Explicitly, all möbius holomorphic maps: $$ z \mapsto \...
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Show that any biholomorphic map from the upper half plane to itself is a Mobius transform

Show that any biholomorphic map from $U = \{z \in \mathbb{C} : Im(z)>0\}$ to $U$ is a Mobius transformation. I know this statement to be true, just having a hard time directly proving this, I ...
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1 vote
1 answer
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Prove conjecture about classification of circle pairings

I need help proving or disproving a conjecture related to circle pairings, which I'm trying to prove for my bachelor final project. I first present some needed terminology and context. A circle ...
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what can we say about the bounds of an inequality which is a mobius transform of x while knowing a given inequality of x?

Given a known inequality such as |x-p/q|< 1/q for some given integers p/q what can we say about the inequalities met by the mobius transform of x--> (ax+b)/(cx+d)?? would similar conditions be ...
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Under fractional linear transformation with two fixed point, Steiner circles are invariant.

I think I know how to prove the first kind of Steiner circles (circles through tow fixed points) are invariant. But I don't know how to prove the second kind (Apollonius) are also invariant. I think ...
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Find Möbius Transformation over an annulus

Find the Möbius transformation that applies the domain $$\Omega=\{z\in\mathbb{C}:\Re{z}>0, |z-2|<1\}$$ onto an annulus $$A(0;p,1)\text{ for some }p\in(0,1)$$ I don't even know where to start, I'...
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Is bijective holomorphic mapping from the open unit disk to itself always a Mobius transformation?

I am proving that for $f\in\mathcal{H}(\{|z|<1\})$, $|f'(0)|\leq2f(0)$ if $f(z)\in\{x>0\}$ for every $|z|<1$. I will apply a Mobius transformation $T$ which $a\mapsto 0$ so that $T\circ f$ ...
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Determine for which pairs $\eta, \lambda > 0$ the mappings $h_\eta, h_\lambda$ are conjugate in $\text{Mob}(\mathbb{H)}$.

I have (what I believe to be) a partial proof of the problem below. I do not; however, know how to treat the $\eta = 1, \lambda \neq 1$ case. For any $\eta > 0$ let \begin{equation} h_\eta(z) = - \...
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Show that every element of $\text{Mob}^+(\mathbb{H})$ is the product of two inversions.

I believe that showing that every element of $\text{Mob}^+(\mathbb{H})$ is the product of an even number of inversions is quite straightforward (barring a few lemmas here and there). I do not; however,...
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7 votes
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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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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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