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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $I_{... 1answer 412 views ### How can I show the unit circle in z-plane maps to a line? For example, it is known that if |c| = |d|, then the linear fractional transformation:$w(z)=\frac{az+b}{cz+d}, ad-bc \neq 0$should map the unit circle ($z=e^{i \theta})$to a line. How would I ... 1answer 248 views ### Mobius transformation that maps interior of a circle to a half plane bijectively Construct a Mobius transformation that maps the interior of the circle$\{z \in \mathbb{C}:|z-3|=2\}$bijectively onto the half plane$\{z \in \mathbb{C}:Re(z)<-1 \}$. I drew the two graphs but I ... 1answer 188 views ### Working with Mobius Transformation I want to find the transformation that maps$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$, the real ... 0answers 84 views ### How is this picture regarding Möbius transformation formed? The following picture is from the Wikipedia article on Möbius transformation. It is explained under the picture that Pre-images of the unit circle are circles of Apollonius with distance ratio$c/...
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 ...