Questions tagged [conformal-geometry]

A conformal structure is one that captures the idea of angles, but not lengths. Conformal geometry is the study of geometries with conformal structures, with special focus on transformations that preserves the angles (while possibly changing lengths). Examples of conformal transformations include the Mercator projection from cartography, and the Möbius transformations of the Riemann sphere.

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conformal mappings transformation [closed]

can someone help me with my problem region is D={z:|z-1|>1,Rez>0,Im>0} and need transform this with w = (2z-1)/(z+2) Anyone have any ideas? Thank you all
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Find conformal mapping from open strip onto open quarter disk.

I'm trying to find a conformal mapping $f:A \rightarrow B$ from the open strip $$A = \{z \in \mathbb{C}| Re(z) < 0,0<Im(z)<1\}$$ onto the open quarter disk in the first quadrant given by $$B=\...
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'Irregular' conformal mapping of square onto circle?

Assume that I want to find the conformal mapping from a square onto the unit disk. In the regular case, where the four edge points are positioned along cardinal directions, this mapping seems to have ...
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Covering map in complex analysis

Question: Let $A_{r}$=$\lbrace z:1<|z|<r\rbrace$ , $f:A_{r_{1}}\mapsto A_{r_{2}}$ is a covering map of degree $d<\infty$. prove that $r_{2}=r_{1}^d$. My attempt: (We assume covering map is ...
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term for distance preserving up to scale

Is there a common/established term for distance preserving up to scale? E.g. Consider two Riemannian manifolds $M \subset \mathbb{R}^m$ and $N \subset \mathbb{R}^n$ equipped with metrics $g_M$ and $...
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Looking for a specific conformal mapping

Let $\mathbb{H}=\{z: \text{Im } z>0\}$ be the upper half plane and $0<a<1$. Find a conformal map from $\Omega_1=(\mathbb{H}\cap\mathbb{D})\backslash\{yi:y\geq a\}$ to $\Omega_2=\mathbb{H}\cap ...
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Show u(r, θ) is a solution to the Dirichlet Problem for the unit disk

Show that $u(r,\theta) = \frac{1}{\pi}\arctan\left(\frac{1-x^2-y^2}{(x-1)^2+(y-1)^2-1}\right)\\$ where $\arctan(t) \in [0,\pi]$ is the solution to Dirichlet's problem for a unit disk for the piecewise ...
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Find the rate at which distances decrease in stereographic projection

I want to map a 3D space onto the inside's surface of a sphere. The 3D space is represented by points (x,y,z) where the z axis is the height. The first thing I did was to use the following equation ...
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calculating Electrostatic potential between two planes using conformal mapping

I have a problem I'm finding difficulty to solve. My problem is : An insulated region whose cross-section is in the shape of a quarter of a circle, separating two flat conducting planes. The ...
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Conformal mapping between intersection of circles to unit disk

I want to construct a conformal mapping of the region $\{z \in \mathbb{C}: |z - 1/2| < 1 \text{ and } |z + 1/2| < 1 \}$ to the unit disk. I think I would use the Mobius transformation $-\frac{z -...
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Conformal mapping onto lower half plane

How to conformaly mapping region $$D = \{|z|>1,\Im z > 0\}$$ on lower half plane? Any suggestions? Thanks in advance.
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Finding a homeomorphism of unit disk in complex plane that interchanges two given points and leaves all points of boundary fixed

I'm trying to solve problem 21 from chapter 1 of M. A. Armstrong's Basic Topology: Let $C$ denote the unit circle in the complex plane and $D$ the disc which it bounds. Given two points $x,y \in ...
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Does function $f(z)=-i\sqrt{z}$ map unit disc to upper disc?

I think that $\sqrt{z}$ is not defined on $[-1,0]$ , but this is the function I got as a result in problem to map conformally unit disc on upper unit disc... Can anybody help me?
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How to apply Circular Inversion Geometry?

With three osculating circles, Descartes' formula relates the incircle's radius in terms of the circumcircle's and the three circle's radii or better their reciprocals, called curvatures: $$(\kappa_1 +...
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Conformal transformation

This is a problem in Volkovyskii's book. I have a little difficulty finding the right result. Who can show me how to find the conformal mapping how affects the exterior of the parabola $y^2=2px,\,\,(...
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Continuous deformation from Cartesian to polar coordinates

Suppose I have the map $$ (x,y) \rightarrow(y\cos(x),y\sin(x)), \text{valid for } (x,y)\in\left[-\pi,\pi\right)\times\mathbb{R}^+ $$ This is going from Cartesian to polar coordinates with the ...
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Conformal Mapping $\mathbb{C}\backslash$ $\{z :|Im(z)| \leq -Re(z)\}$ to Upper half plane

Finding it difficult to find a conformal mapping from the set $\mathbb{C}\backslash$ $\{z :|Im(z)| \leq -Re(z)\}$ to the upper half plane. Any advice will be very helpful I know I can use $f(z) =...
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Conformal Mapping $w=z^2$

I have to map $|argz≤\frac{π}{4}|$ under the transformation $w=z^2$ ,the answer mentioned is right side of $w$ plane how ever Iam getting $u≤\frac{π}{4}$ could someone help me out with this
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Is any 1-form conformally closed in 2 dimensions?

On $\mathbb{R}^2$, say, given an arbitrary 1-form $u = u_x dx + u_y dy$, does there exist a (non-zero) function $f(x,y)$ such that $f u$ is an closed 1-form, i.e. $$ \partial_x(f u_y) = \partial_y (f ...
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How are these two properties of Conformal Killing Vectors compatible?

Let $(M,g)$ be a semi-Riemannian manifold. A Conformal Killing Vector (CKV) $Y$ is any vector field obeying the Conformal Killing Equation: $$\nabla_\mu Y_\nu+\nabla_\nu Y_\mu=g_{\mu\nu}\dfrac{2}{d} \...
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Extension of conformal mapping of parallelogram onto a half-plane to an elliptic function?

Let $D$ be the interior of a closed parallelogram in the complex plane with one of its vertices at the origin, and $f(z)$ be a conformal mapping of $D$ onto a half-plane $H$ ($f$ is a bijection). ...
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Conformal mapping between a square and a unit disk?

I am in need of a specific (simple, if possible) complex bijection mapping that would map a square onto the unit disk, including an explanation/examination of "why it works". I need it as an example ...
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Are there drawbacks replacing quaternions by Conformal Geometric Algebra in an implementation

I am working on a 3D IK engine and stumbled across Conformal Geometric Algebra. I initially did find it a bit confusing but then again quaternions still occasionally perplex me. I am strongly ...
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Approximate formulas for the linear transformations

The upper half-plane is mapped onto the unit disk so that the point $z=hi\,\,\, (h>0)$ passed into the center of the circle. Find the length $\Gamma$ of the image of the segment $[0, a]$ of the ...
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Analytic continuation of a conformal map across the unit circle

I know that if $f$ is a conformal mapping of $\mathbb{D}$ onto some domain $D$ such that $\partial D$ is a Jordan curve, then $f$ has a continuous extension up to $\partial \mathbb{D}$ such that $f(\...
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Mapping between doubly connected regions

Suppose we have two annuli. The only way to have conformal map between them is to have same radius ratio. Now lets take two annuli A(1,2) and A(1,8). I want to prove that conformal map between two ...
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Möbius: Find the general form

I am studying about supplementary questions of the theory of linear transformations. A bilinear transformation with one fixed point is said to be parabolic. Please, can you help me to solve this ...
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Complex function mapping

Suppose $f(z)$ is an entire function, And let $\zeta$ be jordan arc s.t endpoints of the arc $z_1=0$ and $z_2=\infty$. Prove that function is constant if $f(\mathbb{C})\cap\zeta$=empty I thought this ...
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Mobius: Find the transformation

I am studying about supplementary questions of the theory of linear transformations. A bilinear transformation with one fixed point is said to be parabolic. Please, can you help me to solve this ...
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1answer
29 views

Proof of subordination principle for holomorphic functions on $\mathbb{D}$

I am trying to prove a very simple theorem that uses the general idea in complex analysis that if $f:\mathbb{D}\to\mathbb{C}$ is holomorphic, then the quantity $|f'(0)|$ is somehow responsible for how ...
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how to tell if a conformal mapping between regions is unique

How can you tell if a conformal mappimg between regions is unique? I have a conformal mapping from {z : |z|<2, |Arg(z)|< pi/6} to {z : Re(z)>0, Im(z)<0} as f(z) = -(iz^3 - 8)/(iz^3 + 8) but ...
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Schwarz's Lemma

Suppose that $f$ is analytic on the unit disk $D$ ,$f(0)=0$ and $f(D)⊂[-1,1]×[-0.01i,0.01i]$ . Prove that $|f'(0)|<1$ . In order to use Schwarz's lemma we have to map the rectangle onto the unit ...
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univalent function $f\colon \bar{\Bbb C}\to \bar{\Bbb C} $ that preserves circles [closed]

$\bar{\Bbb C}$ is the extended complex plane. Let $f\colon \bar{\Bbb C}\to \bar{\Bbb C} $ be a univalent function that preserves circles(if $K$ is a circle in $\bar{\Bbb C}$, then so is $f(K)$). ...
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Formula for action of Möbius transformation on the hyperboloid model

The group of Möbius transformations are isomorphic to the group of orientation-preserving isometries of hyperbolic space. The 3-dimensional hyperboloid model is a model of hyperbolic space. What's the ...
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conformal map $\{z : |z|<1, |z-1| >1 \} \rightarrow\{w : \operatorname{Im}(w)>0\}$

First of all, I know The conformal map $\{ z: |z|<1\} \rightarrow \{w : \operatorname{Im}(w)>0\}$ is $w = \frac{z \overline{z_0} - Az_0}{z-A}$ with $|A|=1$, $\operatorname{Im}(z_0)>0$. ...
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Modulus of an intersection of two annuli with big moduli

Suppose we have two round and concentric annuli $A = D_A^1\setminus D_A^2$ and $B = D_B^1\setminus D_B^2$ in $\mathbb{C}$ formed by two pairs of concentric discs $D_A^1, D_A^2$ and $D_B^1, D_B^2$. We ...
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A question about the coordinate expression of the Weyl tensor

The Weyl tensor is defined as the traceless component of the curvature tensor. So should $$W_{abcd}=R_{abcd}-\frac{1}{n^2}g^{pr}g^{qs}R_{pqrs}g_{ac}g_{bd}$$ The coordinate expression, given here ...
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An intuitive approach to Conformal Mappings

I'm trying to develop an intuitive approach to conformal maps. I know some of the basic maps (upper half plane to unit circle, upper half plane to 1st quadrant, interiior of unit circle to outside ...
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Conformal map from the right half plane to the right half plane

Find a conformal map from the right half plane onto itself such that $z=1$ goes to $z=2$. My thoughts: Is this as simple as $z\mapsto 2z$? I feel like there is something big that I'm missing.
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Conformal map from $\mathbb{D}$ to right half plane

The question is to find a conformal map from $\mathbb{D}$ to $\{z\in\mathbb{C}:Rez>0\}$ such that $z=1$ goes to $z=0$. My thoughts: We know that map $f(z)=\frac{1+z}{1-z}$ maps $\mathbb{D}$ to ...
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Conformal Mapping of $\mathbb{D}$ onto itself taking $x$ to $y$

I want to find a conformal mapping of the unit disk $\mathbb{D}$ onto itslef that takes 1/2 to 1/3. Here is my attempt: We know that $f(z)=\frac{z-a}{\bar{a}z-1}$ with $|a|<1$ maps $\mathbb{D}$ ...
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Why $\frac{1}{z}$ is conformal at $0$

In Rudin Example 10.4 it is said that $f(z) = \frac{1}{z}$ is holomorphic at $\mathbb{C} \setminus \{0\}$ (which i checked through Cauchy-Riemann equations). What is going on at $0$? How can I ...
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Show that a holomorphic function $f$ with $f ' \not= 0$ is conformal

Show that a holomorphic function $f$ with $f ' \not= 0$ is conformal. I've come across this problem but I couldn't know how to solve it. I know that a holomorphic function means that it's complex ...
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Need Help! Conformal Mapping - First Fundamental Form

Here $\tilde \gamma$ is given in terms of $\tilde u$ and $\tilde v$ so in the last line using chain rule mustn’t there be $\sigma_{\tilde u}$ but Pressley writes only $\sigma_u$ and writes $cos\theta$ ...
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Struggling to visualise conformal mapping

I'm really struggling to visualise why certain conformal mappings give me these new geometries. In particular, I'm struggling to understand the figure I attached above: the author of the figure claims ...
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Sources for proving the bound on the third Bieberbach (schlicht) coefficient

I am trying to prove that for schlicht functions, that is, univalent functions $f$ from the unit disc to $\mathbb{C}$ of the form $$f(z) = z + a_2 z^2 + a_3 z^3 + \cdots,$$ we always have $|a_3| \leq ...
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If conjugation by a matrix preserves the matrix norm then the matrix must be conformal?

Let $A$ be an $n \times n$ real invertible matrix, and suppose that $\| X\|^2=\| AXA^{-1}\|^2$ for every $n \times n$ real matrix $X$. Is it true that $A$ must be conformal? (It is easy to see that ...
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Conformal map and upper-half plane

Let $$V := \left \{ z: \Im(z) > 0 \right \} $$ be the upper half-plane. For $a \in V$ we define $$h_{a}(z) = \frac{z-a}{z-\bar{a}} $$ for $ z \neq \bar{a}$. So that $h_{a}$ is a conformal map $...
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Finding a Möbius transformation from $\{z=x+iy\in\mathbb{C}:x+y>0\}$ to $D(1,4)$.

Find a Möbius transformation $T$ from $\{z=x+iy:x+y>0\}$ onto the disk $D(1,4)$, such that $T(1)=2$ and $T(0)=-3$. The proof given is as follows: Since $-i$ is symmetric to $1$ with respect to ...
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Conformal map from a 7-sided polyhedron to a square pyramid.

I have a right-angled square pyramid, $A$, whose height and base-length is $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...

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