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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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Computing all possible conformal factors on the sphere

Proposition to prove. Let $\tau\colon \mathbb S^n\to \mathbb S^n$ be a conformal map, meaning that $\tau^\star g= \Lambda^2 g$ for a scalar field $\Lambda$. (Here $g$ denotes the standard metric ...
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Gaussian curvature with Laplacian

In a lot of papers and books, I have seen the following expression of Gauss Curvature in $2$-dimensional surfaces with a conformal metric $$\overline{g} = e^{2u}g$$ $$K - \overline{K} e^{2u} = \...
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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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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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Mapping Property of $f(z)=\frac{z-i}{z+i}$.

Consider the map $f(z)=\dfrac{z-i}{z+i}$. What is the image of Circle with radius $ r $ under $f$ I got $x=\dfrac{r^2 -1 }{r^2 -2r\cos(\theta) +1} $ and $y=\dfrac{-2r\sin(\theta)}{r^2 -2r\cos(\...
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1answer
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Does the orthogonal complement determine the inner product up to scaling?

Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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Show that $f(z)=\frac{1}{2\pi}\int\limits_0^{2\pi}f\left(\frac{e^{i\theta}+z}{1+\overline{z}e^{i\theta}}\right)d\theta$

Let $f$ be analytic on domain $\Omega$ which contains the closed unit disk $\overline{\mathbb{D}}$. Show that (a) $$f(0)=\frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})d\theta$$ (b) Use part (a) to ...
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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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Complex Analysis: Finding image of a domain

(https://i.stack.imgur.com/fnIsU.jpg) I am beginner in Complex Analysis and I am stuck in this set of question. I can see these all are composition of map but how to find image? I have no idea how to ...
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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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The quadrature of the circle: comparing Archimedean and Ulam spirals

There are two closely related arrangements of the natural numbers that allow to show patterns in the distribution of some sets of numbers (multiples of 2, 4, 8, square numbers, prime numbers): the ...
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Is there a way to use Mobius transformation to prove existence of Laurent Series

I'd like to prove the existence of a Laurent series for a function with a pole. The kind of proof I have in mind must use an inversion. The key idea is that if $$f\left(z\right)=\sum_{n=-\infty}^\...
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How to show that every conformal map on extended $\Bbb{C}$ is Mobius

I have no problem in showing that a Mobius transformation is a conformal map. I can't find anyway a reference or a simple proof that every conformal map of extended $\Bbb{C}$ can be written as a ...
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Is there a conformal mapping from the surface of a cube to the surface of a spherical cube that preserves edges?

Is there a conformal mapping (with certain singularities noted below) from the surface of a cube to the surface of a spherical cube that preserves edges? Note that this also implies that vertices and ...
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What are the Faber polynomial coefficients?

The Faber polynomial recurrence relation is given as $$\phi_{m+1}(z)=\frac{1}{c}(z \phi_m(z)-m c_m-\sum_{m=0}^{M}c_m \phi_{M-m}(z) \ )$$ with the initial values: $\phi_1 = \frac{1}{c}(z-c_0)$ and $\...
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Trying to understand the conformal transformation law of Ricci curvature

I'm trying to understand the conformal transformation law of Ricci curvature. It states that if the metric $g$ now changes to $e^{2\phi}g$, then $$Ric(e^{2\phi}g)=Ric^g-(n-2)Hess^g(\phi)-\Delta_g(\phi)...
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Image of a region bounded by lines $x-y <2$ and $x+y>2$, under mapping $w=1/z$

I got $z = x+iy$ and $w=u+iv=\frac1 z = \frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}$ $u(x,y) = \frac{x}{x^2+y^2}$ and $v(x,y) = -\frac{y}{x^2+y^2}$ also $x(u,v) = \frac{u}{u^2+v^2}$ and $\;y(u,v) = -\frac{...
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Analytical conformal mapping of region between two non-concentric ellipses to an annulus?

As titled. What is the analytical form of the conformal mapping for the doubly-connected region between two embedding, but NON-concentric ellipses to an annulus? According to conformal mapping ...
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1answer
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Codifferential / divergence of differential form under conformal metric change

I have a question related to this and a second post. I want to calculate the codifferential under a conformal metric change, $g_\psi = e^{2\psi} g$. By Besse's book on Einstein manifolds, or an ...
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a conformal mapping using the tangent function?

I am dealing with a 2D coordinate transform, which is based on the tangent function, this way: x becomes p = tan(x) y becomes q = tan(y) the choice is due to tan() being able to "map" the finite ...
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1answer
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Conformal map from upper half plane to the slit unit disk

I am trying to find a conformal map from $\mathbb{H}=\{z:Im(z)>0\}$ onto $\mathbb{D}$ take away $(-1,0]$, where $\mathbb{D}=\{z:|z|<1\}$ (the slit unit disk). I have found that $f(z)=\frac{z-i}...
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1answer
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Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?

Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\nabla$, such that $\...
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Complex Analysis: Difference between the image of the Joukowski-Mapping and $\sin z$ or $\cos z$?

i am currently studying complex analysis, conformal mappings in particular and i am trying to get a good grasp about the different mapping properties of elementary functions such as $$e^z, \log z, \...
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1answer
71 views

Explicit Riemann Mappings for the Complement of a Plane Curve

The Riemann Mapping Theorem says that if $U$ is a simply connected open subset of $\mathbb C$ that is not $\mathbb C$, then there is a conformal isomorphism between the open unit disk $\mathbb D$ and $...
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How does one represent generalized polynomials in Conformal Geometric Algebra C(4,1)

I am interesting in representing arbitrary curves using conformal geometric algebra. I have a special interest in spatial loops. Also, I would like to represent characteristic polynomials of matrices ...
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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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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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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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Perspective Mappings Between Quadrilaterals

Given a quadrilateral represented by $(X_n, Y_n)\;$ I would like to obtain a specific point $\;(A, B)\;$ in the same quadrilateral when it changes perspective knowing only the vertex $\;(X'_j, Y'_j)\;$...
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Solving the Dirichlet Problem for an infinite strip

I have been looking into the Dirichlet problem and conformal mappings, but am unsure as to how to find a solution $u(x, y)$ for the Dirichlet problem given this information: The region is $U = \{\ x+...
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Riemann mapping from disk $\mathbb{D}=B(0,1)$ to $\mathbb{D} \cup B(2-\epsilon,1)$

If $\epsilon>0$ and $f_{\epsilon}$ is the Riemann mapping from $\mathbb{D}$ (unit disk) to the union of $\mathbb{D}$ and the unit disk centered at $2-\epsilon$, made unique by specifying $f_{\...
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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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Mapping a curve-sided quadrilateral to a rectangle

I am currently investigating different ways of solving the Laplace equation $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial z^2} = 0 $$ numerically on the domain $\Omega$ shown as ...
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Find the image of $\{z\in\mathbb{C}:|z-2|<2$ and $|z-1|>1\}$ under the map $z\mapsto \frac{1}{z}$

As the title explains, I'm trying to solve a question which asks me to find the image of $\{z\in\mathbb{C}:|z-2|<2$ and $|z-1|>1\}$ under the map $z\mapsto \frac{1}{z}$. I find it really hard ...
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Finding a conformal map from the intersection of two regions to a unit disk [duplicate]

I'm trying to solve a problem which asks me to find a conformal mapping from the intersection of $\{z\in \mathbb{C}: |z-i|< \sqrt2$ with $|z+i|>\sqrt2\}$ onto the open unit disk. I'm really ...
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1answer
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Space of univalent mappings $f: \mathbb{D} \to \mathbb{C}$ has no nesting

Let $S$ be the space of univalent (i.e. injective) mappings from the disk $\mathbb{D}$ to the plane $\mathbb{C}$ normalized so that $f(0) =0$ and $f'(0)=1$. So $$f(z) = z+a_2z^2+a_3z^3+\cdots.$$...
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1answer
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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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1answer
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Analytic continuation of Schwarz-Christoffel mappings?

I have not studied carefully the topic of analytic continuation. Here is a question that popped up in my mind. Let $f: \mathbb{D} \to \Omega$, where $\mathbb{D}$ is the open unit disk, and $\Omega$ ...
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1answer
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How can I construct this conformal mapping?

Let $\mathbb{D}$ denotes the unit disk, then construct a conformal mapping that map the set $S=\mathbb{D}$ \{(-1,$-\frac{1}{2}$]$\bigcup$[$\frac{1}{2}$,1)} onto $\mathbb{D}$ itself. I know some ...
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Extremal distance

I'm reading Conformally Invariant Processes in the Plane by Lawner and I have a doubt, he takes $R_L=(0,L)\times(0,i\pi) \subset \mathbb C$ and define $\partial_1=[0,i\pi]$, $\partial_2 = [L,L+i\pi]$, ...
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Jordan curve and Conformal maps

Let $\mathbb D$ be the unitary open disk, $D$ a bounded domain(open and connected) with boundary a jordan curve and $f$ a conformal map from $\mathbb D$ to $D$, is it true that we can extend $f$ as an ...
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conformal coordinates on a Riemann surface

Let $\Sigma$ be a Riemann surface with complex structure $j$ and a volume form $dvol_\Sigma$. I read somewhere that one can take the so-called 'conformal coordinates' $z=s+it$ so that $j\partial_s = \...
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What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?

I have two biosignals recording the same phenomenon with different methods. Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
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1answer
51 views

Find Conformal Mapping Between Circles

How would I go about finding the conformal mapping bewtween $|z| = 2$ and $|z-1| = 1$? I have done the following so far: Step 1: map regions between circles onto a strip: $$\frac{az + b}{cz +d} = \...
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How to represent a conformal transformation using spinors?

In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan ...
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29 views

Finding transformation in $\mathbb{R}^2$

I have to find the transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right. More precisely, the problem is as follows: ...
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183 views

Must every holomorphic function $f: \mathbb{D} \to \mathbb{D}$ have a fixed point?

(a) Prove that if $f: \mathbb{D} \to \mathbb{D}$ is analytic and has two distinct fixed points, then $f$ is identity. (b) Must every holomorphic function $f: \mathbb{D} \to \mathbb{D}$ have a ...
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1answer
51 views

Finding a Particular Möbius Transformation from $D \to D$

I'm well aware that the Möbius transformations that take the unit disk to itself, $f:D \to D$, are given by $$ f(z) = \frac{e^{i \theta}(z-\alpha)}{1-\bar{\alpha}z}, $$ where $\theta\in [0,2\pi)$ ...