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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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How the line element change in a complex change of variables?

So I'm learning conformal field theory and having a hard time to prove the conformal Ward identity. From the lectures notes from John Cardy, he express the integral $$ \delta S = \frac{1}{2\pi} \...
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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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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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Non-conformal metrics on vector bundles where $\nabla g=\omega(\cdot) g$

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M \ge 2$), equipped with a metric $g$ and a connection $\nabla$, such that $\nabla_X g=\omega (X) g$ for every vector field $X$ on $M$. ($\...
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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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66 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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Understanding Moonshine and Heterotic E8xE8

Recently I have become familiar with the conjectured relationship of monstrous moonshine and pure (2+1)-dimensional quantum gravity in AdS with maximally negative cosmological constant and, it’s being ...
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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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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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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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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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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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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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Showing $\sin(z)$ is elliptic via conformal maps

One could use Euler's identity to show that $\sin(z)$ is an elliptic function, however, if we define the $\arcsin(z)$ as $$\arcsin(z) = \int_0^z \frac{1}{\sqrt{1-w^2}}dw$$ we would like to show ...
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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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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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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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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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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 $\theta\in [0,2\pi)$ ...
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Checking whether $w = f(z)$ is conformal and its mapping?

I was thinking about the conformal map $w =f(z)$ in which $f$ satisfies $w = f(z) = \frac{df}{dz} - e^z$. Is ths conformal, how can i find the conformal mapping of $z$ plane? Similarly what will be ...
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How to see the image of the conformal map tangent?

Prove that : $w=\text{tan}(z)$ map the region $\{-\frac{\pi}{2}<Rez<\frac{\pi}{2} \}$ to the region $w=u+iv$, where two rays $\{ u=0,|v|\geq 1\}$ are excluded. I know this conformal map can be ...
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Riemann mapping theorem. Why using rotations?

When proofing the Riemann mapping theorem that every simply connected domain $\mathbb{C} \neq D \subset \mathbb{C}$ is conformal equivalent to the unit disk $\mathbb{D}$, we can construct a Möbius ...
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Conformal mapping of periodic bladed boundary

I am new to conformal mapping and I am trying to find the mapping of the vertical boundary with tilted cuts to a horizontal boundary with perpendicular cuts (See the image). Any suggestion is ...
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1answer
44 views

Fitting a conformal / holomorphic function

Suppose we have some 2D points $x_i$ (which we may take to be complex numbers) and some corresponding 2D points $y_i$. We seek a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x_i)\approx ...
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Conformal mapping interpretation

It says on Alfors, Complex Analysis page 73, chapter 2.3 that: Suppose that an arc $\gamma$ with the equation $z = z(t), \alpha \leq t \leq \beta$, is contained ina region $\omega$, and let $f(z)$ be ...
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Notation regarding generalized Minkowski space

In section 12 of the book Surfaces in classical geometries: A treatment by moving frames by Gary R. Jensen, Emilio Musso and Lorenzo Nicolodi (see preview here), Möbius geometry is described. They ...
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Why does orientation preserving maps respect CR?

From this post it is written that "orientation preserving conformal maps respect CR. The matrix of the map must be a constant multiplied by some matrix of rotation that has a positive determinant. ...
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Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
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Transformation law for the Ricci curvature

In 4 dimensions, for a conformal change of metric $g=e^{2u}g_0$ the Ricci curvature tensor $\operatorname{Ric}$ satisfies the transformation law \begin{equation}\tag{1} \operatorname{Ric}_g = \...
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Conformal energy and first fundamental form

In my reference in mesh processing I came across the following equation Where $\bf{I}$ is the first fundametal form of some surface patch $\bf{x}$, I don't know the exact meaning of the notation $\bf{...
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62 views

Circles Riemann sphere

I was able to prove that the circle passing through the north pole of the Riemann sphere is mapped into a straight line. How can I prove that the circle on the Riemann sphere corresponds to a circle ...
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Are conformal maps between Riemannian manifolds real-analytic?

Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there exist real-analytic ...
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Inversion in a sphere preserves circles (proof)

Inversion in the unit sphere, for a vector $x$, is defined by $$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$ How can we prove that a circle's inversion in the sphere is also a circle? (I ...
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Is $\frac{w-1}{w+1}$ a conformal mapping from $\mathbb{C}\setminus(-\infty,0]$ onto $\mathbb{C}\setminus\big((-\infty,-1]\cup[1,+\infty)\big)$?

Is the meromorphic function $g$ defined by $g(w) := \frac{w-1}{w+1}$ a conformal mapping from the singly slit plane $\mathbb{C}\setminus(-\infty,0]$ onto the doubly slit plane $\mathbb{C}\setminus\big(...
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In what $precise$ sense is Minkowski space asymptotically flat?

I've brought this question over from the physics stack exchange, where it didn't generate interest. We say a manifold $(M,g)$ is conformally compact if it is the interior of some $(\overline M, \...