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 ...

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3D mesh segmentation simple algorithm

I am developing the algorithm reported in this article: Lest square conformal mapping Here is presented an algorithm to flat a 3d mesh on the parametric space, but i don't understand the ...
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Is there any generalization of Riemann Mapping theorem?

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally ...
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Is any smooth deformation of a metric in dimension 1 conformal?

Consider $(S^1, g)$ where $S^1$ is the unit circle and g is a metric. Now consider the metric $$ \tilde g := f g $$ where f is a smooth positive function. Since in 1 dimension this is the only smooth ...
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Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
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when can a surface conformally equivalent to the sphere be isometrically immersed?

Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a ...
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Conformal map and linear independence over $\mathbb{Q}$

I would like to know whether the imaginary parts of $n$ complex numbers $z_1,\dots z_n$ sharing the same real part $r$ are linearly independent over $\mathbb{Q}$ if and only if no conformal mapping ...
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Conformal mapping, which geometric objects are these?

Given the mapping $$w = \frac{z + i}{z - 1} $$ find the images on the $w$ plane of $$i) |z - i| = |z - 1|$$ $$ii) x^2 + y^2 = 1$$ $$i) |z + i| = |1 + i|$$ $$ii) ???$$ Polar form, maybe? Which ...
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$\{z : 0 < |z| < 1\}$ and $\{z : r < |z| < R\}$ are not conformally equivalent.

Prove that $D=\{z : 0 < |z| < 1\}$ and $A=\{z : r < |z| < R\}$ are not conformally equivalent if $r > 0$. I am trying to apply the Riemann mapping theorem to show but am not able to ...
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44 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
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Find an open connected set $G \subset \mathbb{C}$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$…

Find and open connected set $G \subset \mathbb{C}$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$ for all $z \in G$. Can you make $G$ maximal? Are $f$ and $g$ ...
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20 views

Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of ...
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Image of circle under linear fractional transform

Given the LFT for a complex $z$, \begin{align*} \phi:z\mapsto \frac{2z+1}{z+2}. \end{align*} I'm asked about the image under $\phi$ of $C:=\left\{\left\lvert z+\frac25\right\rvert = \frac25\right\}$. ...
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35 views

Show that the roots of P' lie in the same half plane as the roots of P

The problem statement is: Part(a) Assume P(z) is a non-constant polynomial with all of its roots in some half plane H. Show that the derivative P'(z) must also have all of its roots in H. Part (b) ...
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Mapping unit circle with the map $\omega= \frac{z\ln z}{z^{2}+1}$

In an exercise involving conformal mappings, the task is to determine the line in the $\omega$ plane traced by the point $\omega= \frac{z\ln z}{z^{2}+1}$, as $z$ traces the unit circle. I am not sure ...
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111 views

Conformal reparametrization

We consider $$\sigma (u,v)=(f(u)\cos v, f(u)\sin v, g(u))$$ Picking $u=\theta , v=\phi , f(\theta )=\cos \theta , g(\theta )=\sin \theta$ we get that the first fundamental form is $$d\theta^2+\cos^2 ...
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Conformal mapping from square to disk as inverse of hypergeometric function

I'd like to write a little program that transforms a fractal generated in the square $(-1,1)^2\subset\mathbb C$ conformally to the unit disk $|z|<1$. I know that conformal mappings from the unit ...
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Conformal map from a subset of the unit disk to the unit disk

I need to find a conformal map from $\Omega = B(0,1)\setminus \overline{B(\frac 1 2, \frac 1 2)}$ to $\mathbb{D} = B(0,1)$. Is the idea here to "straighten" both of the circles into generalized ...
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33 views

Conformal mapping from Gaussian grid to rectangular grid

I have data in a Gaussian grid - https://en.wikipedia.org/wiki/Gaussian_grid In this Gaussian grid coordinates will be defined in terms of longitude and latitude). I am going to be transforming this ...
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1answer
65 views

Conformally mapping an ellipse into the unit circle

I'm currently studying for a complex analysis final and I don't think I've really developed the intuition for conformal mappings yet. I'm attempting a problem from Ahlfors: map the outside of the ...
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Number of independent harmonic cross-ratios

How to derive the formula for number of independent harmonic cross-ratios for N points, the answer is $ \dfrac{N(N-3)}{2} $.
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describing the contor lines of constant x and y in the W plane of $f(z)=\frac{1}{z-1}$

Let $f(z)= \frac{1}{z-1}$, Finding the real and imaginary parts $u=\frac{x-1}{(x-1)^2+y^2}$ and $v=\frac{-y}{(x-1)^2+y^2}$. I have to describe the contour lines of constant $x$ and $y$ in the W plane. ...
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Mapping the region $\Gamma_{z}$ using the conformal map $ \omega=\frac{-2z}{z^{2}+1}$

Suppose we have an analytic function $$ \omega=\frac{-2z}{z^{2}+1}$$ and the region $\Gamma_{z}$ given by $$\Gamma_{z}:=\left \{ z \in \mathbb{C}| \Im \left ( z \right )\geq 0 \wedge \left | z \right ...
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Schwarz Christoffel mapping on the unit disk.

We can write the conformal map from the upper half plane to the polygon by $f(z)=\alpha\int_0^z \prod_{k=1}^n (\zeta-A_k)^{-\mu_k}\,d\zeta+\beta$. From the above, I want to deduce the conformal map ...
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Mapping of the circle $|z-1-i|=\sqrt{2}$

I have a problem I think it is a little hard or at least has some points need to be considered to solve it. I know that if the transformation function is with the form of $f \circ g$, I should ...
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54 views

Sign of first eigenvalue of conformal Laplacian

Let $(M^n,g)$ be some manifold of dimension $n \geq 3$. The conformal Laplacian is given by $L=-4 \frac{n-1}{n-2} \Delta+ R$, where $R$ is the scalar curvature of $M$ and $\Delta= ...
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40 views

Is my reasoning of why this power mapping is onto the upper half-plane correct?

Let $z_0$ be any complex number, $t$ any real number and $α∈(0,2π)$. Find a conformal mapping from the sector $\{z:0≤arg(z−z_0)≤α\}$ onto the upper half plane, $\{w:Im(w)≥0\}$ such that $z_0$ is ...
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the transformation of mapping the right half plane onto |w|<2

I have a question about finding the transformation function. Can you please find the transformation which maps the right half plane $R(z)>0$ onto the circle $|w|<2$? Thank you.
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harmonic ratio of every four points in conformal mapping

i know the concept of harmonic functions or the conformal mapping and also i know some characteristics of the conformal mapping( i think ). i have already faced a theorem: "the conformal mapping keeps ...
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Checking explicitly that a conformal mapping maps a half plane to the unit disk

We can verify that \begin{align*} f(z) = -i\frac{z - 1}{z + 1} \end{align*} maps the unit disk to the upper half plane by the following method. First compute the inverse, \begin{align*} z = ...
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Prove that if $f(z)$ is analytic at infinity, then $\lim_{z \to \infty}{ f'(z)}= 0$

I don't really know how can i prove that. I know a function $f(z)$ is analytic at $z=\infty$ means $f(\frac{1}{\xi})$ is analytic at $\xi = 0$. In particular: $$\lim_{z \to \infty}{f(z)} = \lim_{\xi ...
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Riemannian metric conformal to another metric

Suppose $M$ is a surface embedded in $\mathbb{R}^3$, then it has the natural induced Euclidean metric, denoted by $\textbf{g}$. Suppose $\tilde{\textbf{g}}$ is another Riemannian metric on $M$, we ...
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Conformal Map from Intersection of Two Discs and Half-Plane

I have one of those "find the map" problems that is really giving me a lot of trouble. Let $B_1(1)$ be the ball of radius $1$ centered at $1$. We have the following domain: $\mathbb{D} \cap B_1(1) ...
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Understanding how to compute the polygonal image of this Schwarz-Christoffel mapping?

The problem statement reads: This function $\large (−z)^\frac{2}{3} (resp., (1−z)^\frac{2}{3})$ is determined as to be real and positive when $z=x<0$ (resp. when $z=x<1$) and analytic in the ...
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Is the conjugate of $z$ a conformal map?

Let $f(z) = \overline{z} $. At which points is $f(z)$ conformal? I believe it is not conformal since $f$ is not analytic: It does not satisfy the Cauchy-Riemann equations. Is this correct?
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Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
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Rigorous definition of a “generator” for a transformation group

EDIT : I think that the whole question can be summarized as « how do we know that every conformal transformation can be written under the form $e^{tX}$ for some operator $$X ? » I'm reading the ...
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Neumann and Dirichlet Conditions for Schwarz-Christoffel Map

I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping. I'm able to map the polygon to the upper-half plane using ...
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Is there a conformal mapping that sends the upper semi-circle to the positive (or negative) real line,

and the real interval [-1,1] to the other half of the real line? I am considering the upper semi-circle ${|z|=1, 0<arg(z)<\pi}$, with the line [-1,1] that closes the loop. I want to map the ...
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Finding a harmonic function on half-disk that is equal to $1$ on the semi-circle and $0$ on the diameter

I first showed that the mapping $$z + \frac{1}{z}$$ sends the upper semi-disk, $\{|z|<1, \operatorname{Im} z >0\}$, along with the real line from $-1$ to $1$, to the whole of the real line in ...
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Is this a continuity / connectedness argument or is it an orientation-preservation argument?

Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle. We know the fact that the upper half plane (UHP) ...
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Is it correct to say that conformal mappings (not just the class of linear transformations) preserve orientation?

We know that conformal mappings preserve angles and orientation of any two intersecting curves in the z-plane. Is this fact alone enough to conclude that the region (domain?) to the right of some ...
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Is this the right way to find the image of the interior of a half-disk, under the mapping z+1/z

I am trying to find the image of the interior of the half disk {|z|<1, Im z>0} under the mapping $$z + \frac{1}{z}$$ and the problem statement also asks to find the images of specific points A, ...
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Can I think of the conformal mapping w = (z+1/z) as a linear fractional transformation?

The mapping $$w = z + \frac{1}{z}$$ looks linear in $z$. However, it would not be in the form $$\frac{Az+B}{Cz+D}$$ since putting the two terms together gives $$\frac{z^2+1}{z}$$ So my ...
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Mobius Transformations and Circular Arcs

Suppose that a region $D$ has as a boundary $\partial D$ which consists of two circular arcs with the same end points (say $a$ and $b$). I want to show that the following Mobius transformation: $w = ...
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24 views

Inverse function of a conformal mapping

I'm trying to prove that if $f$ is a conformal mapping at $z_0$, then it has an inverse $g$ that is conformal at $w_0=f(z_0)$. I proved the existence of $g$ using the Inverse Function Thereom. Since ...
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What's wrong with this naive idea of conformal mapping? [closed]

...if I have a unit quarter-circle sector in the first quadrant, then the mapping $z^4$ maps the sector to the unit disk conformally. Why doesn't this mapping do the job? $z^2$ gives the half ...
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Mapping the Poincaré disk to hyperbolic surfaces in $\mathbb{R}^3$.

Take any hyperbolic surface with constant curvature in $\mathbb{R}^3$, such as Dini's surface, or a hyperboloid of constant curvature. If I understood things correctly, for any such surface, we ...
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Finding the Mobius transform

Just needing confirmation on my assumption for the problem in the figure below. I'm assuming since the line passes through two points on the circle that this can be mapped to a wedge by w = ...
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scalar curvature under conformal deformation of a two - dimensional Riemannian manifold

I am currently stuck with an identity that I'd love to derive myself. Suppose $(M,g)$ is a surface (a two - dimensional Riemannian manifold) without boundary. Let $\tilde g = e^{2u} g$ be a conformal ...
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Distance preserving transformations of the complex plane

Show that the most general transformation fixing the origin and preserving distances is either a rotation, or a rotation followed by a reflection in the real axis, for a transformation $f: \mathbb{C} ...