# 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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### 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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### 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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What's the ... 0answers 31 views ### 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$|A|=1$,$\operatorname{Im}(z_0)>0$. ... 0answers 34 views ### 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 ... 0answers 24 views ### 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 ... 0answers 36 views ### 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 ... 0answers 21 views ### 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. 0answers 25 views ### 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 ... 1answer 40 views ### 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$\mathbb{D}$... 2answers 49 views ### 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 ... 1answer 56 views ### 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 ... 0answers 26 views ### 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$cos\theta$... 0answers 30 views ### 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 ... 0answers 27 views ### 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 ...
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 ...