# 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 to find conformal mapping?

I know how to find conformal mapping but here I am a little confused that how can I estimate conformal mapping from the given range and domain information? A proper guide will be appreciated what ...
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### mapping complement of half disk to unit disk

Let $K=\{ Im z \geq 0, |z| \leq 1 \}$ be the closed set of the intersection of the closed disk with upper half plane. I need to find a conformal map that takes complement of $K$, i.e. $\mathbb{C} - K$ ...
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### conformal map from a disk minus a radial segment to unit disk

I need to find a conformal map from the disk minus a radial segment $\Omega$ to the unit disk $\mathbb{D}$, where $\Omega = \{ z \in \mathbb{C}: |z|<1, z \notin [\frac{1}{2},1) \}$. I have tried ...
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### conformal map $\{ Re(z) > 0 : |z-1|>1 \}$ onto unit disk

How many conformal mapping we apply to map the exterior of the disk $\{z : |z-1|>1 \}$ in the right half plane to the unit disk?
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### On the proof about the dimension of the conformal group of a manifold

I have been reading the book "Transformation Groups in Differential Geometry" by S. Kobayashi. More concretely, I am trying to understand the proof of the Theorem 6.1 of Chapter IV. Theorem ...
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### Finding the explicit expression of a conformal map

I want to find a conformal map $f$ between $D = \{z\in\Bbb C : |Arg(z)|<\frac \pi4\}\setminus[1,+\infty)$ and $D(0,R)$, for $R$ a real positive number that I have to compute as well. It also has to ...
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### Schwarz-Christoffel formula for a half-plane

I can't understand the example that was given in the book Schwarz-Christoffel Mapping by Tobin Driscoll and Lloyd Trefethen. It's formula 2.5 at page 12. By using Schwarz–Christoffel formula for a ...
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### Show that a torus is conformally equivalent to a plane

With the metric of the torus given by $ds^2=a^2d\theta^2+(b+asin\theta)^2d\phi^2$, I'm asked to find the conformal transformation which proves that a torus is conformally equivalent to a plane. I must ...
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### Is there an analytic function whose imaginary (or real) part 1) equals zero on the real axis 2) equals 1 on the imaginary axis?

The boundary conditions must only be satisfied in the first quadrant. I don't know how to even start to solve this problem, since non of my originally thought simple analytic functions satisfied these ...