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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To find the image of a strip under a given mapping
Question : Find the image of the strip $0<\operatorname{Im} z<1$ under the mapping $w=(z-i) / z$ ?
My Approach : Writing $z=x+iy$, the given mapping is
$$w = u + i v = \frac{x^2+y(y-1)- i x}{x^...
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Conformal map from the region outside a semicircle to the region outside a disk of radius $\frac{R}{\sqrt 2}$ centred at the origin
Can I really get a conformal map from the region outside a semicircle, $\mathbb{C} \setminus S_R = \left\{(x_1, x_2):|x_1|^2 + |x_2|^2 = R^2, x_2 \leq 0 \right\}$ to the region outside a disk $D$ of ...
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Questions & explanation verifications about Stein and Shakarchi complex analysis chapter 8 proposition 1.1
I'm an undergrad taking my first complex analysis course, and I ran into something I wasn't completely clear about in Stein & Shakarchi's complex analysis book right at the beginning of the ...
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Mapping that fixes a point after a rotation
What is the mapping that rotates the entire complex plane through an angle $\theta$ about a given point $z_{0}$.
My approach : The mapping $w_{1}=e^{i \theta} z $
rotates the plane through the angle $\...
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Show that a biholomorphic map f is conformal
Let $f: \mathit{D} \to \mathit{D'}$ be biholomorphic map between two domains $\mathit{D}$ and $\mathit{D'}$. By considering the equation $f(f^{-1}(w)) = w$ (for $w \in \mathit{D'}$) show that $f$ is ...
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Conformal mapping of a doubly-connected polygonal domain onto an annulus
Currently, I have a domain D that is bounded by two polygons $C_1$ and $C_2$ (with $C_2$ completely 'inside' $C_1$). Now I want to map this domain onto $D'$ with the outer circle $C_1'$ (with unit ...
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Generating area-preserving maps of the plane
What's the most efficient way of generating area-preserving maps of the plane? I know that Hamiltonian flows are area-preserving, but are all smooth, well-behaved, orientation-preserving area-...
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Conformal mapping of biaxial symmetric shapes into an (unit) circle [closed]
I would like to know if there is some kind of coordinate mapping (perhaps conformal mapping) that can map the perimeter of the circles that cross each other (like shown in the picture) into a (unit) ...
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Expression for Finite Blaschke Products
We know that every proper analytic map from the unit disk to itself is a finite Blaschke product, i.e.
$$f(z) = e^{i\theta}\prod_{i=1}^n \dfrac{z - a_i}{1 - \bar{a_i}z}$$
for some $\theta$ and some $...
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Does it make sense to talk about a square or rectangular cyclide when we invert a square or rectangular torus?
In Conformal Tiling on a Torus, the author John M. Sullivan provides a conformal parameterization of the torus. The conformality is associated to the so-called aspect ratio $s = \sqrt{\frac{R^2}{r^2}-...
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Why is conformal mapping important for Riemann Sphere
The mapping of the complex plane to the Riemann sphere is conformal. As a result, many sources contend that it's this property of conformal that makes the concept of Riemann Sphere non-trivial. ...
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Mapping the upper half-plane onto rhombus
Book's question: Map the upper half-plane $\Im z>0$ onto a rhombus in the $w$-plane with angle $\alpha\pi$ at the vertex $A=0$ and side $d$. The correspondence of the points is given by $A=0\to z=0$...
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evaluate integral with the kernel of Green function (fundamental solution of Poisson equation) of infinite domain using conformal map
I must find the following definite integral
$$\phi(x,y) = \int_{0}^{\alpha} f(\theta)\frac{-1}{2\pi} \ln \sqrt{(x-\rho\cos\theta)^2+(y-\rho\sin\theta)^2} \rho d\theta$$
for a given $f(\theta)$ which ...
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Prove $\sqrt{ z − p} − i \sqrt{p}$ sends exterior of $y ^2 = 4px$ to half-plane $x>0$ (Complex analysis)
How to solve Q4 in Exercises 13.6 of the book "Complex Analysis" by Ian Stewart? Here's the problem:
With a suitable choice of the square root, show that
$f(z) =
\sqrt{
z − p} − i
\sqrt{p}$
...
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(locally) symmetric spaces where every conformal transformation is an isometry
By an argument using Liouville's theorem for conformal maps Conformal automorphism of $H^n$ it can be shown that every conformal automorphism of a hyperbolic manifold is an isometry.
Are there any ...
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Conformal group of the Weeks manifold
The Weeks manifold https://en.wikipedia.org/wiki/Weeks_manifold is an arithmetic hyperbolic 3-manifold known for having the smallest volume of any closed orientable hyperbolic 3-manifold (if you ...
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Is the group of conformal automorphisms of a finite volume hyperbolic manifold finite?
Let $ M $ be a Riemannian manifold. Let $ Iso(M) $ be the group of isometries of $ M $.
Let $ Conf(M) $ be the conformal group of $ M $ (the group of all diffeomorphisms of $ M $ that preserve the ...
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Hyperbolic and Conformal Geometries applied to Relativity Physics - Are there other geometries?
If we analyse the light spheres in two inertial frames $K_1$ and $K_2$ we have
$$x_1^2+y_1^2+z_1^2-c_1^2t_1^2=0$$
$$x_2^2+y_2^2+z_2^2-c_2^2t_2^2=0$$
(making no assumptions as the constancy of $c$ at ...
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Is $f(z)=z+\frac{1}{10}(1-z)^3$ a maps which sends the unit disk to itself?
I was currently viewing the paper Rigidity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary, by Daniel M. Burns and Steven G. Krantz. In this paper they prove the following statement.
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Conformal mapping from a wedge to the upper half complex plane
I found (L. J. Laslett, "On Intensity Limitations Imposed by Transverse Space-Charge Effects in Circular Particle Accelerators", Proceedings of The 1963 Summer Study on Storage Rings, ...
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Condition for non-negative sectional curvature
Is there any equivalent condition for a metric to have all sectional curvatures less or equal to zero?
Basically, I have done some conformal perturbation on a Riemannian metric of negative sectional ...
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Conformal mapping from quadrics to the plane
I am writing a raytracer (a type of 3D engine) to render quadrics, and I am working on rendering a one-sheeted hyperboloid defined by the zero set of $x^2-y^2+z^2=1$. I need 2D coordinates $a \in [0, ...
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How to find conformal map from half plane x<-y to circle
I don't quite get the procedure of finding a conformal map.
For example the following exercise: Give a conformal map that pictures the half plane x<-y onto the disk with center 2i and radius 1.
I ...
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Is there a conformal map between euclidean and nil geometry?
I'm looking for a conformal map (not necessarily injective or surjective) from ordinary 3-dimensional euclidean space (or a region of it) to some example of nil geometry; I would be most interested if ...
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Boundary expansion of conformal map
Let $B$ be a finitely connected bounded planar domain, and suppose $z_0$ is a boundary point of $B$ for which there is an arc $\gamma : [0,1) \rightarrow B $ (injective continuous function) satisfying ...
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How to find a conformal mapping which maps the region between |z + 3| < √ 10 and |z − 2| < √ 5 onto the interior of the first quadrant?
Yesterday I spent some time trying to find a conformal mapping which maps the region between |z + 3| <
√10 and |z − 2| <
√5
onto the interior of the first quadrant.
I started with determining ...
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Conformal map from righthalfplane without line segment onto unit disk
For my problem I need to find a conformal map from $\Omega = \{ z \in \mathbb{C} \ | Re(z)>0\} \setminus [0,1]$ to $ \mathbb{D} = \{z \in \mathbb{C} \ | \ |z|<1 \}$. I also need to map two to ...
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(informal) Is the boundary of locally conformally flat Manifold also locally conformally flat [closed]
Just a quick informal Question. I was wondering if the boundary of an LCF Manifold is also LCF ?
By definition a LCF manifold is one in which, for each point, there exists a neighborhood around that ...
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Conformal equivalence of metrics: different definitions in discrete and continuous case
I currently study discrete conformal maps and read the paper "Discrete conformal maps an ideal hyperbolic polyhedra" by Bobenko, Pinkall and Springborn. Consider the following definitions:
...
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Calculation about ellipsoid and unit sphere are conformal
$S^2$ is the topological sphere. Consider two differet metric on it, respectively denote $(S^2,g_1), (S^2, g_2)$. The two metric are conformally equivalent if
$$
g_1=\lambda g_2
$$
where $\lambda:S^2\...
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Connected components of image of non-degenerate boundary component
This is a follow up to my previous question, asked and answered here: Connected components of conformal image of boundary
I omitted this by accident from the last question, so I have created this ...
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Connected components of conformal image of boundary
Let $f : G \rightarrow \mathbb{D}$ be a biholomorphism (a holomorphic map with holomorphic inverse), and suppose $G$ is a bounded open subset of the plane. Let $C$ be a compact connected subset of the ...
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Conformal mapping from triangle to upper half-plane
I try to understand the answer to the following question as I want to deepen my knowledge about conformal mapping: Conformal mapping from triangle to upper half plane.
I do not have enough knowledge ...
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Conformal map between simple and doubly connected domains
I know that, according to the Riemann mapping theorem, for any non-empty simply connected open subset $U$ of the complex plane $\mathbb{C}$ which is not all of $\mathbb{C}$, then there exists a ...
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projective transformation that preserves angle at a point
I try to find all projective transformations on $\mathbb RP^2$ that fix the point $[0,0,1]$, and preserve the angle between any two lines through $[0,0,1]$.
Using SageMath to calculate Jacobian I ...
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Find conformal map $h$ from the circular sector common to circles $ | z-1| = 1$ and $ |z-i| = 1$ to the right-half plane. Find $h^{-1}(1)$
I am trying to first construct a fractional linear transformation, to map the sector to a half or full circle. From there it seems straightforward using rotations, translations, and standard mappings ...
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Conformal map on the unit circle
I would like to find a conformal map from the unit circle to the unit circle so that point(a,0) is mapped onto (0,0), while every point on the circumference is mapped onto itself.
Is there an easy way ...
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Best Recursive Subdivision Tiling Mapping Function
I am trying to create subdivision tilings inspired by the work of Brian Rushton (eg. page 77 of this paper). The challenge is to subdivide tiles according to a certain rule, and then apply this rule ...
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Understanding Conformal Equivalences
In the book "Computational Conformal Geometry" by David Gu, the author writes:
"Riemann uniformization theorem states that all surfaces in real life can be conformally mapped to one of ...
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Circle-preserving implies angle-preserving?
The question:
Can someone help me analyze the correctness of this statement, and/or in what environments (or under what modifications) it could be correct?
If $f:E_1\mapsto E_2$ maps circles and ...
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Can any conformal mapping on a bounded multiply-connected domain be approximated by polynomials?
Mergelyan's theorem states that a necessary condition for a function, which is analytic inside a compact set $X$ in the complex plane $\Bbb C$ and continuous on the boundary of $X$, to be uniformly ...
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Bounds for analytic map from unit disk to right half-plane
As discussed at Bounds on a mapping from unit disc to left half plane, for analytic map $f$ from unit disk to left or right half plane with $f(0)=\pm 1$ (depends on left or right half-plane), we must ...
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A problem with conformal mapping and complex analysis
Map onto the unit circle $|w|<1$ the interior of the circle $|z|<2$ with the circle $|z+1|\leq 1$ thrown out with a cut on the real axis $\left \{ y=0; 0\leq x\leq \frac{1}{2} \right \}$
My ...
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Deriving the coordinate formula for the Gauss curvature under conformal map using the moving frames method
As an exercise, I wanted to derive this formula for the Gaussian curvature (with n=2) under a conformal map:
$$\tilde{K}=e^{-2\rho}K-e^{-2\rho}\Delta\rho$$
with the method of moving frames.
Given two ...
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Transformations on a piece of paper
Consider that I have a piece of paper, and I do a drawing on such paper. Now, consider the set of all possible "transformations" I can apply to this piece of paper that do not tear it. For ...
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Extract Spherical coordinates of panorama from camera pose information and panorama (+pixel) size for correct position in a 3D space.
I am working with spherical image representation. I have the following camera pose information:
[pose.rotation.x, pose.rotation.y, pose.rotation.z, pose.rotation.w],
[pose.translation.x, pose....
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Isometries of the Disc with respect to hyperbolic metric
This is a Problem 3 part b from Chapter 8 from the book Complex Analysis by Stein and Shakarchi.
Show that if $\phi : \mathbb{D} \to \mathbb{D}$ preserves the hyperbolic distance then $\phi$ or $\...
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How to prove this identity holds for conformal mappings?
I have recently encountered a problem in complex analysis. Suppose we have domains $D$ and $\tilde{D},$ and let $\phi: D \to \tilde{D}$ be a conformal mapping. Suppose we also have functions $f,g\in C^...
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Showing that the Schwarz-Christoffel Integral is a biholomorphic mapping
I want to show that the Schwarz-Christoffel Integral
$$
I(z)=\int^z_0\dfrac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}
$$
is a biholomorphic mapping from the upper half complex plane to the rectangle. If $I$ is ...
- 1
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0
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Conformal mapping $|z-z_0|\leq R$ onto $|w|\leq 1$
I'm trying to probe that the region $|z-z_0|\leq R$ maps conformally onto the unit circle $|w|\leq 1$ under the bilinear transformation:
$$w=f(z)=\frac{R(z-\alpha)}{R^2-(z-z_0)(\overline\alpha-\...
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