What is Conformal mapping: Definition and 57 Discussions

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let


{\displaystyle U}


{\displaystyle V}
be open subsets of



{\displaystyle \mathbb {R} ^{n}}
. A function



{\displaystyle f:U\to V}
is called conformal (or angle-preserving) at a point




{\displaystyle u_{0}\in U}
if it preserves angles between directed curves through



{\displaystyle u_{0}}
, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types.
The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds.

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  1. Ebanflo

    I Is any attention being given to Conformal Cyclic Cosmology?

    Conformal Cyclic Cosmology, or CCC, is a hypothesis put forward by Roger Penrose in the early 2000s. My understanding of physics is lacking so my explanation will not be that clear, but I will summarize it here. Essentially, the existence of a previous spacetime, or "aeon," is postulated. This...
  2. nomadreid

    I Video (analytic continuation) seems to mix 4-D & 2-D maps

    The question here is not asking for links to help understand analytic continuation or the Riemann hypothesis, but rather help in understand the bits of hand-waving in the following video’s explanations : https://www.youtube.com/watch?v=sD0NjbwqlYw (apparently narrated by the same person who does...
  3. M

    I Conformal Mapping Wedge to Plate

    Hi PF! Does anyone know the conformal map that takes a wedge of some interior angle ##\alpha## into a half plane? I'm not talking about the potential flow, just the mapping for the shape. Thanks!
  4. hoalacanhdk

    Application of Conformal Mapping in Physics

    Hi, I'm studying about Conformal Mapping in Complex Analysis and see its applications in Heat transfer, Fluid and Static Eletrocity. But it is said that this subject is very useful in many branches of Physics. Can you tell me about that? Thanks.
  5. tommyxu3

    I Make a Pseudo-Riemannian Metric Conformal

    Hello everyone: I studied in differential geometry recently and have seen a statement with its proof: Suppose there is a Riemannian metric: ##dl^2=Edx^2+Fdxdy+Gdy^2,## with ##E, F, G## are real-valued analytic functions of the real variables ##x,y.## Then there exist new local coordinates...
  6. D

    I Applications needing fast numerical conformal maps

    What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask. As part of an engineering project I've implemented...
  7. Zeeree

    Mobius transformation for the first quadrant

    Homework Statement Find the images of the following region in the z-plane onto the w-plane under the linear fractional transformations The first quadrant ##x > 0, y > 0## where ##T(z) = \frac { z -i } { z + i }## Homework EquationsThe Attempt at a Solution [/B] So for this, I looked at the...
  8. B

    A How can conformal mapping be used to convert curves between different maps?

    I know the concepts of conformal mapping and complex mapping but I didn’t see none explanation about how apply this ideia and formula for convert a curve, or a function, between different maps. Look this illustration… In the Cartesian map, I basically drew a liner function f(x) = ax+b...
  9. B

    Quadratic Conformal Mapping with Parameters | QP

    I found this formula for doing a quadratic conformal map with parameters: I think there's probably a nice Einstein notation representation of this above but I haven't figured it out yet.. But anyway the mapping is like below: I don't know enough about General Relativity to know how this...
  10. J

    Complex Analysis: Conformal Mappings

    I am looking for conformal transformations to map: 1. Disk of radius R to equilateral triangular region with side A. 2. Disk of radius R to rectangular region with length L and width W. 3. Disk of radius R to elliptic disk with semi-major axis a and semi-minor axis b. Thanks!
  11. PhysicsKid0123

    Conformal mapping definition

    "Definition: A map ƒ: A ⊂ ℂ→ ℂ is called conformal at z0, if there exists an angle θ ∈[0,2Pi) and an r > 0 such that for any curve γ(t) that is differentiable at t=0, for which γ(t)∈ A and γ(0)= z0, and that satisfies γ ' ≠0, the curve σ(t) = ƒ(γ(t)) is differentiable at t=0 and, setting u =...
  12. D

    Conformal Mapping: Sketch Regions & Find Mapping

    Hi, I need to sketche ach of the following regions: R = {z :|z| < √2, 7π/16 < Argz<9π/16}, R1 = {z :|z| < 16, Rez>0} and write down a one-one conformal mapping f1 from R onto R1. Here is my sketch https://onedrive.live.com/redir?resid=4cdf33ffa97631ef%2110238 But I'm finding hard to find the...
  13. M

    Other Conformal Mapping: Textbook for Electrodynamics Learning

    What's a good textbook to learn conformal mapping in electrodynamics?
  14. K

    Conformal mapping from polygon with circle segments

    I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping. Anyone has any tips?
  15. J

    Conformal Mapping: Transforming Polygons to Circles?

    Is there a conformal mapping that transforms regular polygons (e.g. triangle and square) to circle?
  16. KleZMeR

    Conformal Mapping: Find Points of Z Plane for f(z)=-(1-z)/(1+z)

    Homework Statement With The map f(z) = -(1-z)/(1+z) where z=x+iy, and f(z) maps z onto w = u + iv plane. show for which points of the z plane this map is conformal. Homework Equations The Attempt at a Solution I have read a lot about this subject, and I think I...
  17. S

    Conformal Mapping: Find Function to Map Between Two Parallel Lines

    Homework Statement Find function that maps area between ##|z|=2## and ##|z+1|=1## on area between two parallel lines. Homework Equations The Attempt at a Solution I don't know how to check if my solution works for this problem? I used Möbious transformation...
  18. D

    Conformal mapping of an infinite strip onto itself

    Homework Statement Find a conformal mapping of the strip ##D=\{z:|\Re(z)|<\frac{\pi}{2}\}## onto itself that transforms the real interval ##(-\frac{\pi}{2},\frac{\pi}{2})## to the full imaginary axis.The Attempt at a Solution I tried to map the strip to a unit circle and then map it back to the...
  19. P

    Conformal mapping w=1/z - question.

    Hi, Homework Statement I'd like to show that the mapping w=u+iv=1/z tranforms the line x=b in the z plane into a circle with radius 1/2b and center at u=1/2bHomework Equations The Attempt at a Solution z*w=1=(b+iy)(u+iv) → 1=|(bu-yv)+i(bv+yu)| → u2+v2=1/(b2+y2) Now, a circle with radius 1/2b...
  20. diegzumillo

    Electrostatics - method of conformal mapping (from Landau)

    I'm studying Landau's Electordynamics of continuous media and, although I like how succinct it is, sometimes it is too succinct! I'm having trouble with a particular passage, so I'll just try to summarize the section up until the part I don't understand. The topic at hand is electrostatic field...
  21. N

    Conformal Mapping and flow normal to ellipse

    Hi, Given that the flow normal to a thin disk or radius r is given by \phi = -\frac{2rU}{\pi}\sqrt{1-\frac{x^2+y^2}{r^2}} where U is the speed of the flow normal to the disk, find the flow normal to an ellipse of major axis a and minor axis b. I can only find the answer in the...
  22. C

    Conformal Mapping - Can't Prove Analyticity

    Homework Statement We have the conformal map w = f(z) = z + K/z. Prove this mapping is indeed conformal. Homework Equations z = x + iy A map w = f(z) is conformal if it is analytic and df/dz is nonzero. f(z) = u(x,y) + iv(x,y) The Attempt at a Solution df/dz = 1 - Kz^-2 =/= 0 for finite...
  23. M

    Conformal Mapping for Transforming Regions: Finding a Function

    Hello folks, I am trying to find a conformal mapping transform function that maps the following region in z-plane into interior of a unit circle in w-plane: |z-i|<\sqrt{2}\text{ ...AND... }|z+i|<\sqrt{2} Many thanks in advance for help & clues. Max.
  24. C

    Conformal Mapping: Is Non-Analytic Point Conformal?

    A theorm I took down in class says: Consider the analytic function f(z). The mapping w=f(z) is conformal at the point z0 if and only if df/dz at z0 is non-zero. However, if df/dz does not exist at that point z0, is that point still a conformal mapping? That would make the function...
  25. A

    Hi,I need a conformal mapping that changes the superellipse to an

    hi, I need a conformal mapping that changes the superellipse to an easier shape. if anyone send me any helpful thing (relative article, idea) I will be so pleased.
  26. A

    Conformal mapping fast help

    Exam tomorrow and I am lacking understanding of conformal transformations and their applications. Can someone therefore point the main properties of conformal mappings that are used to make the conclusions in the following type of exercises: the mapping f(z) = 1 + 1/z maps the unit circle...
  27. L

    Solving Laplace's equation using conformal mapping

    I'm trying to use conformal mapping to solve for a function u(x,y) satisfying Laplace's equation ∇2u = 0 on the outside of the unit circle (i.e. the complement of the unit disk), with boundary conditions: u = 1 on the unit circle in the first quadrant, u = 0 on the rest of the unit circle...
  28. T

    Conformal Mapping: Part II - Finding u and v for Given Values of x and y

    Homework Statement part ii of http://gyazo.com/0754ea00b2a4ea4a4d171906f6bf28bf Answers http://gyazo.com/821f370c502cd20210925f8498d18fa1 Homework Equations I did part i. I had to spot that 1/(x+iy)^2 = 1/(x^2+y^2)^2... (I subbed y = y-1) is this a standard result? Should...
  29. D

    MHB Conformal Mapping of Strip -1 < Im(z) < 1

    Describe the image of the strip $\{z: -1 < \text{Im} \ z < 1\}$ under the map $z\mapsto\dfrac{z}{z + i}$ So I know that $-\infty < x < \infty$ and $-1 < y < 1$. Then $$ \frac{x + yi}{x + i(y + 1)} $$ Now if I take the the line y = -1, I have $$ \frac{x-i}{x} $$ Then find out what happens...
  30. A

    Electrostatic Conformal Mapping Problem

    Homework Statement The transformation z=1/2(w + 1/w) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface. (b) Use...
  31. A

    Electrostatic Conformal Mapping Problem

    Homework Statement The transformation z=\frac{1}{2}(w + \frac{1}{w}) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface...
  32. R

    Conformal mapping between two half space

    Hi all, Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance. Regards, Tony
  33. A

    Conformal mapping of images

    Hi everybody, I was looking at the following link: http://www.dimensions-math.org/Dim_CH5_E.htm The section 6 deals with conformal mapping of the image for different kinds of transformations. I tried to reproduce them in mathematica for the transformation z \rightarrow z^2. I followed the...
  34. A

    Conformal mapping application- electrostatics

    Homework Statement Consider the transformation: w = i[(1-z)/1+z)] Find the electrostatic potential V in the space enclosed by the half circle x^2 + y^2 = 1, y =>0 and the line y = 0 when V = 0 on the circular boundary and V = 1 on the line segment [-1,1]. Homework Equations w = u +...
  35. A

    Conformal Mapping: How Do I Map the Region Above the x-axis?

    What I'm trying to do is to apply conformal mapping and map the area bounded by the x-axis and a line at 60 degrees to the x-axis to the region above the x-axis. I think the basic goal of what I'm trying to do is to map \pi/3 to \pi. My problem is I really have no idea where to go from there...
  36. A

    Capacitance calculation by using conformal mapping

    Hi all How to calculate the capacitance of a sphere-plane system by using conformal mapping? Thanks
  37. C

    Conformal Mapping: Transform a Circle to a Rectangle

    Hello! Please I need some help with this: Is it possible to transform a circle into a rectangle? If so what would be the expressions of x' and y' in terms of x and y. Thank you in advance!
  38. S

    Conformal Mapping of Upper Half Plane from Unit Half Disc | Homework Solution

    Homework Statement Find a conformal mapping f of the set V to the upper half plane H+ = {z | Im(z) > 0 where V = {z: |z| < 1 and Im(z) > 0} Homework Equations None, really. It's worth noting that V is the unit half disc. The Attempt at a Solution I have a Mobius transformation S...
  39. E

    Double Check My Conformal Mapping Steps for Quarter-Plane

    Homework Statement Can someone double check my mapping steps I've taken? The domain includes the quarter-plane Re(z)>0 and Im(z)>0, with a branch cut from the origin to the point \sqrt{3}exp(pi*i/4) The Attempt at a Solution I want to map the region from the quarter-plane to the...
  40. F

    Finding Center and Radius of Circle in Conformal Mapping

    Can you tell me is my solution true of the next problem. Find center w_0 and radius R of the circle k, in which the transformation w=\frac{z+2}{z-2} converts the line l:\text{Im} z+\text{Re} z=0. Solution: 2 \to\infty -2i=(2)^*\to w_0...
  41. E

    Complex variables conformal mapping trig identity

    Homework Statement map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation} on some domain which contains z=e^{i\theta}. \theta between 0 and \pi Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The...
  42. E

    Describing Biholomorphic Self Maps of Punctured Plane

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  43. P

    Conformal mapping problem(bilinear type)

    Homework Statement Hi all. We are asked to transform the shaded area in below figure to between two concentric circles, an annulus. Where these circles' center will be is not important, just transform the area to between any two concentric circles. As you see in figure, shaded area is whole...
  44. C

    Conformal Mapping: Exterior Circle to Interior Hexagon

    Homework Statement I'm trying to find a function that map the exterior of a circle |z|>1 into the interior of a regular hexagon. Homework Equations The Attempt at a Solution I have tried mapping the exterior to the interior circle. Then mapping interior circle to the upper plane which then I...
  45. I

    Mapping for Potential Distribution in a Straight Line Capacitor?

    Hi, my question is what mapping to use for the problem in the picture attached. I need to be able to find the potential distribution etc by mapping from the x-y plane (as pictured) to a straight lines plane capacitor, which would be pretty straightforward, but I can't find this map in any...
  46. N

    Complex numbers: Conformal mapping

    Homework Statement Hi all. I have seen a conformal mapping of z = x+iy in MAPLE, and it consists of horizontal and vertical lines in the Argand diagram (i.e. the (x,y)-plane). On the Web I have read that a conformal map is a mapping, which preserves angles. My question is how this...
  47. L

    Conformal Mapping L to Sector: Find Angle α

    Let L:=\{z:|z-1|<1\} \cap \{z:|z-i|<1\}. Find a Mobius transformation that maps L onto the sector \{z: 0< arg(z) < \alpha \}. What is the angle \alpha? no idea of how about to set up the problem The intersection of the two circles forms a lens shaped region L with boundary curves, let's...
  48. L

    Conformal mapping of sn(z,m) function

    Hi all friends, I am working on tracer in the oil field. In attempt to understand the analytical solution of tracer breakthrough in 5-spot pattern I'm reading the paper of Brigham and Abbasadez. For obtaining of potential field, they used some mapping methods from z plane to w plan, with...
  49. O

    Complex potential / conformal mapping

    Hi everyone, (I hope I'm posting it in the right place, please feel free to move this thread to the appropriate place) My high school graduation project is about the application of the theory of complex variables in physics. Specifically, I am learning about the complex potential, its...
  50. S

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    I'd like to map the open unit circle to the open ellipse x/A^2 + y/B^2 = 1. How would I go about doing this? I really have no idea how to go about doing these mappings. I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back...