PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
conformal mapping (Definition)

A mapping $ f \colon \mathbb{R}^m \to \mathbb{R}^n$ which preserves the magnitude and orientation of the angles between any two curves which intersect in a given point $ z_0$ is said to be conformal at $ z_0$. A mapping that is conformal at any point in a domain $ D$ is said to be conformal in $ D$.

An important special case is when $ m = n = 2$. In this case, we may identify $ \mathbb{R}^2$ with $ \mathbb{C}$ and speak of conformal mappings of the complex plane. It can be shown that a mapping $ f \colon \mathbb{C} \to \mathbb{C}$ is conformal if and only if $ f$ is a complex analytic function.

If $ m = n$, then we can study invertible conformal mappings. It is clear from the definition that the composition of two such maps and the inverse of any such map is again an invertible conformal mapping, so the set of such mappings forms a group. In the case $ m = n > 2$, the group of conformal mappings is finite dimensional and is generated by rotations, translations, and spherical inversions.

This notion of conformal mappings can be generalized to any setting in which it makes sense to speak of angles between curves or angles between tangent vectors. In particular, one can consider conformal mappings of Riemannian manifolds. It can be shown that, if $ (M,g)$ and $ (N,h)$ are Riemannian manifolds, then a map $ f \colon M \to N$ is conformal if and only if $ f^* h = s g$ for some scalar field $ s$ (on $ M$).



"conformal mapping" is owned by rspuzio. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: quasiconformal mapping, conformal mapping theorem, angle between two lines

Also defines:  conformal

Attachments:
example of conformal mapping (Example) by Johan
proof of criterion for conformal mapping of Riemannian spaces (Proof) by rspuzio
simple example of composed conformal mapping (Example) by pahio
inversion of plane (Topic) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: field, scalar, Riemannian manifolds, tangent vectors, inversions, translations, rotations, generated by, finite dimensional, group, inverse, composition, clear, invertible, complex analytic function, complex plane, domain, point, intersect, curves, angles, orientation, preserves, mapping
There are 30 references to this entry.

This is version 13 of conformal mapping, born on 2003-04-28, modified 2006-11-03.
Object id is 4219, canonical name is ConformalMapping.
Accessed 14020 times total.

Classification:
AMS MSC: 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
 53A30 (Differential geometry :: Classical differential geometry :: Conformal differential geometry)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)