Talk:Conformal map

'mapping' vs. 'map'

Should we move this page to conformal mapping? (That is now just a redirect page pointing here.) Michael Hardy 23:56, 20 Feb 2004 (UTC)

Why mapping is better than map? Tosha

The reason is reflected in today's reversion of an edit putting this page in the Geography project. For some reason, "mapping" connotes the mathematical context, such as conformal mapping. The substantive "map" has a strong tie to geographical study.Rgdboer (talk) 21:30, 17 April 2008 (UTC)

Orientation Issue

Is this page consistent in talking about preservation of orientation? Charles Matthews 08:54, 14 Sep 2004 (UTC)

No, Charles. I don't believe this page's definition of a conformal mapping is remotely correct. As I understand it, conformal mappings are mappings that are locally complex-differentiable, i.e., mappings that locally preserve angles. I'll try to research and fix this. Norbornene (talk) 02:26, 7 September 2017 (UTC)

conjugate

A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation or its conjugate.

If 'conformal' means 'preserves angles', then conjugates mobius tranfomations are not conformal - they reverse all the angles.

Example of use

I have a concern about the example recently added by 69.140.68.72: since one of the major points about a conformal map is that it preserves angles, it seems a pity to use an example which clearly changes an angle ... I don't have sufficient knowledge of applications to suggest an alternative - anyone? Madmath789 17:23, 13 June 2006 (UTC)

I clarified what was going on. Conformal mappings preserve angles, but only for points in the interior of their domain, and not at the boundary. Thus, the map $z\to z^{2}$ from the interior of the first quadrant (x>0, y>0) to the upper half plane (y>0) converts a 90 degree angle into a 180 degree one, but this map is not conformal at the origin, which is on its boundary, as there its derivative is zero. Oleg Alexandrov (talk) 18:01, 13 June 2006 (UTC)

Conformal equivalence

The phrase "conformal equivalence" is employed without giving its definition. Can someone add this? —Preceding unsigned comment added by 99.237.37.181 (talk) 22:19, 16 December 2007 (UTC)

I've never heard the term used before, but conformal equivalence seems to make sense here, where stereographic projection is the conformal transformation in this case. Can I get a second opinion before adding the wikilink please? Ben (talk) 15:22, 16 January 2008 (UTC)

Added the wikilink. Ben (talk) 21:03, 21 January 2008 (UTC)

"Higher-dimensional Euclidean space"

I'm having trouble with the section "Higher-dimensional Euclidean space". Isn't in addition to the types mentioned another conformal map possible? By letting an 2d-conformal map act on two of the dimensions and leaving the component perpendicalur to this plane unchanged? --Pjacobi (talk) 14:36, 16 June 2008 (UTC)

That's not going to be conformal, unless the two dimensional map is an isometry. Think about the 1+1 dimensional case. In $\mathbb {R} ^{2}$ if you consider a map $f(x,y)=(g(x),y)$ where $g$ is conformal in one dimension (which means very little), then the map $f$ will generally not be conformal. Oded (talk) 14:42, 16 June 2008 (UTC)

Thank you very much. I had the wrong mental image. --Pjacobi (talk) 17:31, 16 June 2008 (UTC)

Pictures

I like pictures in this article made by Christian.Mercat, which were removed by anonimus user. Should I revert this edit ? --Adam majewski (talk) 09:52, 21 February 2009 (UTC)

Please consider putting them somewhere else! They are nice but don't look serious enough for an encyclopedia article. --Anonymous

Thx for answer. (:-) I think that images are great and shows what is very hard to understand from words/equations. Regards--Adam majewski (talk) 06:36, 24 February 2009 (UTC)

Thanks talk! I liked them as well :-) Well, you know, like math should be boring, serious things are on the black board only in Finland :-) Never mind. I've put them here: Conformal pictures. Christian.Mercat (talk) 12:36, 22 April 2009 (UTC)

I think it would be appropriate to select just one of those pictures, together with the original, and display both on this page somewhere (probably not near the top), attached together in a multiple image box with all captions and footer. That would help convey an intuitive feel for what conformal transforms look like, what is and isn't preserved. I disagree with anon, they look professional. However, I don't think it is appropriate to keep such a larger number of those pictures here, and I think the Conformal pictures article should be deleted afterward. Cesiumfrog (talk) 00:35, 23 November 2010 (UTC)

Laplace equation

The transform of a solution of the Laplace equation again is a solution only in two dimensions. In general the transform must be multiplied with s^-(d-2)/2 to get a solution, where s is the local scale factor. This shouldn't be swept under the carpet.radical_in_all_things (talk) 16:40, 26 May 2012 (UTC)

I noticed this too. I have reworded a little, now at least the paragraph doesn't make a false claim. Of course, it could use some editing explaining why we are resticting to two dimensions and what is necessary to do in higher dimension Otherwise one might wonder why bother with conformal maps in higher dimension in the first place. --Giuseppe Negro (talk) 14:27, 12 June 2014 (UTC)

Physics

In the "uses" section of this article, somebody has mentioned the (quite true) fact that conformal transformations are important in general relativity. However, they've then gone and spoiled things by claiming that this is somehow related to the existence of a "force"; moreover, they claim that conformal transformations are used somehow to make general relativity describe the state of the universe before the big bang.

Needless to say, both of these things are manifestly false. If nobody has any objection, I'd like to remove them. — Preceding unsigned comment added by 82.31.22.37 (talk) 22:30, 24 November 2012 (UTC)

In General Relativity, conformal maps are the simplest and thus most common type of causal transformations. Physically, these describe different universes in which all the same events and interactions are still (causally) possible, but a new additional force is necessary to effect this (that is, replication of all the same trajectories would necessitate departures from geodesic motion because the metric is different). It is often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the big bang.

As written, I think the article is correct (but obviously lacking in sources, although the same can be said for the rest of this article and of most others on mathematics).

Because conformal transformations are designed to preserve the causal structure of the manifold, they do not alter the validity of any world line but they do alter its status as a geodesic or otherwise. That is, in general, the geodesics in a conformally transformed spacetime do not correspond to geodesics of the original (physical) spacetime. So you might ask what *do* they correspond to? The answer is to "other (non-geodesic) trajectories". What is the physical interpretation of a non-geodesic trajectory? It is the world-line of a particle that is being influenced by an external force (such as: the fuselage of an accelerating rocket, or, an electric charge in an EM field). The conformal transformation doesn't change which trajectories (and hence which interaction events) are/aren't causally permitted/prohibited, but it does change which ones will occur naturally (under solely inertial motion). So yes, a conformal transformation is equivalent to adding an additional unspecified force. And I think that also happens to be the simplest correct picture for explaining the physical significance of a conformal transformation.

There is a topic called conformal cosmology, which, among other things, can be applied to removing that singularity which prevents tracing trajectories earlier than the big bang. (You'll probably find something in this vein if you search for papers coauthored or supervised by SM Scott.) The idea is not so much to extrapolate prior to the singularity for its own sake, but merely to regularise the behaviour just at the surface of the singularity, probably for the purposes of clarifying (or classifying) some of the properties of that singularity (as compared/contrasted to other singularities in other spacetime solutions). This is only one minor example, it is not of any particularly great importance, and would easily be overstated, but nonetheless is a genuine topic of present study. Not only is it a legitimate example of current research applying conformal cosmology, but it can be stated in less than ten words in a manner which is technically valid and which also conveys to the non-specialist reader a feeling for the motivation of (and the gist of the topic of) some of the research applications for conformal transformations. The reader can at least get something from this example phrased this way, no matter how lacking their background is, and without the article needing to go into any depth. Now if you have a better example, which succesfully conveys to the average reader in only ten words a motivation for using conformal maps in GR, I for one want to hear it.

Cesiumfrog (talk) 04:08, 25 November 2012 (UTC)

External links modified

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The bot found an archived version but the formulas do not appear. The following has been removed:

• Conformal Mapping Module by John H. Mathews

An archived version with functional formulas may be replaced in External links. — Rgdboer (talk) 02:59, 13 August 2017 (UTC)

External links modified

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The following was removed as unhelpful:

• Java applet by Christian Mercat to deform pictures (link broken, applet available under ComplexMapStill.jar); MacOSX Java applet that deforms the video flux from the webcam.

This application of Conformal mapping is too particular (video flux from webcam) to support the article. — Rgdboer (talk) 02:51, 13 August 2017 (UTC)

Similar form

The following was removed but may be useful for § Complex analysis:

A conformal map is called that because it preserves the shapes of things (at an infinitesimal scale). The term is based on the Latin prefix com- (together, with, near) and the Latin noun forma (shape, appearance)

"conformal – definition and meaning". Wordnik.com. Retrieved 2013-09-05.

"English etymology of conformal". myEtymology.com. Retrieved 2013-09-05.

"conformal – Memidex dictionary/thesaurus". Memidex.com. 2013-06-26. Retrieved 2013-09-05.

The contributor inserted these dictionary citations in the section dealing with alternative angles, now § Pseudo-Riemannian geometry. — Rgdboer (talk) 01:09, 7 October 2017 (UTC)

Section "Pseudo-Riemannian geometry"

This section is not sourced. It uses concepts that are not defined here nor in linked articles. Some of these concepts are even nonsensical, such as "any [Jacobiam matrix with non-zero determinant] lies in a particular planar commutative subring and has a polar decomposition determined by parameters of radial and angular nature".

It seems that this section has been written by a fan of the old-fashioned terminology of split complex numbers, dual numbers and other hypercomplex numbers, who do not really understand the subject of the article.

A section on (pseudo)-conformal maps in Pseudo-Riemanian geometry would be useful here, if based on reliable sources. This gibberish is of no help for writing such a section.

So, I'll delete this section per WP:TNT. D.Lazard (talk) 11:04, 19 February 2021 (UTC)

Some precise definition

It would help at least one definition from many equivalent ones of conformality is given for Rn to Rn maps for all n. 213.28.228.243 (talk) 15:16, 8 December 2022 (UTC)

The definition in the 2nd sentence of the article seems precise to me.

Let $U$ and $V$ be open subsets of $\mathbb {R} ^{n}$. A function $f:U\to V$ is called conformal (or angle-preserving) at a point $u_{0}\in U$ if it preserves angles between directed curves through $u_{0}$, as well as preserving orientation.

What issue do you have with it? –jacobolus (t) 19:26, 8 December 2022 (UTC)