Talk:Stereographic projection

message: Article merged: See old talk-page here. Joshua R. Davis (talk) 03:04, 12 December 2007 (UTC)
thread_title: Merging

message: Hi, all. I'm doing a major revision of this article, because I feel that it confuses the two kinds of stereographic projection (plane tangent to pole vs. plane through equator), doesn't give enough formulas, doesn't connect enough to Riemann sphere, etc. Just letting you know. Joshua R. Davis 16:43, 26 April 2007 (UTC)
replies: text: Hi Joshua. You're welcome to work on it. I've not touched this article, but link to it also for higher dimensional polytope projections (polychoron --> 3-sphere --> 3-space hyperplane) although not described here. On the two projection types (pole vs equator), they are identical transformations, just differing by a factor of 2 in scaling. Tom Ruen 18:01, 26 April 2007 (UTC)
replies: text: Hah! I hadn't done the calculations and did not notice the simple scaling factor of 2 (by similar triangles). Still, we need to be clear about which version we're using, especially as it relates to the Riemann sphere. I think the polytopes will work in nicely. Joshua R. Davis 19:52, 26 April 2007 (UTC)

text: This confusion can be clarified by mentioning that one is a transverse aspect of the other. The polar has the point of projection 90 degrees offset from the point of projection of the equatorial. Most cartographic books refer to one as the polar aspect, and the other as the equatorial aspect. I believe this should be the way they are referred in this article Dmgerman 05:31, 4 July 2007 (UTC)
replies: text: "Aspect" is not a term in common use in mathematics; saying that one is the "transverse aspect" of the other does not clarify anything for me, personally. I think that we should try to minimize the specialized jargon in the early sections of the article. On the other hand, it would be useful to mention polar and equatorial aspect in the cartography section. Joshua R. Davis 12:49, 4 July 2007 (UTC)
thread_title:  Revising 26 April 2007 

message: This is a reference I need to add the photographic uses section:
 @misc{ margaret95perspective,
  author = "F. Margaret",
  title = "Perspective Projection: the Wrong Imaging Model",
  text = "Fleck, Margaret M. (1995) Perspective Projection: the Wrong Imaging Model,
    TR 95-01, Comp. Sci., U. Iowa.",
  year = "1995",
  url = "citeseer.ist.psu.edu/margaret95perspective.html" }
and update the text to quote her reasons why the stereographic is superior to the equi-solid. Although I would prefer a better one (but she has been cited in the literature).
Also, I need to find a list of commercially available stereographic fisheye lenses. I am almost sure they have been produced in the past. Dmgerman 05:29, 4 July 2007 (UTC)
thread_title:  Work in progress 

message: In my opinion the image in the definition is not the best. It will be much better to have
the lower  image of the projection that is tangent to the circle, as it is usually projected.Dmgerman 06:28, 4 July 2007 (UTC)
replies: text: This is not agreed upon. You might want to read the parallel discussion at Talk:Riemann sphere. There I offer evidence that the equatorial projection is more common in the math literature. (Futhermore, for purposes of the Riemann sphere the equatorial projection is manifestly superior.) The tangent projection may be more common in your literature; if so, then we should compromise somehow. Joshua R. Davis 12:55, 4 July 2007 (UTC)
thread_title: Image in the definition

message: I seems so, but I am not sure that stereographic does not also mean stereoscopic. Perhaps a note on this regard would be useful (if that is the case).
Dmgerman 06:58, 4 July 2007 (UTC)
thread_title: Is stereographic often confused with stereoscopy?

message: In a recent edit, Quota described the projection thusly: "Its intent is to show a view of the sphere as seen from a specific viewpoint. ['projections' are not "inuititive"; views are.]" I think that the wording "view...from a specific viewpoint" suggests that it is an ordinary viewing projection --- i.e. perspective projection --- which it certainly is not. Furthermore, I agree that the word "view" is more intuitive than "projection" for non-mathematicians, but the wording is/was "picturing"/"picture", which seems at least as friendly as "view". So I have changed most (but not all) of Quota's edit. If there is an objection, then we can discuss it. Joshua R. Davis 21:59, 27 October 2007 (UTC)
thread_title:  Edit to intro 

message: :Image:Globe panorama03.jpg is scheduled to be Wikipedia:Picture of the day for May 13, 2008. If some people here could check out the caption at Template:POTD/2008-05-13 and make improvements, it would be greatly appreciated, because I'm afraid I totally didn't get it and so I have no idea if what I lifted from the article even makes sense. Thanks. howcheng  {chat} 07:14, 7 May 2008 (UTC)
thread_title:  POTD notification 

message: * If you stereographically project a plane onto a sphere, and then project back onto the plane using a different angle (equivalently, by rotating and relocating the sphere) you wind up with a mobius transformation. The fact that stereographic projections can be done at higher dimensions  generalises the mobius transformations, too.
:* If you project a plane onto a sphere and then back the sphere onto the plane, but using the opposite pole, then the resulting transformation of the plane is a Circle inversion. --CiaPan (talk) 06:51, 17 January 2012 (UTC)
replies: text: Some of this could be added to the Properties section. The stuff on Mobius transformations would be better at Riemann sphere, I think. Mgnbar (talk) 14:35, 5 July 2008 (UTC)
thread_title:  Couple of points 

message: "On the other hand, it does not preserve area, especially near the projection point."
I think it's the reverse.
it does not preserve area, especially far away the projection point.
:It could be rephrased, but I think it is essentially correct as stated.  The "projection point" here is usually the North Pole, near which a given spherical area projects to an arbitrarily large planar area.  So things get as bad as they can possibly get near the projection point.  siℓℓy rabbit (talk) 23:54, 29 December 2008 (UTC)
replies: text: It is confusing, should drop the "especially", just say it doesn't preserve area! I'll change it. Tom Ruen (talk) 00:02, 30 December 2008 (UTC)

text: Better. Thanks! Tom Ruen (talk) 00:28, 30 December 2008 (UTC)

text: This is NOT area preservation property. Area preservation means a figure F and its image p(F) have equal areas: .  And it is generally NOT true—area A ratio  of a spherical figure F and its planar image p(F) in the stereographic projection grows to infinity as F gets closer to the projection point, so one can find two figures F and G of equal areas arbitrarily close to each other and still get  arbitrarily large, providing the figures are close enough to the sphere's 'north pole' and F is a bit closer to it than G. --CiaPan (talk) 06:25, 27 January 2012 (UTC)
thread_title:  Area preservation 

message: There`s an imaged labeled "Stereographic projection of a Cantellated 24-cell". I'm not an expert but I think it is a Schlegel diagram, and not a stereographic projection...
Dunno how to embed images but here it is http://en.wikipedia.org/wiki/File:Cantel_24cell2.png
-Etienne  —Preceding unsigned comment added by 76.71.239.211 (talk) 01:32, 19 February 2009 (UTC)
replies: text: Indeed, Schlegel diagram can be constructed by a perspective projection but not a stereographic one, because only vertices of polytope are on the sphere. Now, it lacks in agreement.  Temaotheos (talk) 03:39, 20 February 2013 (UTC)
thread_title: Stereographic projection of a Cantellated 24-cell.

message: Under the heading "Wulff net", the following claim is made:
:The angle-preserving property of the projection can be seen by examining the grid lines. Parallels and meridians intersect at right angles on the sphere, and so do their images on the Wulff net.
But keeping the projections of parallels and meridians perpendicular does not imply angle preservation.  For example, any equatorial right cylindrical projection projects the parallels and meridians onto a rectangular grid, but the only one that actually preserves angles in general is the Mercator.  In the others, the angles of the parallels and meridians are preserved, but oblique angles are bunched up against the meridians or the parallels.  So while the stereographic projection does preserve angles, the perpendicular images of the parallels and meridians do not demonstrate that.
-- Elphion (talk) 04:41, 2 February 2013 (UTC)
thread_title:  Misleading claim 

message: Hello! This is a note to let the editors of this article know that :File:Stereographic projection SW.JPG will be appearing as picture of the day on January 23, 2016. You can view and edit the POTD blurb at Template:POTD/2016-01-23. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. — Chris Woodrich (talk) 23:55, 4 January 2016 (UTC)
replies: text: As usual, looks great, Chris Woodrich. Thanks! Strebe (talk) 05:27, 23 January 2016 (UTC)
replies: text: In line with the change in the article, can we change the blurb for this image for when it is next featured as "picture of the day"? The stereographic projection maps all circles to circles, a much stronger statement than merely small circles to circles.--Leon (talk) 22:18, 27 February 2016 (UTC)
thread_title:  [[:File:Stereographic projection SW.JPG]] to appear as POTD soon 

message: The stereographic projection is not the only projection that maps small circles to small circles: every conformal map does this. See this for an example of another conformal map projection that has this property. However, the stereographic projection is the only projection that maps circles to circles, as attested here. Please restore my edit.--Leon (talk) 21:41, 27 February 2016 (UTC)
replies: text: The change thus far is okay, but aren't lines circles of infinite radius? Can we at least do away with the clause in brackets?--Leon (talk) 21:52, 27 February 2016 (UTC)

text: Does this dispute boil down to whether lines in the plane are regarded as circles of infinite radius? If so, then it seems best to clarify that point and link to Generalised circle, rather than ignore it. Mgnbar (talk) 00:44, 28 February 2016 (UTC)

text: Leon: “Small circle” does not refer to infinitesimal circle. It refers to a plane section of a sphere. Mercator does not map these to circles. Strebe (talk) 02:48, 28 February 2016 (UTC)
replies: text: Assuming the article Circle of a sphere is correct, then I guess I am wrong. But the article was misleading for someone unfamiliar with the definition. I reads better now.--Leon (talk) 09:15, 28 February 2016 (UTC)
thread_title:  Mapping circles to circles 

message: The main page asserts that:
: “The conformality of the stereographic projection implies a number of convenient geometric properties. Circles on the sphere that do not pass through the point of projection are projected to circles on the plane...”
I do not think this is quite right. Stereographic projection is both conformal and indeed sends circles to circles (or to straight lines). But the latter property does not follow from the former.
As a counter-example, Mercator’s projection is conformal but does not send circles to circles.
--Peter Ells (talk) 08:27, 6 July 2018 (UTC)
replies: text: You're right. And not just this one fact; actually no one property described below “The conformality of the stereographic projection implies a number of convenient geometric properties.” is directly implied by conformality.  So I'm removing the sentence from the article. --CiaPan (talk) 11:01, 6 July 2018 (UTC)

text: Thanks --Peter Ells (talk) 22:23, 6 July 2018 (UTC)
thread_title:  Conformality does not imply circles property 

message: My interest in stereographic projection (SP) arises from study of the astrolabe, which was invented by the ancient Greeks. They certainly had a practical knowledge that stereographic projection is both conformal and projects circles to circles (or straight lines). See the excellent diagram of a tympan on the astrolabe page.
According to the main article, the conformal property of SP was first proven by Edmund Halley; and according to a YouTube video, the circles property was proven by Bernhard Riemann.
My question is: Is there any evidence that the ancient Greeks could prove these properties, which they certainly knew and had a deep interest in?
Thanks, --Peter Ells (talk) 23:04, 6 July 2018 (UTC)
replies: text: What you call a 'circles property' can be proven with Euclidean geometry, just by triangles' similarity and following proportionality of their respective edges. The proof is quite long, but it consists of simple geometric steps. So I would say ancient Greeks could prove that. However, that doesn't mean they did it (and even if they did, that doesn't mean they did it that way). --CiaPan (talk) 18:57, 8 July 2018 (UTC)
thread_title:  How much could the early Greeks prove? 

message: User:Sharouser keeps adding a link to SPIC, which is just a redirect to this article. But old versions of SPIC have actual content, albeit at near-stub level. And other users keep replacing those versions with redirects to this article.
Presumably the solution is for Sharouser (or anyone) to write a decent SPIC article, probably in draft space, then get it into main space, and then introduce a link from this article to SPIC. Right? Mgnbar (talk) 10:46, 4 September 2019 (UTC)
replies: text: Ah, I see that it has been discussed for three days at Wikipedia talk:WikiProject Maps. Thanks. Mgnbar (talk) 10:56, 4 September 2019 (UTC)
replies: text: Coincidentally, I've just tagged the article for a proposed split, linked to that project page discussion.  Thanks for demonstrating my point that this will not have been seen by all interested editors.  Following a properly-notified discussion, it should be possible to implement the consensus without dispute.  Lithopsian (talk) 11:00, 4 September 2019 (UTC)
thread_title:  Stereographic projection in cartography 

message: There is currently an edit dispute about a comma in the complex analysis section. A period was changed to a comma. I reverted the comma, not realizing that it replaced a period, which was even worse. That's my mistake.
My position is that there should be neither a period nor a comma there. The period is grossly ungrammatical, while the comma is moderately ungrammatical. Shall I explain my position, or may I just remove the comma? Mgnbar (talk) 21:15, 18 June 2021 (UTC)
replies: text: IMHO do as you wish, I'm not going to edit-war. (Also, I do not know English punctuation well enough to argue either way, I just knew the period was misplaced there.) However I intend to invite I also invited the IP-user who entered the comma to join this discussion. (User talk:131.180.230.195) :) --CiaPan (talk) 16:54, 19 June 2021 (UTC)
replies: text:  Updated --CiaPan (talk) 17:17, 19 June 2021 (UTC)
thread_title:  Punctuation dispute 

message: Most of the preceding discussion relies on the alleged fact that a circle is projected either to a circle or a line. It is far to be an evidence. So, if the assertion is wrong, it must be removed. If it is true, a citation is required, and also a sketch of the proof (this is much mare important than, say, the explicit formulas for the projection that are given in details although their  derivation is purely straightforward). So. I'll tag the assertion with  and . D.Lazard (talk) 11:05, 21 April 2022 (UTC)
replies: text: I don't think that anyone is actually disputing that claim here, and sources should be easy to find, but okay. Mgnbar (talk) 11:15, 21 April 2022 (UTC)
replies: text: The alleged fact is a fact. I've added a reference to Ahlfors. The proof observes that any circle on the sphere lies in a plane a_1x_1 + a_2x_2 + a_3x_3 = 1, and hitting this with the stereographic transformation, with the restriction x_1^2 + x_2^2 + x_3^2 = 1, yields the equation of a circle or a straight line -- i.e., a circle in the extended plane. -- Elphion (talk) 13:13, 21 April 2022 (UTC)

text: An interesting approach to the problem: I don't know the proof, so the claim is dubious. On a contrary, I can say I know the proof and I'm going to remove . And no, I'm not going to provide a sketch, because that would be WP:OR. Anyway, even though the proof is based on extremely simple facts, like similarities of triangles and resulting aspect ratios, it's too long to put it into an article (it's about 12 KB in plain text, plus all the LaTeX markup to be added, plus translation to English). --CiaPan (talk) 16:48, 21 April 2022 (UTC)

text: – this was known to the ancient Greeks (the main part of the proof is the same as a theorem about circular cross sections of cones from Conics) and while no proofs remain from Greek times, we have proofs (using synthetic geometry) by medieval Islamic scholars (and then again repeatedly later); this article could include a synthetic proof either inline – it’s not too long and not too distracting – or in a footnote, but it would take someone drawing a nice figure or two. Alternately, if you start by showing that the stereographic projection is a sphere inversion, then it’s a one-liner (assuming you accept that sphere inversions map circles to circles). Alternately, it’s easy to show via coordinates: just massage the equation for a point on your circle after stereographic projection until it looks like some form you accept as unambiguously the equation for a circle (e.g. using center–radius form). –jacobolus (t) 03:12, 26 June 2022 (UTC)

text: The other important property of the stereographic projection that it preserves angles (conformality). This was also probably known to ancient Greek mathematicians/astronomers (and medieval Islamic/Indian mathematicians), but the earliest known proofs are from the late 16th century. Neugebauer includes a straightforward proof in A History of Ancient Mathematical Astronomy, which I have put online here; someone could make a nice SVG version of that figure and rewrite the proof for Wikipedia, and it could fit in this article. –jacobolus (t) 05:15, 26 June 2022 (UTC)
replies: text: These links may be helpful: Möbius plane (section 2), Inversive geometry (section 2.2) --Ag2gaeh (talk) 09:35, 26 June 2022 (UTC)
thread_title:  Circles are projected onto circles?