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Article merged: See old talk-page here. Joshua R. Davis ( talk) 03:04, 12 December 2007 (UTC)
How can we know it is Hipparchus proved that tow remakable properties of stereographic projection at first? how did he prove them?where can i find relevant reference?
thanks
66.159.177.102 14:31, 11 March 2007 (UTC)
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)
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)
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)
This section is very poor and wrong. Any stereographic is conformal, not only the polar aspects. It should be rewritten. Also, there is no point on saying that it preserves area in an infinitesimal region around the point of projection.
Dmgerman 06:36, 4 July 2007 (UTC)
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)
81.208.53.251 14:06, 21 September 2007 (UTC)Hall, I work with radars and my question is: given a point P on the Earth surface (let it be a sphere, the transformation from the WGS-84 ellipsoid to a Conformal sphere is a different complicated matter) and a point T with same projection on the surface but having altitude h, will P and T share the same projection on the stereographic plane? Let's say the stereographic plane is tangent to the Earth sphere far from the point P, it would be desirable that all the points (like T) along the vertical share the same projection, but if we keep the same transformation method, connecting T with the antipode and THEN intersecting the stereoplane, we'll have different projections P' and T' from P and T. Someone knows about? Thanks, bye 81.208.53.251 14:06, 21 September 2007 (UTC) Paul Netsaver
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)
If the equal area projection is not stereographic, what exactly is it? Mikenorton ( talk) 16:39, 20 November 2007 (UTC)
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)
"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 locally approximately preserves areas everywhere, in the sense that two small regions close together will have approximately the same ratio of areas in the image as in the domain. But not if they're far apart. Michael Hardy ( talk) 02:52, 19 February 2009 (UTC)
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)
Under the heading "Wulff net", the following claim is made:
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)
(I have revised that sentence in the article to reflect this.) -- Elphion ( talk) 05:22, 2 February 2013 (UTC)
At http://en.wikipedia.org/wiki/Circle#Equations
It says
An alternative parametrisation of the circle is:
x = a + r \frac{1-t^2}{1+t^2}\, y = b + r \frac{2t}{1+t^2}.\,
In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.
I only half understand this point. (and i would like to understand it)
Could we add it here ( at stereographic projection) as example (then we can link to this example on the circle page) — Preceding unsigned comment added by 213.205.251.130 ( talk) 18:44, 31 July 2014 (UTC)
x = a + r \frac{2t}{1+t^2}, y = b + r \frac{1-t^2}{1+t^2},
Could a knowledgeable editor please review the recent edits to Portal:Mathematics/Selected article/39? Is the addition correct? Is it so important to the subject that it is worth mentioning in a summary of this length? -- John of Reading ( talk) 08:28, 27 December 2014 (UTC)
The article features an example worded this way:
I get that sin(50°)=0.766. But where does the 0.321 or 0.557 come from? I feel like there should be an expansion of where these numbers come from and how they are they are chosen. And to make it more practical (without sacrificing the arbitrary), perhaps specify that this point might be an island or something. And explain why this feature looks like it's in the upper hemisphere but the text indicates that it's expected to be in the lower hemisphere.
I realize that this isn't exactly a cartography class, but if the contents of the article features an example, shouldn't that example be something that the reader can process? D. F. Schmidt ( talk) 17:48, 26 April 2015 (UTC)
I'm not sure that anything could help me comprehend this short of a complete class or a very thorough overview of what to do and what not to do and why this is even a thing. In short, I was probably never in the intended audience of this article. But if I ever was, I still don't understand. If I had to guess, though, no one else seems to have any issues understanding. D. F. Schmidt ( talk) 05:21, 14 May 2015 (UTC)
The first picture shows an orthographic projection, which is not a stereographic one. I removed it, but my edit was reverted by CiaPan . The picture is misleading and should be removed.-- Ag2gaeh ( talk) 09:22, 11 September 2015 (UTC)
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)
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)
Tomruen, why the insistence on “full” sphere in the description of the image you added? A clipped stereographic projection of a full sphere is semantically identical to A stereographic projection of a partial sphere. However, it is more confusing, both because “clipped” is an idiosyncratic description (“cropped” would be more precise, but even then, many readers wouldn’t grasp the significance) and because people would wonder why the article claims a “full” sphere when it evidently is not. Can we please not do this? Strebe ( talk) 19:00, 8 December 2016 (UTC)
Maybe I'm sleepy or something, but:
— Tamfang ( talk) 07:22, 14 January 2018 (UTC)
A routine for bc to automate the stereographic projection. Assumes 0 at south pole and ∞ at north pole. Polar/rectangular conversion is already included in both subroutines.
# Definitions to enable working in degrees pi = 4*a(1) define sin(z) { return s(z*pi/180); } define cos(z) { return c(z*pi/180); } define cot(z) { return cos(z)/sin(z) } define atan(z) { return a(z)*180/pi; } # The actual conversion routines # x = planar horizontal (real) axis # y = planar vertical (imaginary) axis # p = spherical zenith angle # t = spherical azimuth # Planar to spherical define from_plane(x,y) { print "p = ",180-(2*atan(1/(sqrt(x^2+y^2)))),"\n"; print "t = ",atan(y/x),"\n"; } # Spherical to planar define from_sphere(p,t) { print "x = ",cot((180-p)/2)*cos(t),"\n"; print "y = ",cot((180-p)/2)*sin(t),"\n"; }
The main page asserts that:
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)
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)
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)
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)
It's very similar to the stereographic projection, with the destination plane running through z=-1.
It uses the tangential plane at P (or line on a circle), and intersects the d-plane at a line (except for the point at infinity and Q).
I can't remember it's name - it would make a good 'see also' link.
Darcourse ( talk) 12:54, 28 January 2022 (UTC)
For the record, for a circle, if the projection line is y=0, and the intersection point for a point P is (x,0), then the line (x,-1) through P is tangent to the circle. Darcourse ( talk) 12:11, 9 February 2022 (UTC)
@Mgnbar; That's correct. With your points, we have then that triangle (P,(0,1),(0,y)) and triangle (P,(x,-1),(0,-1)) are similar (examine the line (P,(0,-1))).
Honestly, neither File:Stereographic projection in 3D.svg nor File:Stereographic_projection_grid.jpg are great images to illustrate the property that the stereographic projection maps circles to circles (treating lines as circles passing through the point at infinity). The first image only demonstrates that for the equator. The second one kinda shows that but it's hard to tell that the circles on the sphere are in fact circles. The other images on the article, File:CartesianStereoProj.png and File:PolarStereoProj.png, are ancient Mathematica screenshots and aren't great either. Apocheir ( talk) 00:43, 21 April 2022 (UTC)
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 {{ citation needed}} and {{ dubious}}. D.Lazard ( talk) 11:05, 21 April 2022 (UTC)
alleged fact that a circle is projected either to a circle or a line– 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)
Right now this article focuses on the projection from the "north pole" which sends the north pole to and sends the "south pole" to the origin. I think that the opposite projection should be preferred, for a few reasons:
1. When projecting a 1-sphere (circle) onto 1-space (line), the traditional convention is to measure angles starting from the " pole" , and to consider anticlockwise angles to be positive (in the complex plane, from toward , where 1 can be thought of as the identity rotation). The stereographic projection of these unit-circle points or unit-magnitude complex numbers is typically taken from the " pole" onto the -axis, yielding the point . In terms of an angle measure , the stereographic projection is the half-angle tangent or .
2. If we try to project unit quaternions (3-sphere) onto the "pure imaginary space" (3-space), we should expect to project from the point and send the identity rotation to the origin.
3. The typical modern way of thinking about points on a sphere is from the outside (like a geographic map) rather than from the inside (like a star chart). The common mathematical convention for thinking of the order of coordinates and the orientation of Euclidean planes and space makes it so that the stereographic projection from the "south pole" which sends the north pole to the origin preserves orientation for shapes on the sphere, whereas the opposite projection currently featured in this article reverses orientation. We might think of this as projection from the south pole coming up towards us as we look "down" from the other side, whereas for the projection from the north pole, we are still looking down, but now the projection points away from us, and leaves us looking at projected shapes oriented as they would be from the interior of the sphere.
4. The convention for the stereographic projection between Poincaré disk model and hyperboloid model associates the inside of the disk with the +t branch of the hyperboloid, and is a projection from the "south pole" which sends the "north pole" on the hyperboloid ↔ the origin in the disk.
5. The most common examples of the stereographic projection in cartography/geodesy, especially when used as a generic illustration of the projection, show the north pole at the center.
6. The north-pole-centered projection is extremely common in the scientific literature, I would guess dominant though I haven’t done a serious survey (both forms are common, as are several inferior variants that do not map the equator to the unit circle, and I have notice in skimming that plenty of recent works uncritically pull formulas directly from Wikipedia without considering which they should prefer).
7. Spherical coordinates used in mathematics books typically use longitude and colatitude (or polar angle) , where the latter starts at at the “north pole” ( direction) and measures out to at the “south pole” ( direction). It’s clearer if radius in the stereographic projection is instead of
What do others think? – jacobolus (t) 03:06, 21 June 2022 (UTC)
Your points #1, 2, 4 seem to be arguments for setting up the projection from (-1, 0, 0)– personally I just write coordinates in order :-) – jacobolus (t) 21:41, 21 June 2022 (UTC)
Mgnbar: “don't know what examples you're talking about”– This article should be about the stereographic projection in general, not only of the 2-sphere. First ("example 1") in the article should come a discussion of the stereographic projection of the 1-sphere (circle), both in terms of Cartesian coordinates on a unit-magnitude circle and in terms of angle measure . Those maps are and respectively. It is important to project from the “ pole” because then the identity rotation maps to the origin. Mapping the identity rotation to infinity is confusing. (And also doesn’t align with e.g. Weierstrass substitution, tangent half-angle formula, Circle#Equations, Parametric equation#Circle, Pythagorean triple#Rational points on a unit circle, etc.) Many important aspects of any discussion of the stereographic projection rely on understanding from the 1-dimensional case. Next (already included) there should be a discussion of the stereographic projection of the 2-sphere, but added to this should be some mention of the projection in terms of spherical coordinates in 3-space and polar coordinates on the plane. After that ("example 2") there should be some discussion of the stereographic projection of the 3-sphere representing versors (unit quaternions) onto the 3-space of “pure imaginary” quaternions, with the identity mapping to the origin. This map is Or in terms of the axis–angle representation where is a unit-magnitude “pure imaginary” quaternion representing an axis and is the half-angle of the rotation (because quaternions sandwich-multiply to rotate vectors, angles are applied twice), we have the stereographic projection These are confusingly called “modified Rodrigues parameters” in the academic literature I have seen. Then after that ("third example") should come the n-sphere transformation and should also discuss the stereographic projection of hyperbolic space from the hyperboloid to the Poincaré ball model. I am concerned that if the 2-sphere projection uses the opposite convention from all of these other stereographic projections which should also be discussed in the same article, it will be potentially confusing. – jacobolus (t) 20:12, 23 June 2022 (UTC)
jacobolus: This article should be about the stereographic projection in general, not only of the 2-sphere.
I don't agree entirely. The 1- and 3- dimensional cases are interesting, but that's not why most people come here. They're looking for an explanation of the 2-dimensional case, and the vast majority will have no interest in anything further. That's what the principal subject of this article should be about (and what it currently is almost entirely about). While it makes sense to mention generalizations, in my opinion most of what you discuss above belongs in a separate article. Certainly the first example in this article should be the 2-dimensional case, since that is by far and away what most people understand by "stereographic projection".
The material you present above is interesting math, but what I see happening is yet another case of a math article pursuing generality so energetically that it will be baffling to most readers -- a fault that affects far too many WP math articles. We should be writing for a general audience, not (at least at first) for the mathematical audience.
-- Elphion ( talk) 22:13, 23 June 2022 (UTC)
vast majority will have no interest in anything further– this is impossible to determine, but is also historically dependent: other wikipedia articles that want to talk about the half-angle tangent or general stereographic projection or geometric proofs of basic facts about the stereographic projection (important historically and very common in geometry books at least up through the 19th century) or spherical trigonometry proven using Cesàro’s method or Lexell’s theorem via stereographic projection (etc. etc.) are unlikely to link here because that material is not covered. Whereas you are likely to get links from e.g. Astrolabe and Stereographic map projection and Pole figure, because that material is covered. (The scope of the current article is something along the lines of “chapter of a mid-20th century undergraduate analytic geometry textbook”.). Many of the links to here from elsewhere on Wikipedia (e.g. the link from Möbius transformation) are not actually going to adequately satisfy readers’ curiosity / dispel their confusion, because the relevant material is missing here. What should be asked is not “what do current inbound readers expect to find” and more “in an ideal Wikipedia what should be the scope of an article with the title stereographic projection and under what article title does each topic and sub-topic most clearly fall. When the scope of an article grows too big, sections can be summarized and split off into their own articles. But that doesn’t mean leaving those topics out entirely. – jacobolus (t) 23:02, 23 June 2022 (UTC)
Would you want to standardize on projecting from (-1, 0, ..., 0)I think in the 2-sphere case, embedded in 3-space, it’s worth using coordinate names x, y, z and projecting onto the equatorial x–y plane because this is very common throughout Wikipedia and all sorts of technical literature. In my own work I like to order these (z, x, y), but I can also see how that could cause its own confusion. – jacobolus (t) 22:02, 25 June 2022 (UTC)
As an example of the kind of place where a “south pole centered” stereographic projection is much clearer, I just added this image to Gudermannian function. – jacobolus (t) 21:16, 28 June 2022 (UTC)
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Article merged: See old talk-page here. Joshua R. Davis ( talk) 03:04, 12 December 2007 (UTC)
How can we know it is Hipparchus proved that tow remakable properties of stereographic projection at first? how did he prove them?where can i find relevant reference?
thanks
66.159.177.102 14:31, 11 March 2007 (UTC)
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)
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)
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)
This section is very poor and wrong. Any stereographic is conformal, not only the polar aspects. It should be rewritten. Also, there is no point on saying that it preserves area in an infinitesimal region around the point of projection.
Dmgerman 06:36, 4 July 2007 (UTC)
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)
81.208.53.251 14:06, 21 September 2007 (UTC)Hall, I work with radars and my question is: given a point P on the Earth surface (let it be a sphere, the transformation from the WGS-84 ellipsoid to a Conformal sphere is a different complicated matter) and a point T with same projection on the surface but having altitude h, will P and T share the same projection on the stereographic plane? Let's say the stereographic plane is tangent to the Earth sphere far from the point P, it would be desirable that all the points (like T) along the vertical share the same projection, but if we keep the same transformation method, connecting T with the antipode and THEN intersecting the stereoplane, we'll have different projections P' and T' from P and T. Someone knows about? Thanks, bye 81.208.53.251 14:06, 21 September 2007 (UTC) Paul Netsaver
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)
If the equal area projection is not stereographic, what exactly is it? Mikenorton ( talk) 16:39, 20 November 2007 (UTC)
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)
"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 locally approximately preserves areas everywhere, in the sense that two small regions close together will have approximately the same ratio of areas in the image as in the domain. But not if they're far apart. Michael Hardy ( talk) 02:52, 19 February 2009 (UTC)
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)
Under the heading "Wulff net", the following claim is made:
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)
(I have revised that sentence in the article to reflect this.) -- Elphion ( talk) 05:22, 2 February 2013 (UTC)
At http://en.wikipedia.org/wiki/Circle#Equations
It says
An alternative parametrisation of the circle is:
x = a + r \frac{1-t^2}{1+t^2}\, y = b + r \frac{2t}{1+t^2}.\,
In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.
I only half understand this point. (and i would like to understand it)
Could we add it here ( at stereographic projection) as example (then we can link to this example on the circle page) — Preceding unsigned comment added by 213.205.251.130 ( talk) 18:44, 31 July 2014 (UTC)
x = a + r \frac{2t}{1+t^2}, y = b + r \frac{1-t^2}{1+t^2},
Could a knowledgeable editor please review the recent edits to Portal:Mathematics/Selected article/39? Is the addition correct? Is it so important to the subject that it is worth mentioning in a summary of this length? -- John of Reading ( talk) 08:28, 27 December 2014 (UTC)
The article features an example worded this way:
I get that sin(50°)=0.766. But where does the 0.321 or 0.557 come from? I feel like there should be an expansion of where these numbers come from and how they are they are chosen. And to make it more practical (without sacrificing the arbitrary), perhaps specify that this point might be an island or something. And explain why this feature looks like it's in the upper hemisphere but the text indicates that it's expected to be in the lower hemisphere.
I realize that this isn't exactly a cartography class, but if the contents of the article features an example, shouldn't that example be something that the reader can process? D. F. Schmidt ( talk) 17:48, 26 April 2015 (UTC)
I'm not sure that anything could help me comprehend this short of a complete class or a very thorough overview of what to do and what not to do and why this is even a thing. In short, I was probably never in the intended audience of this article. But if I ever was, I still don't understand. If I had to guess, though, no one else seems to have any issues understanding. D. F. Schmidt ( talk) 05:21, 14 May 2015 (UTC)
The first picture shows an orthographic projection, which is not a stereographic one. I removed it, but my edit was reverted by CiaPan . The picture is misleading and should be removed.-- Ag2gaeh ( talk) 09:22, 11 September 2015 (UTC)
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)
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)
Tomruen, why the insistence on “full” sphere in the description of the image you added? A clipped stereographic projection of a full sphere is semantically identical to A stereographic projection of a partial sphere. However, it is more confusing, both because “clipped” is an idiosyncratic description (“cropped” would be more precise, but even then, many readers wouldn’t grasp the significance) and because people would wonder why the article claims a “full” sphere when it evidently is not. Can we please not do this? Strebe ( talk) 19:00, 8 December 2016 (UTC)
Maybe I'm sleepy or something, but:
— Tamfang ( talk) 07:22, 14 January 2018 (UTC)
A routine for bc to automate the stereographic projection. Assumes 0 at south pole and ∞ at north pole. Polar/rectangular conversion is already included in both subroutines.
# Definitions to enable working in degrees pi = 4*a(1) define sin(z) { return s(z*pi/180); } define cos(z) { return c(z*pi/180); } define cot(z) { return cos(z)/sin(z) } define atan(z) { return a(z)*180/pi; } # The actual conversion routines # x = planar horizontal (real) axis # y = planar vertical (imaginary) axis # p = spherical zenith angle # t = spherical azimuth # Planar to spherical define from_plane(x,y) { print "p = ",180-(2*atan(1/(sqrt(x^2+y^2)))),"\n"; print "t = ",atan(y/x),"\n"; } # Spherical to planar define from_sphere(p,t) { print "x = ",cot((180-p)/2)*cos(t),"\n"; print "y = ",cot((180-p)/2)*sin(t),"\n"; }
The main page asserts that:
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)
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)
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)
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)
It's very similar to the stereographic projection, with the destination plane running through z=-1.
It uses the tangential plane at P (or line on a circle), and intersects the d-plane at a line (except for the point at infinity and Q).
I can't remember it's name - it would make a good 'see also' link.
Darcourse ( talk) 12:54, 28 January 2022 (UTC)
For the record, for a circle, if the projection line is y=0, and the intersection point for a point P is (x,0), then the line (x,-1) through P is tangent to the circle. Darcourse ( talk) 12:11, 9 February 2022 (UTC)
@Mgnbar; That's correct. With your points, we have then that triangle (P,(0,1),(0,y)) and triangle (P,(x,-1),(0,-1)) are similar (examine the line (P,(0,-1))).
Honestly, neither File:Stereographic projection in 3D.svg nor File:Stereographic_projection_grid.jpg are great images to illustrate the property that the stereographic projection maps circles to circles (treating lines as circles passing through the point at infinity). The first image only demonstrates that for the equator. The second one kinda shows that but it's hard to tell that the circles on the sphere are in fact circles. The other images on the article, File:CartesianStereoProj.png and File:PolarStereoProj.png, are ancient Mathematica screenshots and aren't great either. Apocheir ( talk) 00:43, 21 April 2022 (UTC)
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 {{ citation needed}} and {{ dubious}}. D.Lazard ( talk) 11:05, 21 April 2022 (UTC)
alleged fact that a circle is projected either to a circle or a line– 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)
Right now this article focuses on the projection from the "north pole" which sends the north pole to and sends the "south pole" to the origin. I think that the opposite projection should be preferred, for a few reasons:
1. When projecting a 1-sphere (circle) onto 1-space (line), the traditional convention is to measure angles starting from the " pole" , and to consider anticlockwise angles to be positive (in the complex plane, from toward , where 1 can be thought of as the identity rotation). The stereographic projection of these unit-circle points or unit-magnitude complex numbers is typically taken from the " pole" onto the -axis, yielding the point . In terms of an angle measure , the stereographic projection is the half-angle tangent or .
2. If we try to project unit quaternions (3-sphere) onto the "pure imaginary space" (3-space), we should expect to project from the point and send the identity rotation to the origin.
3. The typical modern way of thinking about points on a sphere is from the outside (like a geographic map) rather than from the inside (like a star chart). The common mathematical convention for thinking of the order of coordinates and the orientation of Euclidean planes and space makes it so that the stereographic projection from the "south pole" which sends the north pole to the origin preserves orientation for shapes on the sphere, whereas the opposite projection currently featured in this article reverses orientation. We might think of this as projection from the south pole coming up towards us as we look "down" from the other side, whereas for the projection from the north pole, we are still looking down, but now the projection points away from us, and leaves us looking at projected shapes oriented as they would be from the interior of the sphere.
4. The convention for the stereographic projection between Poincaré disk model and hyperboloid model associates the inside of the disk with the +t branch of the hyperboloid, and is a projection from the "south pole" which sends the "north pole" on the hyperboloid ↔ the origin in the disk.
5. The most common examples of the stereographic projection in cartography/geodesy, especially when used as a generic illustration of the projection, show the north pole at the center.
6. The north-pole-centered projection is extremely common in the scientific literature, I would guess dominant though I haven’t done a serious survey (both forms are common, as are several inferior variants that do not map the equator to the unit circle, and I have notice in skimming that plenty of recent works uncritically pull formulas directly from Wikipedia without considering which they should prefer).
7. Spherical coordinates used in mathematics books typically use longitude and colatitude (or polar angle) , where the latter starts at at the “north pole” ( direction) and measures out to at the “south pole” ( direction). It’s clearer if radius in the stereographic projection is instead of
What do others think? – jacobolus (t) 03:06, 21 June 2022 (UTC)
Your points #1, 2, 4 seem to be arguments for setting up the projection from (-1, 0, 0)– personally I just write coordinates in order :-) – jacobolus (t) 21:41, 21 June 2022 (UTC)
Mgnbar: “don't know what examples you're talking about”– This article should be about the stereographic projection in general, not only of the 2-sphere. First ("example 1") in the article should come a discussion of the stereographic projection of the 1-sphere (circle), both in terms of Cartesian coordinates on a unit-magnitude circle and in terms of angle measure . Those maps are and respectively. It is important to project from the “ pole” because then the identity rotation maps to the origin. Mapping the identity rotation to infinity is confusing. (And also doesn’t align with e.g. Weierstrass substitution, tangent half-angle formula, Circle#Equations, Parametric equation#Circle, Pythagorean triple#Rational points on a unit circle, etc.) Many important aspects of any discussion of the stereographic projection rely on understanding from the 1-dimensional case. Next (already included) there should be a discussion of the stereographic projection of the 2-sphere, but added to this should be some mention of the projection in terms of spherical coordinates in 3-space and polar coordinates on the plane. After that ("example 2") there should be some discussion of the stereographic projection of the 3-sphere representing versors (unit quaternions) onto the 3-space of “pure imaginary” quaternions, with the identity mapping to the origin. This map is Or in terms of the axis–angle representation where is a unit-magnitude “pure imaginary” quaternion representing an axis and is the half-angle of the rotation (because quaternions sandwich-multiply to rotate vectors, angles are applied twice), we have the stereographic projection These are confusingly called “modified Rodrigues parameters” in the academic literature I have seen. Then after that ("third example") should come the n-sphere transformation and should also discuss the stereographic projection of hyperbolic space from the hyperboloid to the Poincaré ball model. I am concerned that if the 2-sphere projection uses the opposite convention from all of these other stereographic projections which should also be discussed in the same article, it will be potentially confusing. – jacobolus (t) 20:12, 23 June 2022 (UTC)
jacobolus: This article should be about the stereographic projection in general, not only of the 2-sphere.
I don't agree entirely. The 1- and 3- dimensional cases are interesting, but that's not why most people come here. They're looking for an explanation of the 2-dimensional case, and the vast majority will have no interest in anything further. That's what the principal subject of this article should be about (and what it currently is almost entirely about). While it makes sense to mention generalizations, in my opinion most of what you discuss above belongs in a separate article. Certainly the first example in this article should be the 2-dimensional case, since that is by far and away what most people understand by "stereographic projection".
The material you present above is interesting math, but what I see happening is yet another case of a math article pursuing generality so energetically that it will be baffling to most readers -- a fault that affects far too many WP math articles. We should be writing for a general audience, not (at least at first) for the mathematical audience.
-- Elphion ( talk) 22:13, 23 June 2022 (UTC)
vast majority will have no interest in anything further– this is impossible to determine, but is also historically dependent: other wikipedia articles that want to talk about the half-angle tangent or general stereographic projection or geometric proofs of basic facts about the stereographic projection (important historically and very common in geometry books at least up through the 19th century) or spherical trigonometry proven using Cesàro’s method or Lexell’s theorem via stereographic projection (etc. etc.) are unlikely to link here because that material is not covered. Whereas you are likely to get links from e.g. Astrolabe and Stereographic map projection and Pole figure, because that material is covered. (The scope of the current article is something along the lines of “chapter of a mid-20th century undergraduate analytic geometry textbook”.). Many of the links to here from elsewhere on Wikipedia (e.g. the link from Möbius transformation) are not actually going to adequately satisfy readers’ curiosity / dispel their confusion, because the relevant material is missing here. What should be asked is not “what do current inbound readers expect to find” and more “in an ideal Wikipedia what should be the scope of an article with the title stereographic projection and under what article title does each topic and sub-topic most clearly fall. When the scope of an article grows too big, sections can be summarized and split off into their own articles. But that doesn’t mean leaving those topics out entirely. – jacobolus (t) 23:02, 23 June 2022 (UTC)
Would you want to standardize on projecting from (-1, 0, ..., 0)I think in the 2-sphere case, embedded in 3-space, it’s worth using coordinate names x, y, z and projecting onto the equatorial x–y plane because this is very common throughout Wikipedia and all sorts of technical literature. In my own work I like to order these (z, x, y), but I can also see how that could cause its own confusion. – jacobolus (t) 22:02, 25 June 2022 (UTC)
As an example of the kind of place where a “south pole centered” stereographic projection is much clearer, I just added this image to Gudermannian function. – jacobolus (t) 21:16, 28 June 2022 (UTC)