Talk:Stereographic projection

Merging

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

History

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

I put that assertion there and I think I got that information from Heinrich Dörrie's book of problems of elementary mathematics, which is clearly not a primary source. I suspect it can be found among Thomas Heath's translations of works of ancient Greek mathematicians. Michael Hardy 21:10, 11 Feb 2004 (UTC)

This assertion is widespread in popular sources and among mathematicians who chose to write about history, and is false. Circle-preservation was from Apollonius of Perga, Conics. For historical material, read the edition by Taliaferro, not the one by Heath. The reason is that Heath did more than just translate: he shuffled the propositions around to conform to his own ideals about how the proofs should be structured. He is up-front about this in the preface but most readers seem to ignore it. Apollonius is also the source for straight lines being a kind of circle. This follows from his definition of circles as the locus of a point related to two other points by a distance ratio.

Conformality does not seem to have been demontrated until the seventeenth century.

66.159.177.102 14:31, 11 March 2007 (UTC)

According to Lee (Conformal projections based on elliptic curves, 1976) the first to demonstrate its conformality was Leadbetter in 1728.Dmgerman 16:19, 4 July 2007 (UTC)

Revising 26 April 2007

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)

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)

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)

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)

"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)

Work in progress

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)

Image in the definition

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 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)

Uses in cartography

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 agree that it is poorly written, but I do not see how it is wrong. The current wording does not claim that non-polar aspects are not conformal; it does not discuss non-polar aspects at all. As for area preservation, the equatorial aspect preserves area around the equator, while the polar aspect preserves it around the (opposite) pole. If you're interested in the area around the pole, then infinitesimal area preservation is useful, at least mathematically. But perhaps it's not useful for cartography, because you have to rescale everything anyway based on the radius of the Earth? Joshua R. Davis 13:02, 4 July 2007 (UTC)

Hi Joshua. There is no area preservation at all. As soon as you move away from the center of projection the scale is changed, hence area is never preserved. What is preserved is scale (for the equatorial aspect along any parallel). So perhaps it is a problem of semantics. With respect to the claim (non-polar aspect being conformal) it is misleading. If nobody objects I'll change it in the next daysDmgerman 17:11, 4 July 2007 (UTC)

I agree that the dispute here is semantic. I don't know what "scale" means. Just to clarify, there is no "finite" area preservation anywhere, in that (almost) no region is mapped to one of the same area. However, there is "infinitesimal" area preservation at the equator (in the equatorial aspect), in that along the equator $X^{2}+Y^{2}=1$, so

$dA={\frac {4}{(1+X^{2}+Y^{2})^{2}}}\;dX\;dY=dX\;dY,$

which means that the stretching/shrinking factor of area goes to 1 as one approaches the equator. A similar result holds at the pole in the polar aspect. Joshua R. Davis 22:01, 4 July 2007 (UTC)

I see your point. yes, it is an issue of definition. For map makers, area preservation means that two different regions of the sphere have the same area in the sphere iff they have the same area in the map representation (no matter where the region is located). Infinitesimal area preservation in one point (or line) is not useful for cartography. Given that this is a section about cartography we should use that definition.Dmgerman 23:09, 4 July 2007 (UTC)

Is stereographic often confused with stereoscopy?

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)

Stereographic projection from targets above the surface

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 general, P and T will have different projections. If P is close to the tangency point (between the sphere and the plane), then P and T will have very similar projections; if P is close to the projection point, then they will have wildly different projections! One way to compute this yourself is to put P and T into Cartesian coordinates (x, y, z) and then use the formulas given in the Definition section of this article (but doubled, because you're using a tangent plane instead of an equatorial plane). The formulas don't care whether P or T are on any particular sphere; they work for all (x, y, z) where z is not 1.

By the way, this is not a help page; you may find people willing to help more at Wikipedia:Reference desk/Mathematics, or in a math help forum on the web or Usenet. Joshua R. Davis 15:40, 21 September 2007 (UTC)

Edit to intro

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)

Equal area lower hemisphere projection

If the equal area projection is not stereographic, what exactly is it? Mikenorton (talk) 16:39, 20 November 2007 (UTC)

The equal-area projection can be written in Cartesian coordinates as

$(X,Y)=\left({\sqrt {\frac {2(1+z)}{x^{2}+y^{2}}}}x,{\sqrt {\frac {2(1+z)}{x^{2}+y^{2}}}}y\right).$

This is quite different from stereographic projection as defined in the article. It is not realized by any projection along straight lines from any point. It maps the lower unit hemisphere to a disk of radius ${\sqrt {2}}$, in a way that preserves area but not angles. I suspect that this projection has many names in the cartography and geometry literature, but I don't know them.

On an unrelated note, I like that geology figure you put in. Joshua R. Davis (talk) 17:33, 20 November 2007 (UTC)

Thanks for the clarification Joshua. I'm afraid geologists tend to refer to them inaccurately as stereographic, because we plot data and manipulate it in similar ways on both the equal angle and equal area projections using Wulff and Schmidt nets. We use the equal area projection in Structural geology because it allows us to contour linear orientation data. I'll try to expand both the Geology and Crystallography sections. Mikenorton (talk) 18:00, 20 November 2007 (UTC)

I happen to have some structural geologists as friends. They sometimes do use "stereographic" for equal-area projection, even though they know that it's not technically stereographic. I think they find the distinction pedantic, and I've got no problem with that in conversation. But it makes sense to be pedantic, or let's say precise, in Wikipedia.

For many months I've been meaning to make an article on the equal-area projection, Schmidt nets, etc. But I never knew what to put in it, beyond the basic math. Do you want to collaborate on it? Joshua R. Davis (talk) 22:06, 20 November 2007 (UTC)

Sure, I'll try to explain why and how we use them if you can do the maths.Mikenorton (talk) 12:05, 22 November 2007 (UTC)

For the record, the equal-area projection in question is the Lambert azimuthal equal-area projection. Joshua R. Davis (talk) 03:17, 12 December 2007 (UTC)

POTD notification

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)

Couple of points

• 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)

• great circles on the sphere intersect one another at diametrically opposite points. If you project the sphere's "equator" onto the plane (the equator is the trace of the circle whose pole is the projection point), all great circles are those circles which intersect that equator at diametrically opposite points.

• if we take the equator as our unit circle, any two great circles will intersect in two points that are negative inverses of one another circle inversion —Preceding unsigned comment added by Paul Murray (talk • contribs) 03:41, 5 July 2008 (UTC)

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)

Area preservation

"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)

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)

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

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)

This is NOT area preservation property. Area preservation means a figure F and its image p(F) have equal areas: A(p(F)) = A(F).
And it is generally NOT true—area A ratio A(p(F))/A(F) 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 A(p(F))/A(p(G)) 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)

Stereographic projection of a Cantellated 24-cell.

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)

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)

• One could ask User:Rocchini what projection he used. I'd guess it's "perspective". —Tamfang (talk) 09:20, 19 January 2018 (UTC)

Misleading claim

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)

(I have revised that sentence in the article to reflect this.) -- Elphion (talk) 05:22, 2 February 2013 (UTC)

a circle is the stereographic projection of ???.

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)

That's the Weierstrass substitution. That article begins with a visual depiction of the stereographic projection. So maybe link to that article from Circle, to help readers understand? Mgnbar (talk) 19:19, 31 July 2014 (UTC)

Thanks added a link to Tangent half-angle substitution but now i am wondering the circle page says "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. "

Should that not be the y-axis ? 213.205.251.130

You are correct. The circle is projected onto a line parallel to the y-axis. So please edit Circle again.

We could treat stereographic projection of the circle more here, at Stereographic projection. But it is such an important special case, that is already treated by Tangent half-angle substitution. So I think that this article doesn't need more about it. Do you agree?

When posting on Wikipedia talk pages, please sign your posts with four tildes: ~~~~. They are a special code that causes your username, etc. to appear. It helps us keep track of who is saying what. Thanks. Mgnbar (talk) 09:33, 1 August 2014 (UTC)

I don't think this needs something at stereographic projection, (now it has links to Tangent half-angle substitution), but the formula's at circle needs correcting, I think the right formula's are

   x = a + r \frac{2t}{1+t^2}, 
   y = b + r \frac{1-t^2}{1+t^2},

changed them at circle but can you check these formulas are correct? (I am quite sure about them)WillemienH (talk) 11:52, 2 August 2014 (UTC)

There are two consistent ways to do this:

• The "cos" or "x" term is associated with the "1 - t^2", while the "sin" or "y" term is associated with the "2t". The projection is onto a vertical line.
• The "cos" or "x" term is associated with the "2t", while the "sin" or "y" term is associated with the "1 - t^2". The projection is onto a horizontal line.

In my experience, it is almost always done in the first way. I've seen it done in the second way only twice: in the first diagram of Tangent half-angle substitution and in the text about "x-axis" that started this thread of discussion. So I propose that we standardize on the first way. Mgnbar (talk) 14:49, 2 August 2014 (UTC)

I have changed the circle page to the second option, I prefer that option because the article refered to "parallel to the x-axis" and with this way the "infinity" point is at the top of the circle (0,1), I find this a nicer way than using the y-axis and the point (0,-1), hope you don't mind WillemienH (talk) 18:05, 2 August 2014 (UTC)

My point is that you could have fixed it just by changing "x-axis" to "y-axis". And then you'd be consistent with standard mathematical practice (in my experience). But I have not backed up that claim, and small inconsistencies like this are not a big deal, so I'm fine with it. Cheers. Mgnbar (talk) 20:28, 2 August 2014 (UTC)

Review needed

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)

I am not an expert, but this does not seem right. A Möbius transformation of the complex plane is an inverse stereographic projection followed by the action of some SO(3)-element (more generally some SO⁺(3, 1)-element, but this is harder to see) on the sphere and –finally–stereographic projection. (This can be used to find explicit coverings SU(2)→SO(3), see Rotation group SO(3) and (probably also, not too sure here) SL(2, ℂ)→SO⁺(3, 1).)

What is worth mentioning though is that stereographic projection works for all spheres, not just the 2-sphere. YohanN7 (talk) 08:57, 27 December 2014 (UTC)

The general case is excellently described in the lead in Möbius transformation:

Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.

It differs as to what is a forward and inverse stereographic projection. The moving of the sphere corresponds to Lorentz boosts. YohanN7 (talk) 09:06, 27 December 2014 (UTC)

The edit is getting at a valid mathematical idea. It is just quite poorly written. I'm not sure whether I should edit, because...

Is Portal:Mathematics/Selected article/39 a copy of Stereographic projection? Are we now editing two parallel copies of a single article? Even after poking around (e.g. at Portal:Contents), I don't understand what this portal thing is about. Mgnbar (talk) 13:23, 27 December 2014 (UTC)

"Portals are pages intended to serve as "Main Pages" for specific topics or areas" (Wikipedia:Portal). A page such as Portal:Mathematics/Selected article/39 is supposed to be an enticing summary of the Stereographic projection article, leading readers to click the "Read more" link. This is similar to the "Today's Featured Article" box on the main page. Pages like this one usually start out as a copy of the lead section of the corresponding article, but they do tend to drift apart. I think it is always valid to copy the current version of the article lead into the portal page. -- John of Reading (talk) 13:55, 27 December 2014 (UTC)

Thank you for graciously explaining. I should have been able to figure it out on my own. I don't think I'll participate in the editing of portals. Regards, Mgnbar (talk) 16:40, 27 December 2014 (UTC)

Example of Wulff net projection

The article features an example worded this way:

For an example of the use of the Wulff net, imagine that we have two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Suppose that we want to plot the point (0.321, 0.557, −0.766) on the lower unit hemisphere. This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30° clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps:

1. Using the grid lines, which are spaced 10° apart in the figures here, mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0, 1)).
2. Rotate the top net until this point is aligned with (1, 0) on the bottom net.
3. Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.
4. Rotate the top net oppositely to how it was oriented before, to bring it back into alignment with the bottom net. The point marked in step 3 is then the projection that we wanted.

To plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°; spacings of 2° are common.

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)

The 0.321 and 0.557 come from a little trigonometry. I'll try to add a sentence clarifying their relationship with the angles. Thanks for your input.

I don't understand your comment about upper vs. lower hemisphere. The text is consistent: the point is on the lower hemisphere, its z-coordinate is negative, and the point is below the plane z = 0. And its location in the Wulff net agrees. What makes you think that the point is on the upper hemisphere? Mgnbar (talk) 18:08, 26 April 2015 (UTC)

Have you seen the picture provided with the example? Let's say we're discussing a point on the earth, and that the common center between the two panes is the equator and the Greenwich meridian. The point that is being constructed looks like it's somewhere in Russia or elsewhere in eastern Europe, which is decidedly north of the equator, something we tend to refer to as belonging to the upper hemisphere. Whatever the text says, if it doesn't match the image, either the image needs to match the text or the text needs to match the image, or this discrepancy (apparent or real) should be acknowledged and the gap between them should be bridged by an explanation. D. F. Schmidt (talk) 20:51, 26 April 2015 (UTC)

You seem to be visualizing the stereographic projection as "looking down on the sphere". That's not what it is. Look at the first image in the Definition section, and read the first paragraph of the Properties section. The lower hemisphere is mapped to the disk, by "pulling it up toward the plane". (The upper hemisphere is mapped to the infinitely large region outside the disk, by "pushing it out toward the plane", but this is not pictured in the Wulff net section.) Does that make sense? If so, perhaps you could help me figure out how to clarify this in the article. Mgnbar (talk) 00:18, 27 April 2015 (UTC)

No, as I visualize it, the horizontal line is the equator and the vertical line is a meridian (in my previous entry, the prime meridian), the top of the circle being the north pole and the bottom being the south pole. Therefore, because the point (as finally projected) winds up being in the top half--above the equator--I understand that to be on the upper half of the sphere that the projection represents. I feel like it is silly to expect any other understanding of the projection in this image. I won't pretend to be an expert in this projection or cartography in general, but I do feel that the steps (and reasons for the steps, including examples of how to read the projection) should be expounded upon. If the reader is expected to be already familiar with the projection, there isn't much use for the content, is there? D. F. Schmidt (talk) 16:58, 6 May 2015 (UTC)

I think that we've found the source of the confusion. In the Wulff net figures, only the southern hemisphere is shown. The equator is the circular boundary of the Wulff net. The south pole is at the center of the Wulff net. The north pole is not shown (because it is at infinity). I just added some text to the section to make this issue more explicit. Is it sufficient? Mgnbar (talk) 20:15, 6 May 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)

Well, your perspective is valuable, because it helps us improve the article for "typical" readers. So, if you can determine specific points of confusion, or specific remarks that would really help other readers, then please let us know here. Otherwise, you might try looking for textbooks on cartography, structural geology, etc. They might give more detailed and coherent treatments than even a good encyclopedia article can. Best wishes. Mgnbar (talk) 12:02, 14 May 2015 (UTC)

first picture misplaced

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)

A graphical projection maps the 3-space onto a plane. A stereogr. proj. maps only a sphere onto a plane !--Ag2gaeh (talk) 09:45, 11 September 2015 (UTC)

Are you talking about the image in the "Graphical Projection" infobox? You are right that it is orthographic. But that image is about the Graphical Projection series, not about this page. The first image after that infobox is stereographic projection. This usage of infoboxes is very statndard on Wikipedia.

After skimming graphical projection, I'm not convinced that the usage of the term "projection" here conflicts with the usage there. Regardless of that issue, people use the term "projection" with slightly different meanings, and the map described in this article is called stereographic projection by numerous reliable sources. That is its standard name, hence what it should be called on Wikipedia. Mgnbar (talk) 09:59, 11 September 2015 (UTC)

I have no objection against any name of a projection You mentioned. I repeat: the stereographic projection is no graphical projection. So, the infobox with title Graphical projection is misplaced. If the infobox would be named projection it could include the stereographic projection. --Ag2gaeh (talk) 11:09, 11 September 2015 (UTC)

Ah, now I understand better. I don't know much about technical drawing, and I don't have an opinion on whether the definition of "graphical projection" should accommodate stereographic projection or not. So I have raised the issue at Talk:Graphical projection#Stereographic projection?. Maybe we can get some other people to discuss it. Regards. Mgnbar (talk) 11:27, 11 September 2015 (UTC)

If it's not a graphical projection (or related to one) then removing the template would be okay. SharkD  Talk  02:13, 9 May 2017 (UTC)

File:Stereographic projection SW.JPG to appear as POTD soon

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)

Picture of the day

A stereographic projection of the world north of 30°S. The stereographic projection is a function that projects a sphere onto a plane. The projection is defined on the entire sphere, except at the projection point, in this case the South Pole. This mapping is conformal, meaning that it preserves angles. The stereographic is the only projection that maps all small circles such as craters to circles.Map: Strebe, using Geocart

As usual, looks great, Chris Woodrich. Thanks! Strebe (talk) 05:27, 23 January 2016 (UTC)

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)

Mapping circles to circles

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)

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)

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)

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)

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)

“Full” sphere

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)

The other images are clearly hemispheres, while this image was a projected full sphere. Since the projection extends across the entire plane, its impossible to show the full projection. The word clipped is technical fact. A clipped projection of a full sphere is correct. A projection of a partial sphere that happens to be limited to a rectangle is silly. Tom Ruen (talk) 22:26, 8 December 2016 (UTC)

A partial sphere?!

A projection of a partial sphere that happens to be limited to a rectangle is silly. Learn civil discourse. Obviously I think your thesis is “silly”, but it’s pointless and a violation of WP:CIVIL to say so. No, a partial sphere that maps as a rectangle isn’t “silly”. Mixing up computer science jargon with mathematical/geographical descriptions is “silly”. Calling a partial sphere a full sphere is “silly”. It’s easy enough to devise a simple portion of the sphere that, when mapped, can still be cropped to produce the image on display, and therefore claiming what’s there started as a full sphere is “silly” when there is no difference after cropping. It’s also impossible to map the full sphere, as you note, and so claiming it’s a full sphere that’s cropped is “silly”. Could we please not engage in this silliness, Tomruen? Strebe (talk) 01:26, 9 December 2016 (UTC)

It was an honest opinion, say unhelpful rather than silly as you like. The computations were on a full sphere, so it IS a full sphere, even if you can't see all of it. You don't qualify the word "plane" or "line" with "partial plane" or "partial line" because its impossible to draw a full infinite plane or a full infinite line. You don't draw a projective sphere as a circle on a flat surface and call it a "partial sphere" because you can only see the front half of it. There are endless places you could demand qualifiers which are not really helpful to the problem. A qualifier partial might make more sense if something SHOULD be visible in the area presented but is missing, but nothing is missing in this rectangle, and a wider view would show more.

How would you like to distinguish between a hemisphere and a sphere? I added the word "full" to differentiate between the other graphs which were hemispheres. Tom Ruen (talk) 03:08, 9 December 2016 (UTC)

I like User:Mgnbar’s solution, except I’d prefer it be referred to as stereographic projection rather than Wulff net. Or, “most of the sphere” rather than “full sphere”. Strebe (talk) 18:14, 9 December 2016 (UTC)

I worded it that way just because the image is in the section "Wulff net". If it is meant to be a general illustration of stereographic projection, then maybe it should be placed in a different section. In particular, "Properties" contains similar images. I leave that up to you. For what it's worth, I've seen the term "Wulff net" applied only to the hemispherical version, not the extended version, in the literature. Mgnbar (talk) 18:53, 9 December 2016 (UTC)

Looks good. The Stereonet is drawn as a hemisphere, and usually the other hemisphere is represented by antipodes, with two colors for front or back hemispheres. I added the image to the section mainly because it makes it more clear all the arcs shown are complete circles. Tom Ruen (talk) 00:15, 10 December 2016 (UTC)

two mismatches

Depending on the particular projection used, the parallels and meridians may or may not match those usually encountered in geography. For example, the figure at left is constructed using the conventions of the Definition section above. Because the projection point is (0, 0, 1), the Wulff net depicts the southern hemisphere z ≤ 0. The equator plots at the circular boundary of the Wulff net, and the south pole plots at the center of the Wulff net. The parallels are chosen to be small circles about the y-axis, and all of the meridians pass through (0, 1, 0) and (0, −1, 0).

Maybe I'm sleepy or something, but:

• the figure at left appears to me to be centred on the equator.
• the polar axis is called z in the middle of the passage and y at the end.

—Tamfang (talk) 07:22, 14 January 2018 (UTC)

The text is inherently confusing because it uses a non-standard system of parallels and meridians. The "equator" is where z = 0, but the line running across the middle of the plot is where y = 0. The "south pole" is where z = -1, but the meridians converge at y = ±1. I've edited the text a bit. Is it still confusing? Do you want to clarify it more? Mgnbar (talk) 18:17, 14 January 2018 (UTC)

You seem to have a problem with z as an axis of the image plane (as in the diagram in the Definition section); that doesn't bother me a bit.

Before and after your changes, the text says that the image shows the southern hemisphere, so the meridians ought to converge at the middle (the south pole). They don't. Is that what you mean by "a non-standard system of parallels and meridians"?

We can say that the image shows the southern hemisphere, gridded by circles defined by an axis that goes through the equator; or that the image shows (say) the eastern hemisphere, with a grid of the more usual kind. I'd prefer the latter. —Tamfang (talk) 06:53, 19 January 2018 (UTC)

• So I rewrote it.

The Wulff net shown here is the stereographic projection of the grid of parallels and meridians of a hemisphere centred at a point on the equator (such as the Eastern or Western hemisphere of a planet).

Does the passage really need to talk about coordinates? —Tamfang (talk) 07:02, 19 January 2018 (UTC)

The left side of this image is the "bullseye" that we're trying to avoid.

I don't really "have a problem" with this text. I wrote the text. I was trying to explain it to you, so that we could improve it.

Your revision of the text is internally consistent. It uses the standard system of meridians and parallels, in which the meridians converge at z = ±1. But it uses a non-standard projection point, somewhere in the x-y-plane. (By "non-standard" I really just mean, "different from the Definition given in the article".)

The previous version was also internally consistent. It used the standard projection point but a non-standard system of meridians and parallels.

You can do it either way, but something has to be non-standard, or else you don't get the Wulff net. Instead you get the "bullseye" shown on the left side of the figure.

Your revision is more succinct and probably clearer than the old one, so let's go with it. It uses an alternative version of stereographic projection, so I've moved the "Other formulations" section up. Cheers. Mgnbar (talk) 07:33, 19 January 2018 (UTC)

If the Definition says the projection point must be a pole or the projection is not stereographic, then the Definition needs changing too. —Tamfang (talk) 09:09, 19 January 2018 (UTC)

The Definition section begins by going out of its way to address that concern. But it does work a specific example (projection point (0, 0, 1), onto equatorial plane), to keep things concrete and to make explicit formulas possible, before the Other formulations.

If you feel that this treatment gives undue weight to (0, 0, 1), then I'm not opposed to minor reorganization and rewording. But I am opposed to major revision, because the explicit example with explicit formulas is valuable. Mgnbar (talk) 17:22, 19 January 2018 (UTC)

I went ahead and did a "minor reorganization", shoving the Other formulations into the Definition. Possibly some of the wording needs to be adjusted throughout the article. Mgnbar (talk) 18:15, 19 January 2018 (UTC)

Code for the projection

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";
    }

Conformality does not imply circles property

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)

@2PeterElls: 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)

@CiaPan: Thanks --Peter Ells (talk) 22:23, 6 July 2018 (UTC)

How much could the early Greeks prove?

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)

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)

Stereographic projection in cartography

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)

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)

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)

Punctuation dispute

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)

@Mgnbar: 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)

Updated --CiaPan (talk) 17:17, 19 June 2021 (UTC)

What's this projection called?

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)

@Darcourse: It's precisely the same stereographic projection, just with a larger scale. Please note that what you describe would be a homothety from the plane z = 0 to z = –1 wrt point at z = +1, which makes a scale 2:1. That's all. --CiaPan (talk) 17:13, 28 January 2022 (UTC)

@CiaPan: But my projection isn't a projection onto the z=0 plane as (-1,1) isn't reached and everywhere else is doubled up. Darcourse (talk) 02:50, 29 January 2022 (UTC)

I don't understand your projection. For example, I don't know what "d-plane" means. If you want me to think about it, please try to explain it more. Mgnbar (talk) 17:23, 29 January 2022 (UTC)

"Doubled up" suggests you may be thinking of the gnomonic projection, or possibly one of the other perspective projections. But like the other editor said, you're not being clear enough for us to understand you. -Apocheir (talk) 19:25, 29 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)

You mean that you're projecting onto the line y = 0? And you project a point P on the unit circle to a certain point (x, 0)? And x is determined from P by the requirement that the tangent line at P hits the line y = -1 at the point (x, -1)? Mgnbar (talk) 12:55, 9 February 2022 (UTC)

It doesn't seem to be any of the perspective projections in the image linked by Apocheir above.

I see why you compare it to the stereographic projection. It maps the southern hemisphere to the unit disk and the upper hemisphere (except for the north pole) to the rest of the plane. Mgnbar (talk) 17:34, 9 February 2022 (UTC)

If my algebra is correct, then that projection is equal to the stereographic projection. That projection produces x = (y0 + 1) / x0, where P = (x0, y0). Stereographic projection produces x = x0 / (1 - y0). But they're equal because x0^2 + y0^2 = 1. Interesting. Mgnbar (talk) 17:41, 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))).

Lede Image

3D illustration of a stereographic projection from the north pole onto a plane below the sphere

Rays of light implement stereographic projection. A curved grid on the sphere is mapped to a square grid on the plane

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)

[restructuring comment once I noticed the already existing section on the lede image] The image I removed (the light projecting circles into lines) does illustrate an interesting property of the projection: namely, that circles through the point of projection get mapped to circles through the point at infinity (AKA straight lines). My initial comment that the image "adds nothing" is not accurate. But the caption explains the image very poorly, and this particular grid is not why stereographic projection is used. The top of the article is not the proper place for it; it belongs rather in a section discussing the primary property of the projection, namely, that circles on the sphere are mapped to circles in the extended complex plane. -- Elphion (talk) 00:44, 21 April 2022 (UTC)

I don't feel strongly about the images. Yes, the new image may focus the reader's attention on a property that is not most important. However, the new image may be attractive to many readers, in a way that the old images are not. And even if we accept that mapping circles to circles is the most important property (I'm not sure), it's not necessarily the best first thing to say/show to a typical reader. The projection of shadows instantly gives the reader a concrete physical understanding of the projection. That's worth something. Mgnbar (talk) 01:39, 21 April 2022 (UTC)

@Mgnbar: No image will suffice as a proof – similarly to File:Stereographic projection grid.jpg, the plane of the resulting projection will be itself projected onto the image's own plane through a parallel of central projection (depending on whether you do it with a 3D modelling software of actually construct the material model and make a photograph). So the projected circles will be visible as ellipses, not circles. --CiaPan (talk) 16:55, 21 April 2022 (UTC)

Sorry, but I don't understand. Are you saying that making such an image inherently involves a second projection (the one onto the camera's CCD/film)? That's true, but I don't think that this second projection confuses readers. They can read the proposed image as a grid of squares, or a set of ellipses as foreshortened circles. Mgnbar (talk) 18:18, 21 April 2022 (UTC)

Precisely. They can - but they needn't. One can see a distorted net of lines and it's up to them to believe you and me those lines are actually parallel and perpendicular. We can only tell them there is a mesh of squares there, but the image alone doesn't prove it. And same with circles: you can see an ellipse only and there is no proof it actually was a circle. --CiaPan (talk) 19:27, 21 April 2022 (UTC)

Okay, but it's not the picture's job to prove anything. If the picture shows how the shadows work, and the caption explains that the shadows are a square grid, and this helps the reader see how the projection works (without proof), then isn't that good? Most mathematical assertions in Wikipedia are not proven in Wikipedia. Sorry if I misunderstand you. Mgnbar (talk) 19:52, 21 April 2022 (UTC)

The projection of shadows instantly gives the reader a concrete physical understanding of the projection.
I have to disagree. It's a nice picture, and clearly something is being projected, but how and why remain completely unexplained without a fairly wordy caption (significantly more informative than the current caption!). The other image, even if not ideal, gets the idea across much better. -- Elphion (talk) 21:04, 21 April 2022 (UTC)

I don't like the current caption (or the aesthetics of the picture, for that matter). And no picture will capture the many whys that exist for stereographic projection. But the picture captures the how extremely well --- i.e., the projection is exactly along the light rays depicted, and the sculpture maps exactly to its shadow.

The previous lede picture (which is also reproduced in this talk page section) merely depicts a different sculpture and uses more minimalist aesthetics. It suffers from the same "second projection" issue that CiaPan raised. Well, it also shows the "light" rays, but that to me is a detail. I do not see how it is more rigorous or less misleading. Mgnbar (talk) 21:33, 21 April 2022 (UTC)

No, the how is not clear. The brightest spot is the south pole -- it took me a while to realize that the hand is holding a light, and that the illumination is coming from there rather than from the south pole. And where is the light positioned? It's not clear that it's at the north pole. And what shape are the "curves" that are being projected? It's not clear that they are circular arcs -- or why those particular arcs should be projected as straight lines. In the other image, by contrast, it is clear that the projection point is the north pole, that the curve being projected is the equator, and therefore circular, and that the projected image is circular. (Like you, I don't buy the argument that the image projection confuses either image.) The second image is very clever, but it doesn't stand on its own. (So much so that my gut reaction was that it has no business in the article.) -- Elphion (talk) 22:52, 21 April 2022 (UTC)

It is odd that the light is not itself bright. That's a defect in the design. The other objections I find less serious, because the picture is not supposed to be a rigorous definition of the projection. Utterly clear diagrams come later in the article. I think that there is value in having a "realistic" image different from those diagrams. But this particular image is not worth fighting for. Cheers. Mgnbar (talk) 23:43, 21 April 2022 (UTC)

The projection of shadows instantly gives the reader a concrete physical understanding of the projection.
Like Elphion, I have to disagree. That image doesn’t say much useful about the projection; it requires a sculpture whose principles of design are not clear to the observer, and the projection of light’s obscuration and transmission into squares on the plane doesn’t convey anything that helps me understand the behavior of the projection. — Preceding unsigned comment added by Strebe (talk • contribs) 03:06, 22 April 2022 (UTC)

Circles are projected onto circles?

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)

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)

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 $a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}=1$, 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)

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 {{dubious}}. 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)

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)

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)

These links may be helpful: Möbius plane (section 2), Inversive geometry (section 2.2) --Ag2gaeh (talk) 09:35, 26 June 2022 (UTC)

Should this article prefer a projection with the "north pole" at the origin?

Right now this article focuses on the projection from the "north pole" ${\textstyle \langle x,y,z\rangle =\langle 0,0,1\rangle }$ which sends the north pole to ${\textstyle \infty }$ and sends the "south pole" ${\textstyle \langle 0,0,-1\rangle }$ 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 "${\textstyle +x}$ pole" ${\textstyle \langle 1,0\rangle }$, and to consider anticlockwise angles to be positive (in the complex plane, from ${\textstyle 1}$ toward ${\textstyle i}$, where 1 can be thought of as the identity rotation). The stereographic projection of these unit-circle points ${\textstyle \langle x,y\rangle }$ or unit-magnitude complex numbers ${\textstyle x+yi}$ is typically taken from the "${\textstyle -x}$ pole" onto the ${\textstyle y}$-axis, yielding the point $\langle 0,y/(1+x)\rangle$. In terms of an angle measure ${\textstyle \theta =\operatorname {atan2} (y,x)}$, the stereographic projection is the half-angle tangent ${\textstyle \theta \mapsto \tan {\tfrac {1}{2}}\theta }$ or ${\textstyle \langle \cos \theta ,\sin \theta \rangle \mapsto \tan {\tfrac {1}{2}}\theta }$.

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 ${\textstyle -1}$ and send the identity rotation ${\textstyle 1}$ 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" ${\textstyle \langle -1,0,\ldots ,0\rangle }$ which sends the "north pole" ${\textstyle \langle 1,0,\ldots ,0\rangle }$ 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 ${\textstyle \theta }$ and colatitude (or polar angle) ${\textstyle \varphi }$, where the latter starts at ${\textstyle 0}$ at the “north pole” (${\textstyle +z}$ direction) and measures out to ${\textstyle \pi }$ at the “south pole” (${\textstyle -z}$ direction). It’s clearer if radius in the stereographic projection is ${\textstyle \tan {\tfrac {1}{2}}\varphi }$ instead of ${\textstyle \tan {\tfrac {1}{2}}(\pi -\varphi ).}$

What do others think? –jacobolus (t) 03:06, 21 June 2022 (UTC)

All of my many books presenting the stereographic projection (all predating WP, so your concern in #6 does not apply) without exception project from the NP. This is, in my experience, by far the most common arrangement in other presentations as well, and that is surely what most people will find in outside references. Principally for that reason, I would favor keeping projection from the NP in our presentation. Mathematically, of course, it makes no difference. I think it reasonable to point out in the article that projection from the SP (or indeed any point on the sphere) works too, and that maps with the NP in the center are in fact projecting from the SP. -- Elphion (talk) 20:49, 21 June 2022 (UTC)

I just looked at the 20+ books mentioning the stereographic projection I have at home, and you are right that most of them put the center of projection at the north pole. Two excellent sources that put the projection at the south pole are Morrison (2007) The Astrolabe, the definitive book about the history, construction, and use of astrolabes; and following Morrison, Popko (2012) Divided Spheres. Neugebauer (1975) History of Ancient Mathematical Astronomy also uses a south pole centered projection (but draws figures where S is on “top” and N is on “bottom”). An important book with a south-pole centered projection that should definitely be mentioned in this article is Donnay (1945) Spherical Trigonometry After the Cesàro Method. Schwerdtfeger (1962) Geometry of Complex Numbers uses a south-pole centered projection. The couple of other pure math books I found that use a south-pole centered projection mirroring this Wikipedia article’s north-pole centered projection were not really notable (the projection was just mentioned on 2–3 pages and not elaborated); most of the better pure math treatments (e.g. Needham (1997) Visual Complex Analysis, Hilbert & Cohn-Vossen (1952 [1932]) Geometry and the Imagination) that I found use a north-pole centered projection the same as this article’s. Coxeter (1961) has a projection centered on $\langle 1,0,0\rangle$ and calls his coordinates ${\textstyle x_{0},x_{1},x_{2}.}$ Rosenfeld & Sergeeva (1977) Stereographic Projection call their center of projection ${\textstyle S}$ and start with a synthetic geometry approach, but when they get to coordinates they project the unit sphere through the north pole ${\textstyle \langle 0,0,1\rangle }$ onto the plane ${\textstyle z=-1}$ (this version obnoxiously maps the equator to a circle of radius 2). Another noteworthy (arguably south-pole-centered) version of the projection is Klein (1939 [1925]) Elementary Mathematics from an Advanced Standpoint: Geometry which uses a unit-diameter sphere passing through the origin ${\textstyle x^{2}+y^{2}+z^{2}=z}$ and projects through the origin onto the plane ${\textstyle z=1}$: this projection is just a sphere inversion across the sphere ${\textstyle x^{2}+y^{2}+z^{2}=1}$, which makes the formulas in both directions particularly simple (I have seen this version in a few other places, but I am having trouble remembering precisely where; I have used this version multiple times myself in doing stereographic projection related calculations, as it makes some relationships easier to understand). –jacobolus (t) 18:57, 25 June 2022 (UTC)

This figure from Markushevich (1982) Complex Numbers and Conformal Mappings has the right idea. :-) –jacobolus (t) 18:33, 2 July 2022 (UTC)

Aside: Elphion, what are your favorite books presenting the stereographic projection? –jacobolus (t) 19:36, 25 June 2022 (UTC)

I just looked at a whole bunch of cartography and geodesy books. There are some more south-pole centered stereographic projections. For instance, here is a nice diagram from Lambert (1772). Cartography/geodesy sources I have here generally don’t use cartesian coordinates, but most of them, when describing azimuthal projections, start by defining a system of point-centered “spherical polar coordinates” (like spherical coordinate system with dropped radius) and then defining other projections from there. If we take the north pole to be the chosen center point (as in spherical coordinate system), then the formulas for the stereographic projection amount to a south-pole-centered projection. For example take a look at the figures and formulas in Adams (1919). –jacobolus (t) 21:43, 25 June 2022 (UTC)

For a coordinate version, here’s Lambert (1765), see p. 399. Edit: oh wait, that’s just the 1-dimensional stereographic projection (half-tangent), not the 2-dimensional case. Nevermind. –jacobolus (t) 07:16, 27 June 2022 (UTC)

For obvious reasons, in geotechnical engineering and geology straight down maps to the origin (this might be considered a north-pole centered projection, or perhaps a “zenith-centered” one). It seems like in crystallography some sources use a south-pole centered projection, e.g. Whittaker (1981) Crystallography (see the spin-off pamphlet "The Stereographic Projection". The origin of graphical use of the stereographic projection in crystallography may be Penfield (1901) which uses a south-pole centered projection. –jacobolus (t) 00:16, 26 June 2022 (UTC)

Jacobolus, you raise some good points. Your point #3 has always bothered me a little. Your points #1, 2, 4 seem to be arguments for setting up the projection from (-1, 0, 0). I'm not sure about your point #5; it certainly applies to maps of the Arctic, but maps of the Antarctic do the opposite. As for point #6, in the scientific literature I have seen both versions — even in different areas of geology, for example. So I'm not sure what to do.

I'm not as convinced as Elphion is, of the dominance of the current treatment. Regardless of that, the article does do other treatments later. Mgnbar (talk) 21:23, 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 ${\textstyle \langle z,x,y\rangle .}$ :-) –jacobolus (t) 21:41, 21 June 2022 (UTC)

Part of my reason for asking is that I want to add the 1-sphere (plane rotation) and 3-sphere (3-d rotation) examples, but I’m worried readers will be confused by internal inconsistency. –jacobolus (t) 21:43, 21 June 2022 (UTC)

Sorry, but I don't know what examples you're talking about, so I don't know why confusion would arise. Mgnbar (talk) 00:55, 23 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 ${\textstyle \langle x,y\rangle ,\ x^{2}+y^{2}=1}$ and in terms of angle measure ${\textstyle \theta }$. Those maps are ${\textstyle \langle x,y\rangle \mapsto {\dfrac {y}{1+x}}}$ and ${\textstyle \theta \mapsto \tan {\tfrac {1}{2}}\theta ,}$ respectively. It is important to project from the “${\textstyle -x}$ 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 ${\textstyle a+bi+cj+dk\mapsto {\dfrac {bi+cj+dk}{1+a}}.}$ Or in terms of the axis–angle representation where $\mathbf {e}$ is a unit-magnitude “pure imaginary” quaternion representing an axis and ${\textstyle \theta }$ is the half-angle of the rotation (because quaternions sandwich-multiply to rotate vectors, angles are applied twice), we have the stereographic projection ${\textstyle \mathbf {e} \theta \mapsto \tanh {\tfrac {1}{2}}\mathbf {e} \theta =\mathbf {e} \tan {\tfrac {1}{2}}\theta .}$ 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 ${\textstyle \langle x_{0},x_{1},\ldots ,x_{n}\rangle \mapsto {\dfrac {\langle x_{1},x_{2},\ldots ,x_{n}\rangle }{1+x_{0}}}.}$ 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)

Also worth including is the “stereographic distance” (note: not a metric) between two unit vectors: ${\textstyle d_{s}(u,v)={\dfrac {|u\wedge v|}{1+u\cdot v}}}$ which is important in spherical trigonometry. (cf. e.g. Hestenes, Li, and Rockwood “Spherical Conformal Geometry with Geometric Algebra”) as compared to the chordal distance ${\textstyle |u-v|}$. In the Riemann sphere, the stereographic distance is ${\textstyle d_{s}(z,w)={\dfrac {|w-z|}{|1+{\bar {z}}w|}}}$ compared to the chordal distance ${\textstyle d_{c}(z,w)={\dfrac {|w-z|}{{\sqrt {1+|z|{}^{2}}}{\sqrt {1+|w|{}^{2}}}}}}$. Having the formula for stereographic distance simply be the stereographic projection of ${\textstyle uv=u\cdot v+u\wedge v}$ reduces confusion that would result from a flipping sign. –jacobolus (t) 21:36, 23 June 2022 (UTC)

Please do not edit talk page posts after other users have responded to them. It alters the context of the other users' posts. For example, by adding a seventh point to your list, you make it seem that I have ignored that point in my post above (whereas really I hadn't seen it yet).

Your point #7 is a point in support of your argument, but in my opinion it is quite a weak one. Mgnbar (talk) 00:55, 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)

This is a problem of writing style and organization rather than a problem of scope per se, in my opinion. It takes some deliberation and discipline from editors to keep articles well organized, tightly focused, comprehensive (within their chosen scope), with a flowing narrative, and clearly linked to / extended by other articles. The current article comes nowhere close to covering all of the important features of the 2-dimensional stereographic projection and its history and applications (much less the more general story), and is missing dozens of important external references, while still containing some amount of unnecessary miscellany that distracts the reader and breaks up narrative flow (more or less the problem you are talking about). I agree this is a general problem on Wikipedia and math articles in particular. Please don’t take my handful of sentences above as a proposal for exactly how the article should be worded. But there is also often the opposite problem on Wikipedia (and math articles in particular) where an article with potentially broad scope and a ton of content has nothing but a high-level definition with no technical details. (For a stark example relevant here, spherical geometry is little more than a stub and sphere is only a little better.)

More concretely: In the context of cartography, ancient astronomy, (2-)spherical trigonometry, crystallography, geology, or panoramic photography, the 2-dimensional stereographic projection is certainly what people are thinking of. But in the context of mathematics, the 1-dimensional stereographic projection is more common still (called by a hodge-podge of random names), and the n-dimensional stereographic projection (and more generally also the stereographic projection of either sphere or hyperboloid) come up all the time, and there are plenty of important theorems about these. And in applications to 3-dimensional motion (say in robotics, rocket science, or physical simulation) the 3-sphere stereographic projection is going to be more important.

I agree that some of these topics can potentially be further elaborated at separate articles, e.g. to be called half tangent and modified Rodrigues vector or the like, but that doesn’t mean they should not be included here as well.

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)

Jacobolus, by "1-sphere (plane rotation) and 3-sphere (3-d rotation)" examples you just mean the S^1 and S^3 examples? Your talk of rotation confused me (although I understand in retrospect why you mentioned it, because of the close relationships to SO(2) and SO(3)). Sure, I am happy to include examples in all dimensions. In fact, the "Generalizations" section already does.

I am not opposed to treating the generalizations more than what is in "Generalizations". And doing the S^1 case first would maintain the gentleness, for which Elphion rightly advocates. Would you want to standardize on projecting from (-1, 0, ..., 0), for the sake of Weierstrass and quaternions? Mgnbar (talk) 01:58, 24 June 2022 (UTC)

Yes, I just mean S1 and S3: these are the most natural models of planar and 3-dimensional rotation, respectively (Spin(2) and Spin(3)). Putting the 1-dimensional case first (and possibly per WP:SPINOFF elaborating further about it in a separate article which I am thinking of naming half tangent, along the lines of Paeth (1991)) can lead with some details that are also relevant in more general cases, and can start with flat diagrams that are easy to interpret and compare to later “3-dimensional” diagrams. 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)

When you said that you use coordinates (z, x, y) on 21 June 2022, I thought that you were joking. Mgnbar (talk) 21:35, 28 June 2022 (UTC)

Not a joke. Cf. https://observablehq.com/@jrus/planisphere –jacobolus (t) 21:44, 28 June 2022 (UTC)

The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection. If twice the area of the blue hyperbolic sector is ψ, then twice the area of the red circular sector is φ = gd ψ. Twice the area of the purple triangle is the stereographic projection s = tan 1/2φ = tanh 1/2ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos φ, sin φ). The purple point has coordinates (0, s).

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)