# Talk:Stereographic projection

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## 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)

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. 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 (talkcontribs) 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)