# Talk:Lambert conformal conic projection

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During a space shuttle mission you have noticed a map of the world and a location of the shuttle. You noted that shuttle flight path appears to be on a frequency graph with high and lows vs. and smooth line. Taking in account the tilt of the earth, what accounts for the irregular plotted flight path???

I assume space shuttle goes up in the space before orbits. It does not travel like an aircraft does (who intends to travel the great circle for a shortest distance to arrive at its destination on the earth). Observed from the earth, it is affected by Coriolis force, and its path is deflected, although it is supposed to be straight up to the sky in an inertial world. Does map projection have anything to do with space shuttle's path? --Natasha2006 15:36, 17 April 2007 (UTC)

## What are the parameters?

...?--190.56.85.26 17:13, 11 July 2007 (UTC)

Not reflection to this discussion but to the article: straight lines on a lamberts projection do present great circles, although great circles are not exactly straight lines, but have a very gently concave curve towards the parallel of origine. —Preceding unsigned comment added by 87.208.17.51 (talk) 19:20, 27 October 2007 (UTC)

Coriolis force does not affect an orbiting body. There must be some medium, such as air molecules, to transfer the force. A balloon would be affected by coriolis force. The orbiting shuttle is not.

This is incorrect, because the "Coriolis Force" is not actually a real force. It is a fictitious force for book-keeping in a rotating reference frame 199.46.198.232 (talk) 15:49, 13 December 2012 (UTC)

The purpose of a map projection is so that pilots don't have to carry big globes in the cockpit. All map projections have a degree of distortion.

—Preceding unsigned comment added by Dmp717200 (talkcontribs) 14:32, 25 March 2008 (UTC)

Are these formulas correct? Because when implementing them, the result looks quite weird. I think the ${\displaystyle \cot ^{\pi }}$ and ${\displaystyle \tan ^{\pi }}$ terms should actually read ${\displaystyle \cot ^{n}}$ and ${\displaystyle \tan ^{n}}$, respectively. Ie. n instead of pi. Unfortunately, a quick search didn't yield another source using similar formulas, but the resulting map looks like it should with these changes. 84.56.14.232 (talk) 00:12, 11 February 2009 (UTC)

The formulas are incorrect as pointed out above. Wolfram has the exact same formulas but the values in question are raised to the power n instead of pi. [1] --Davepar (talk) 05:23, 9 March 2009 (UTC)

## History section needed!

This article really needs a "history" section. Who the heck was "Lambert"? When & why did he invent this projection? Who used it?

TIA.

SteveBaker (talk) 12:58, 6 July 2009 (UTC)

The image File:Lambert_conformal_conic.svg is misleading by displaying a secant cone which is not the cone on which the Lambert Projection is drawn. This seems to be a common misconception. For instance the webpage the description page refers to says the map is "Mathematically projected on a cone conceptually secant at two standard parallels." The image should be removed. --Theowoll (talk) 18:31, 4 December 2011 (UTC)

The NOAA description is nonsense. The half-angle argument isn’t relevant. You can always construct a Lambert conformal conic for which the scale is correct along the specified two parallels. That conic is just a uniform scaling of some otherwise identical tangent version of the conic. Hence there is no reason it should not be called “secant”.
A more relevant discussion of the topic can be found here. Really, it’s very simple: All conformal conic projections with the same cone constant necessarily differ only in scale. The nominal scale you (arbitrarily) assign to it determines the standard parallels, and therefore whether it is conceptually “secant” or “tangent”. Strebe (talk) 02:41, 14 December 2011 (UTC)
Right, all Lambert conformal conic projections with the same half-angle (same ${\displaystyle n}$) differ only in scale. But only the tangent cone has the same half-angle as the cone on which the projection is drawn (the Lambert cone). A secant cone that belongs to two reference latitudes that give an equivalent map (i.e. up to scaling) has a different half-angle. So this secant cone is in no relation to the Lambert cone. Refering to such an secant cone makes no sense. It only leads to confusion, because it makes people believe that the map is drawn on the secant cone and then unrolled into the plane. That this is wrong is explained in the NOAA FAQ in a completely senseful and relevant way. Wikipedia shouldn't participate in propagating misconceptions. Not only should the image be removed from this article, there should even be a section in the article Map projection, which explains this error. --Theowoll (talk) 22:05, 26 December 2011 (UTC)
For every Lambert conformal conic with two standard parallels, there is a secant cone onto which you can project conformally and then unroll, yielding the same results as Lambert conformal conic with two standard parallels. This must be true because:
• Every Lambert conformal conic with two standard parallels is merely a scaling of some Lambert conformal conic with one standard parallel, and
• A scaled tangent cone properly positioned is a secant cone.
The fact the projection formulæ in this case do not coincide with the form that Lambert gave (and that is usually presented) is not relevant to the conceptual question. The idea that a secant cone must have the same cone constant as the “Lambert cone” arises solely out of the specific form of the projection formulæ, not out of any geometric constraint on the problem. Therefore the distinction you and the NOAA FAQ make is irrelevant. The image is fine. The concept is fine. It just does not suit someone who cannot think in any terms but the canonical formulæ. Strebe (talk) 09:21, 27 December 2011 (UTC)
I take that back. The image is not fine, but not for the reasons you give. It is not fine because the map selects the 30° and 60° parallels whereas the globe appears to select the 15° and 45° parallels. Strebe (talk) 09:29, 27 December 2011 (UTC)
I actually don't care about canonical formulæ. I'm aware that every Lambert conformal projection is given by a stereographic projection followed by exponentiating with the power ${\displaystyle n}$ in the complex plane and a uniform scaling. From this it is clear that all Lambert conformal projections with the same ${\displaystyle n}$ are equivalent, i.e. differ only by a uniform scaling. The powers ${\displaystyle n}$ determine these equivalence classes of Lambert conformal projections. Since uniform scalings leave angles invariant, all cones that you get when you roll up the maps in one equivalence class have the same half-angle ${\displaystyle \phi _{1}=\phi _{2}}$, the latitude of the standard parallel of the tangent cone projection in that equivalence class. But here comes the bad news: A secant cone that passes through two standard parallels of a Lambert projection has a different half-angle than the tangent cone with the equivalent projection with one parallel. So your first sentence is already wrong: When you unroll the secant cone passing through two standard parallels, you can't get the corresponding Lambert conformal map, because the central angles of the circular sectors of the unrolled cone and the Lambert map differ. You can't fix that by a scaling. --Theowoll (talk) 22:59, 27 December 2011 (UTC)
My thesis works under any of three conditions. (a) The map gets transformed after unrolling the cone. Or, (b) Scale is not the same at the two parallels at which the secant cone intersects the globe, but so what? This is what I was getting at. Or, (c) Projecting very oddly onto the cone such that either the entire cone is not used, or the projection onto the cone overlaps itself.
I agree that none of (a), (b), nor (c) is a useful demonstration of the project-onto-cone-and-unroll-the-cone concept. Hence I retract my criticism of the NOAA FAQ page and agree that the graphic should be discarded. Strebe (talk) 19:33, 28 December 2011 (UTC)

## Not a secant cone, but why?

Profile view of a secant cone map projection.

I recently had an edit reversed, and I believe it is because of a misunderstanding about a secant cone projection. This is not what the Lambert projection is, and currently the article states that this is because the secant projection would result in unequal scales on the standard parallels. This explanation is incorrect and is not supported by the cited reference from NOAA. The reference does state two requirements for the Lambert projection. One is that the scales on the reference latitudes must be equal. It is not stated that the secant cone projection is in conflict with that requirement. In fact, it is not.

In the image here a secant cone projection is shown in profile. A right circular cone and a sphere share a common axis. They intersect on two circles, the reference latitudes. From the center of the sphere the sphere surface is projected onto the conic surface. The cone can then be cut on a generator, laid flat, and scaled.

Since the reference latitudes are intersections, they are invariant in the projection, and flattening the surface does not change the scale. There is a 1 : 1 scale along each of the reference latitudes, and these scales remain equal when the map scale is applied.

On these same latitudes, however, the north-south scale is not 1 : 1. Let arbitrarily small arcs of length s be drawn on the sphere surface in the north-south direction, straddling the reference latitudes. They are projected onto the cone, and the images have length s'. At these two latitudes the projection lines intersect the cone at equal angles, α. The projection is not orthogonal, and as s diminishes, the scale in the north-south direction approaches 1 : csc(α). This scale is again equal on both reference latitudes.

The scales on the reference latitudes are equal to each other. However, the north-south and the east-west scales are not equal. That is the other requirement in the NOAA notes. A conformal map must have this property at all points, not only on the reference latitudes. The secant cone projection is not conformal.

I will keep my fingers off the article for a few days, but if my explanation is not refuted with an argument, I will then change it back. --Geometricks (talk) 16:22, 18 September 2014 (UTC)

There’s nothing wrong with any of that. All you’re saying is “If it is a secant cone, then it can’t be conformal.” That’s pointing out that a secant cone violates Condition #1 in the reference. But because Lambert conformal is defined by its conformality (hence the name), this article’s text instead points out that a Lambert conformal conic cannot have a secant case because the secant case of a (conformal) Lambert conic could not have the same scale where the cone cuts into the globe. In other words, it cites the violation of Condition #2. Strebe (talk) 06:17, 19 September 2014 (UTC)
The secant cone projection has an elementary geometric form involving projection rays from a common source connecting points on the sphere to their images on the cone. The point made by NOAA was simply that the Lambert projection is not derived from this model. As you must know, the Lambert projection is far more complicated and has no corresponding spatial relationship. It is conic only in the sense that meridians are concurrent lines, allowing the map to be rolled into a cone. Juxtaposing it with a globe, secant or otherwise, would have no particular meaning.
While not conceding, I did promise to stay out of it if anyone gave me an argument (as opposed to a simple contradiction). This qualifies, so I will leave it be. Strebe, you and others have more time invested in the article, while mine was more in the nature of a drive-by edit. Let me part with some suggestions for editing the formulas. Begin with n since all of the other formulas depend on that parameter. Also, I notice that several of the variables are not defined or explained in any way. Could that be touched up? --Geometricks (talk) 05:37, 20 September 2014 (UTC)
The secant cone projection has an elementary geometric form involving projection rays from a common source connecting points on the sphere to their images on the cone. I think this statement illustrates where the misunderstanding lies. What you say is literally correct, but the entire field of map projections long ago generalized away from literal projections. Hence your statement does not reflect the modern usage and nomenclature of map projections. The map projection literature does talk about the Lambert and other conic projections as “secant cases” even though that’s literally correct only as applied to the perspective conic. Whether “secant cone” has any didactic value to a lay reader is debatable, especially if one does not further loosen the analogy by explaining that the “rays” projecting the image are not straight, but curve upward or downward on their way to the cone surface. At that point you might as well discard the imagery altogether, and indeed Lee (1944) states, “It is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding.” I do not think there is anything wrong with “secant case” applied to conic projections as long as those using the term understand that it means only that the projection has two standard parallels instead of one—in other words, as jargon between experts. As for your proposals for improvement, I’ll look into them as time permits. Thanks for your comments. Strebe (talk) 08:29, 20 September 2014 (UTC)
…although, having written that, it now occurs to me that the cone/cylinder/plane analogy can still be applied sensibly without getting technical by showing the rays from the center of the globe being deflected at the surface of the globe. Since the rays always strike normal to the surface, the analogy would not be much like an index of refraction (unless you show the rays going in reverse from cone surface to globe center), but still, it affords a simple interpretation without error. Strebe (talk) 23:02, 20 September 2014 (UTC)

## File:Lambert conformal conic projection SW.jpg to appear as POTD soon

Hello! This is a note to let the editors of this article know that File:Lambert conformal conic projection SW.jpg will be appearing as picture of the day on September 21, 2016. You can view and edit the POTD blurb at Template:POTD/2016-09-21. 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, 6 September 2016 (UTC)

 Picture of the day The Lambert conformal conic projection is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in 1772. Conceptually, the projection seats a cone over the sphere of the Earth and projects the surface conformally onto the cone. The cone is unrolled, and the parallel that was touching the sphere is assigned unit scale. By scaling the resulting map, two parallels can be assigned unit scale, with scale decreasing between them and increasing outside them. Unlike other conic projections, no true secant form of this projection exists. Map: Strebe, using Geocart Archive – More featured pictures...

Perfect as-is, Chris Woodrich. Thanks for all the hard work! Strebe (talk) 16:53, 7 September 2016 (UTC)