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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.
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800:) 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
1235:…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.
1008:, 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
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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.
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I don’t think the article should contain that explanation. The term “conceptually” is used here because the projection does not arise out of a literal projection of light onto a developable surface. It is purely a mathematical construct. As such there are map projection teachers (see M. Lapaine) who
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How is this projection performed? If you put a light source in the center of Earth and project from there, the resulting projection will be non-conformal (for most cases at least). It becomes especially obvious that this projection in non-conformal when the angle at the top of the cone is set to 180
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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,
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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
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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
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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.
762:. 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”.
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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.
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Since the reference latitudes are intersections, they are invariant in the projection, and flattening the surface does not change the scale. There is a 1Â :Â 1 scale along each of the reference latitudes, and these scales remain equal when the map scale is applied.
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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:
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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.
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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.
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The article simply states that "conceptually, the projection seats a cone over the sphere of the Earth and projects the surface conformally onto the cone," but it doesn't explain how this projection happens. As I wrote in a
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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.
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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
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in a completely senseful and relevant way. Knowledge 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
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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.
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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.
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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? --
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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.
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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. --
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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
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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.
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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
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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.
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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? --
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Every Lambert conformal conic with two standard parallels is merely a scaling of some Lambert conformal conic with one standard parallel, and
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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. --
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This article really needs a "history" section. Who the heck was "Lambert"? When & why did he invent this projection? Who used it?
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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strongly object to even using “developable surfaces” as an analog, given how easy it is to misconstrue what’s actually happening.
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diminishes, the scale in the north-south direction approaches 1 : csc(α). This scale is again equal on both reference latitudes.
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is misleading by displaying a secant cone which is not the cone on which the Lambert Projection is drawn. This seems to be a
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that was touching the sphere is assigned unit scale. By scaling the resulting map, two parallels can be assigned unit
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This article had an importance automatically assigned to it. Once it has been checked by a human, please remove
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I actually don't care about canonical formulæ. I'm aware that every Lambert conformal projection is given by a
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https://web.archive.org/web/20160308084611/http://www.nnrms.gov.in/greennrms/download/NnrmsStandardsDoc.pdf
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On these same latitudes, however, the north-south scale is not 1Â :Â 1. Let arbitrarily small arcs of length
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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Are these formulas correct? Because when implementing them, the result looks quite weird. I think the
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either the entire cone is not used, or the projection onto the cone overlaps itself.
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of the unrolled cone and the Lambert map differ. You can't fix that by a scaling. --
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Right, all Lambert conformal conic projections with the same half-angle (same
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Requested articles/Social sciences/Geography, cities, regions and named places
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An editor has reviewed this edit and fixed any errors that were found.
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Hello! This is a note to let the editors of this article know that
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degrees, which reduces the cone to a plane, for which we obtain a
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are equivalent, i.e. differ only by a uniform scaling. The powers
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http://www.nnrms.gov.in/greennrms/download/NnrmsStandardsDoc.pdf
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on September 21, 2016. You can view and edit the POTD blurb at
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A scaled tangent cone properly positioned is a secant cone.
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for additional information. I made the following changes:
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Explanation of how the projection is performed is needed
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over the sphere of the Earth and projects the surface
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A more relevant discussion of the topic can be found
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256:Articles missing geocoordinate data by country
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40:WikiProjects
1333:conformally
1187:Geometricks
1138:Geometricks
532:—Preceding
489:—Preceding
464:Natasha2006
391:Cartography
30:Start-class
1544:Categories
1465:Report bug
725:The image
708:SteveBaker
440:|autoi=yes
291:Notability
240:Listed at
1448:this tool
1441:this tool
1315:used for
1276:Main Page
538:Dmp717200
112:Geography
103:geography
59:Geography
1454:Cheers.—
1337:parallel
1018:Theowoll
811:Theowoll
802:NOAA FAQ
739:Theowoll
546:contribs
534:unsigned
491:unsigned
457:path???
1485:explain
1401:my edit
1357:Archive
1012:of the
925:scaling
735:webpage
684:Davepar
427:on the
263:Infobox
221:Cleanup
139:on the
1525:Strebe
1374:Strebe
1351:Strebe
1345:secant
1237:Strebe
1213:Strebe
1154:Strebe
1071:Strebe
871:Strebe
846:Strebe
764:Strebe
477:...?--
194:Assess
36:scale.
1349:Map:
1341:scale
1311:is a
704:TIA.
319:Stubs
305:Photo
1529:talk
1511:talk
1378:talk
1329:cone
1305:The
1284:talk
1278:. —
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637:and
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419:High
400:Maps
389:and
387:Maps
359:Maps
324:See
310:See
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226:See
211:and
185:and
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1415:to
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277:Map
131:Mid
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