The single-parallel Lambert is tangent to the ellipsoid at the central parallel. The two-parallel Lambert is secant to the ellipsoid at the two standard parallels. In both cases, wherever the map touches the ellipsoid, the scale factor is 1. If the scale factor is less than 1 at the central parallel, then the two standard parallels exist, and the map is a two-parallel Lambert.
LDPs are an interesting case; they can be single or two-parallel design. What sets them apart is the combined factor, the product of the scale factor and the elevation factor. It will usually be the product of a scale factor that is less than (two parallel) or equal (single parallel) to 1 and an elevation factor that is greater than 1. This combined factor, then, can be 1, less than 1, or greater than 1, depending on the underlying design and the ellipsoid height selected.
It's in the LDPs that terminology becomes important. Loose descriptions of combined factors as scale factors lead to confusion. To me, it's always made sense to use NGS terminology: scale factor, elevation factor, and combined factor, but that's far from standard in the industry. As long as everyone is clear about meanings, it's unimportant, but that's often not the case.
This whole 2022 exercise is a great learning environment. So far, everything in the Lambert group has been duplicable from Stem's Manual 5.
I'm not sure I agree with the above. A single-parallel Lambert does not require the scale to be 1 at the central/standard parallel. It only requires that some grid scale factor be defined.
I would not label LDPs as "special cases". They are nothing more than projections tailored to minimize distortion, with the same defining parameters as other projections of the same type.
No projection definition should contain a combined factor or an elevation factor, only a grid scale factor. The grid scale factor may be determined from a particular elevation/combined factor (depending on terrain, population distribution, etc.), but ultimately it is defined in the same way that a state plane (or any other) projection would be defined - with a grid scale factor that is constant along the central parallel, central meridian, or skew azimuth (for OM projections).
As such, terminology for an LDP should be exactly the same as it would be for a state plane projection of the same type. The only difference is that the LDP will (should) minimize distortion over a particular area through judicious placement and selection of the grid scale factor.
I'm really not trying to be contrarian here. I just see a lot of confusion about LDPs, and far too many folks who think that LDPs are?ÿ voodoo magic, or at least something entirely different than just another projection. Any surveyor worth their stamp should be able to apply and work in an LDP just as they can work in a state plane projection.
I've seen clients reject using LDPs in favor of breaking big projects up into multiple sections with different CSFs applied, with all the attendant problems that you would expect from a major corridor design having 3 different coordinate systems. I've even seen surveyors reject using LDPs, which is just insane.?ÿ?ÿ
To me, it's always made sense to use NGS terminology: scale factor, elevation factor, and combined factor, but that's far from standard in the industry. As long as everyone is clear about meanings, it's unimportant, but that's often not the case.
This whole 2022 exercise is a great learning environment. So far, everything in the Lambert group has been duplicable from Stem's Manual 5.
?ÿNow this I wholeheartedly agree with!
@rover83?ÿ
You raise interesting points. In Stem's derivation of the Lambert projection, he first defines the two Standard Parallels and then uses them to solve for the Central Parallel and the scale factor on the Central Parallel. So he (and those who went before him at NGS) explicitly defined a two-parallel projection. For NC, the Central Parallel turned out to be 35.2527586002 degrees and its scale factor is 0.999872591882.
Now this projection can be exactly duplicated by assuming the Central Parallel to be 35.2527586002 degrees and the scale factor there to be 0.999872591882. Any point under one definition will have the same state plane coordinates that it has under the other definition. Defining the plane in terms of Central Parallel and scale factor instead of Standard Parallels does not change it in any way at all. This is a "one-parallel definition" of a two-parallel projection, with the Standard Parallels implicit and a function of the defined variables.
On the other hand, if we define the Central Parallel to lie at 35.2527586002 degrees north with a scale factor of 1.000000000000, then the two Standard Parallels disappear, and the projection is a single parallel Lambert. Mathematically, the projection plane is tangent to the ellipsoid at the Central Parallel instead of secant to the ellipsoid at the Standard Parallels. In fact, if you use Stem's formulas with the Central Parallel and the Standard Parallels all equal to each other, the calculation will blow up because of zero denominator.
The LDP is special because it often lies above the ellipsoid, neither tangent nor secant, because they never touch. If the LDP combined factor, often referred to as its scale factor, is greater than 1, then this is the case. However, there will be parallels north and south of the Central Parallel where the combined factors are equal; not equal to 1, but equal to each other.
LDPs are also special because an elevation factor is central to their definition. In many cases, it's really questionable as to whether an LDP serves better than using a single combined factor along with State Plane Coordinates. An LDP may be applicable to a larger area and grid azimuths may be closer to observed azimuths, but they still need to be adjusted.
the calculation will blow up because of zero denominator.
If you are referring to the equation using the ratio of ln differences, note that you replace it with n= sin(Phi-sub-1)?ÿ (Snyder,?ÿ _Map Projections_, USGS document PP-1395, p. 108.
@bill93?ÿ
Yes, but notice that this assigns a central parallel rather than calculating it from the two secant parallels. Follow it through to the scale factor calculation.
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@mathteacher?ÿ I was reading two formats (.dec & HMS) of the lattitude.?ÿ Which gave me 09" difference. nm
My understanding so far is they are not constrained by two-parallels in a secant projection, and that it isn't a true tangent (not planar ???ª ) using the GRS80 ellipsoid as the true definition for the central parallel but, instead using some measure of the topography above the ellipsoid to minimize linear distortion throughout the system. Pretty much what Sean was posting
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anyways ..
Yep, that's a good analysis. The reason for abandoning the explicit assignment of the two parallels is to allow for choosing a central parallel that more closely fits the state or zone as a whole. However, the two parallels still exist, albeit in different locations, and they can be implicitly determined just as the central parallel can when the standard parallels are known. NGS just believes that their exact location is not important, and I agree.
As I said earlier, it's a highly technical mapping issue. In terms of working with state plane projections, it is of no consequence.
But in terms of fully understanding the tools we're working with, it's very nice to know.
@mathteacher Thank you for expounding on the differences and similarities between 2-parallel and 1-parallel Lambert projections. What you've outlined was my point. I wasn't attempting to say that there was a performance difference between them, only that within the SPCS the single parallel that will deliver the same projection surface as a 2-parallel definition will not have an even minute value. Since the guidelines for SPCS design require an even minute and the single parallel version of existing 2-parallel projections cannot be to the even minute, the projection shape will necessarily be different, even if slightly. The result of this will be that the convergence angle (or mapping angle) change at a different rate for points that are not along the central meridian. Hopefully the difference will be practically negligible as many surveyors will want to be able to bypass rotation from transformations from SPCS83 to SPCS2022.?ÿ
"... the single parallel that will deliver the same projection surface as a 2-parallel definition will not have an even minute value."
That's very well stated and very true in the sense that either each of two standard parallels can have even minutes. or a single parallel can have even minutes, but all three likely cannot. I would say, though, that the single parallel specification is an easier path to a more nearly optimal projection than is the two-parallel specification. Further, NGS adopting this new design standard is a significant improvement, as is their approval process for fairly large LDPs.
Sometimes I quibble beyond being helpful, but I plead no contest in this case. The discussion has been very helpful to me and I hope to others. I don't do the work that I once did with map projections; GIS software is so good that I don't have to, but it can become a black box if we don't keep up the old skills.
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I like the single parallel Lambert too. My only caution is for surveyors to understand that the SPCS2022 Lambert zones with identical boundaries and the same central meridians as the 27 and 83 zones will have varying grid factors (i.e. scale factors) and varying mapping angles across the zone compared to 27 and 83. Hopefully the convergence variation will be minimal, but it will be different.?ÿ
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You always add so much to these discussions.?ÿ
You always add so much to these discussions.
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In many cases, it's really questionable as to whether an LDP serves better than using a single combined factor along with State Plane Coordinates
It really depends on two tests: the positional tolerance of your measurements, and the planned application of your deliverable.
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But I would say that for 95+% of typical survey work, LDPs are superior to applying a CSF to state plane.
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As a general rule, a well designed LDP will keep distortion to within 50ppm in critical/populated/important areas. That's five hundredths per one thousand feet of measured line. Unless we're measuring the line with just a straight EDM shot, disregarding angles, centering error, etc. (and any other connected lines/observations in the network), our final precisions on that 1000' line will likely be no better than that .05' at 95% confidence.
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Using the ALTA/NSPS specs as an example since they are applicable nationally and tend to be tighter than state specs for land boundaries, that line only needs to be precise to 2cm +50ppm, or ~0.12', to meet specs. So the nominal CSF of 50ppm is less than half of that error ellipse.
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And applying that single CSF to an entire project still doesn't get all lines exactly at ground, it just gets them a little bit closer in some areas (and maybe a little further away in others). It's still only an approximation.
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I'm a card carrying millimeter chaser, but I would submit that when we're that far below the threshold of detectable difference, there is no practical difference between grid and ground, and that the benefits of maintaining fidelity to a geodetic framework far outweigh half an inch on the ground. The caps we set are bigger than that.
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Which brings me (finally) to my second test, the application of our data. Our digital deliverables are being manipulated and interpreted by true geospatial software far more often than in the past. It's really annoying to try and work with data that are not easily transformed, reprojected or unable to have other datasets easily overlaid. The end users tend to really, really dislike it and it is far easier to screw it up.
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For that matter, as a surveyor, so do I. Give me LLH or XYZ ECEF with metadata over grid values (which inevitably come without metadata) any day. Keeping everything in a defined coordinate system lets the end user manipulate it however they see fit. Datum transformations, time shifts, reprojections are all far more easily done when we don't have to manually scale and move coordinates. And if we do need to transform or overlay a scaled system, once they're back in grid, that ground coordinate system is now gone anyways and that original survey product has been rendered useless in a geospatial sense.
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This is going to be even more critical when the new datums drop, with 4D reference frames tied to active control.
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I used to love scaling to ground. It just felt better. I would always publish geodetic, grid and ground values plus the transformation parameters...but LDPs play far better with more software packages and minimize the chance of screwups during a project. On a more selfish level, they make it easier for me to do my job too.
with 4D reference frames tied to active control.
Still trying to understand that. I know what you're saying, just not sure how it gets implemented.
This isn't going to be just another successor itrf at zero epoch but a time dependent data capture that .. unless we have the gps week recorded is irrelevant ??
I mean plate tectonics and velocities are all baked in ? Passive marks become an anachronism ??ÿ
When I first started playing around with map projections and their associated mathematics, I was awed by State Plane projections and the fit that they provided when each line was adjusted by its individual combined factor and each angle was adjusted for convergence. Then I discovered the increased efficiency that using a single combined factor over an area provided, albeit with some loss of good fit.
Then I actually read James Stem's Manual 5 and developed an Excel workbook that created Lambert projections, handled forward and inverse calculations of coordinates from lat/lon, and calculated scale, elevation and combined factors along with convergence angles. For the first iteration, I proved that a Lambert state plane could be duplicated from the location of the central parallel and the scale factor along that parallel. That is what NGS is doing with the 2022 state planes, but I did it out of ignorance; I didn't fully understand the calculation of the latitude of the central parallel from those of the two standard parallels.
It was a short step to Lambert LDPs. But further reading revealed that grid to ground conversions were widely used, even locally by NCDOT. It was hard to see how they could be so bad, and, except for the often-omitted definitions and some other caveats, they don't seem to be. However, duplicating ground coordinates from a TBC least-squares calibration is not a doable kitchen table task for a hacker like me. People climb Mt. Everest because it's there; I do arcane math for the same reason.
I'm still puzzled by all of the derogatory remarks hurled at state plane coordinates. With endless computing power available in both field and office, and the accuracy that fully adjusted state plane coordinates offer, I wonder why so many people think it's such a bad system. As near as I can tell, it's geodetically sound, well defined, well documented. and reliably computerized.
Anyway, I wrote a few papers about LDPs and Wendell was kind enough to publish some of them, but I think they've disappeared from the site. I've attached one that addresses a single parallel Lambert projection by developing one on a measured site and even includes a quiz at the end.
I hope you find it interesting.
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A one parallel Lambert means that from the central parallel north and south the grid scale will be larger than one. Often the ellipsoid surface is above sea level, sometimes quite a bit. This means that in costal areas the grid distances will be longer than ground distances. By a considerable amount near the edges of the zones. Of course it work better as elevations increase, but care must be taken to design the zones.?ÿ
I'm still puzzled by all of the derogatory remarks hurled at state plane coordinates.
I would contribute this is due to ignorance. Over the past 20 years I have done a lot of construction site setup for GPS machine guidance system all over the USA. I find about 50% of the surveyors think they are on SPC, but are not. I have seen several Landfill sites where a surveyor applied a grid to ground scale about coordinate 0,0. Which moved the entire Landfill more than 200ft from SPC. I was the first surveyor to identify this to the owners in 20 years of the Landfill operation. This meant that a drill rig operator was closer the SPC using his IPhone than the original surveyor was using $50k survey gear.
@rover83?ÿ
I'm missing my LDP right now, scale factor and control points across the boundary of two different SPC zones is making my head hurt.
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a surveyor applied a grid to ground scale about coordinate 0,0. Which moved the entire Landfill more than 200ft from SPC.
I would be worried that the coordinates were scaled, and didn't move.?ÿ Scaling from the zone origin is as equally valid, as scaling from any other point.
The red flag was in the surveyors datum note which states:
"NAD83 NY Western zone (Ground)"
FYI, never apply scale factor to coordinates.?ÿ
Scaling from the zone origin is as equally valid, as scaling from any other point.
Metadata!
The red flag was in the surveyors datum note which states:
"NAD83 NY Western zone (Ground)"
FYI, never apply scale factor to coordinates.?ÿ
That's busier than I want to entertain.?ÿ Not sure what happened there.
