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# Lambert meets Limon: An LDP at a CBL with a Quiz at the End

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Among the items on the NGS datum revision agenda for 2022 is an offer to develop Low Distortion Projections (LDPs) as requested by individual states. Covering relatively small areas (NGS has formulated rules for this), they are likely to be tangent projections instead of the secant projections used for both NAD 83 and NAD 27 state planes.

One way to explore tangent LDPs is to develop one and compare some ground measurements to grid distances. Since Calibration Base Lines have published measured distances and sometimes have NGS data sheets for their marks, they are a convenient resource. To be a good candidate, the CBL must also have been measured recently and it must be at an elevation that makes state plane grid distances significantly different from ground distances.

The Limon, Colorado CBL meets those criteria. Its marks have NGS data sheets containing NAD 83(2011) coordinates and the ellipsoid height of CBL 0 is 1610.152 meters. It was re-measured in 2014.

The marks are CBL 0 (KJ0588), CBL 400 (KJ0593), and CBL 1400 (KJ0594). The 1400 mark is listed as destroyed, but the data sheet is still available. Also, a 150 mark is listed in the CBL data, but there is no data sheet for it.

The grid origin for the Limon CBL LDP is CBL 0. The LDP is developed in accordance with the Lambert presentation in NGS Manual 5, modified to accommodate a single standard parallel.

For the projection to be tangent to the ground at CBL 0, the combined factor (CF) at that point has to equal one. The combined factor is the product of the elevation factor (EF) and the scale factor (SF). The elevation factor on the data sheet for CBL 0 is 0.99974744, so the scale factor can be computed:

CF = EF * SF

1 = 0.99974744 * SF

SF = 1/0.99974744 = 1.00025262380

Rounding the scale factor to 1.0002526 makes the distortion at the grid origin -0.020 parts per million with no effect on the Northing and Eastings of the points, so the practical effect is nil. However, the additional digits were helpful in eliminating system bugs, so I kept them.

Table 1 compares the mapping constants for the Limon CBL LDP to those of the Colorado state plane Central Zone 0502. Some of the defining constants for the state plane projection are computed constants for the LDP and vice versa. The table is in the current NGS format for a Lambert state plane, with the exceptions marked by bold print and asterisks.

The Colorado projection is defined by the standard parallels, Bs and Bn; the true projection origin, Bb; the longitude of  the central meridian, Lo; the Northing at the true projection origin, Nb; and the Easting for the central meridian, Eo. Values for these numbers were selected by NGS.

The Limon projection is defined by the latitude of its central parallel, Bo; the scale factor at the central parallel, ko; the longitude of the central meridian, Lo; the Northing for the grid origin, No; and the Easting for the central meridian, Eo. Values for these constants were selected by the author.

 Table 1. Comparison of Constants for Limon CBL LDP and Colorado Zone 0502 Defining Constants Limon CBL LDP Colorado Central Zone 0502 Bs= NA 38:27 Bn= NA 39:45 Bb= 39:02:35.78945* 37:50 Lo= 103:39:50.67931 105:30 Nb= 0.0000 304,800.6096 Eo= 100,000.0000 914,401.8289 Computed Constants Limon CBL LDP Colorado Central Zone 0502 Bo= 39.2684051778** 39.1010150117 SinBo= 0.632954054800 0.630689555225 Rb= 7,838,789.7339 7,998,699.7391 Ro= 7,813,789.7340 7,857,977.9317 No= 25,000.0000** 445,522.4170 K= 12,498,366.6842 12,518,269.8410 ko= 1.00025262380** 0.999935909777 Mo= 6,362,619.2189 6,360,421.3434 ro= 6,375,457 6,373,316

*Computed constant.  **Defining constant. NA means Not Applicable.

The Limon CBL LDP could have been designed by using Bb, the true grid origin, as a defining constant. Using a value such as 39:03 would have cleaned up the constants somewhat. However, doing that while preserving No = 25,000 would have required making Nb a computed constant, equal to 746.7963. Cleaning up one constant can dirty up another.

Thus, there are options and trade-offs inherent in the LDP design process. Taking one more step into the details, note that defining Bb as 39:02 while keeping everything else the same would make Nb negative.

LDP coordinates and more

Table 2 compares the LDP coordinates of the Limon CBL marks to the Colorado state plane coordinates for those marks. Note that, in the LDP, each point has a unique combined factor just as it does in the state plane. This is one of the ways that an LDP differs from the application of a project-wide combined factor. Grid distances computed from LDP coordinates will be very close to ground distances, but they can be made even closer by applying state plane distance computation procedures to them.

 Table 2.  Comparison of Coordinates Limon CBL LDP and Zone 0502 CBL 0 Limon CBL LDP CO State Plane Latitude 39° 16' 06.25864" 39° 16' 06.25864" Longitude 103° 39' 50.67931" 103° 39' 50.67931" Northing 25,000.000 465,705.526 Easting 100,000.000 1,072,818.305 Scale Factor 1.0002526238 0.99994016 Elev. Fact. 0.99974744 0.99974744 Comb. Factor 1.00000000 0.99968762 Distortion, ppm 0.000 -312.380 Convergence 0° 00' 0.0" 1° 09' 28.4" CBL 400 Limon CBL LDP CO State Plane Latitude 39° 16' 19.01815" 39° 16' 19.01815" Longitude 103° 39' 53.69975" 103° 39' 53.69975" Northing 25,393.591 466,097.450 Easting 99,927.580 1,072,737.972 Scale Factor 1.00025263 0.99994035 Elev. Fact. 0.99974717 0.99974717 Comb. Factor 0.99999973 0.99968754 Distortion, ppm -0.270 -312.460 Convergence − 0° 00' 1.9" 1° 09' 26.5" CBL 1400 Limon CBL LDP CO State Plane Latitude 39° 16' 50.90323" 39° 16' 50.90323" Longitude 103° 40' 01.24918" 103° 40' 01.24918" Northing 26,377.150 467,076.846 Easting 99,746.603 1,072,537.218 Scale Factor 1.00025265 0.99994082 Elev. Fact. 0.99974618 0.99974618 Comb. Factor 0.99999876 0.99968702 Distortion, ppm -1.240 -312.980 Convergence − 0° 00' 6.7" 1° 09' 21.8"

Other noteworthy items in Table 2 are the differences in magnitude of the LDP and state plane coordinates, the distortion values, and the convergence angles for the two projections. The LDP is designed for a very small area, so large-magnitude coordinates are not necessary. However, the state plane convention of having Eastings much larger than Northings is maintained. The LDP distortion values indicate that grid distances will be very close to ground distances. Convergence is a function of the difference between a point’s longitude and the central meridian for some given latitude, so the LDP convergence angles are guaranteed to be nearer to zero.

Verifying distances and azimuths

Table 3 shows distances and azimuths for the points on the Limon CBL. Published distances can be verified at https://www.ngs.noaa.gov/CBLINES/BASELINES/co .

 Table 3. Distances and Azimuths Limon CBL Distances in Meters Coordinates Grid Distance, Limon CBL LDP Published Distance Point Northing Easting From CBL 0 From CBL 400 From CBL 0 From CBL 400 CBL 0 25,000.000 100,000.000 -- -- -- -- CBL 400 25,393.591 99,927.580 400.1981 -- 400.1971 -- CBL 1400 26,377.150 99,746.603 1,400.2686 1,000.0705 1,400.2719 1,000.0747 Azimuths Arc-to-Chord (T − t) From To Grid Azimuth Geodetic Azimuth Decimal Grid Az f3 T − t (Seconds) CBL 0 CBL 400 349° 34' 27.4664" 349° 34' 27.5976" 349.5742962 39.26958661 -6.697E-09 CBL 0 CBL 1400 349° 34' 26.8662" 349° 34' 26.8660" 349.5741351 39.27253894 -8.200E-08 CBL 400 CBL 1400 349° 34' 26.5725" 349° 34' 24.6614" 349.5740707 39.27490181 -9.204E-08

Grid distances are within millimeters of the published ground distances. The largest difference, 4.2 mm, is from CBL 400 to CBL 1400. That’s accuracy of about 1 : 238,000.

Examining the azimuths is instructive. Geodetic azimuths change over the course of the line, illustrating that geodetic lines do not have constant azimuths. Grid azimuths, adjusted for convergence, are close to geodetic azimuths.

A shortcut to an LDP

Software that can handle .prj files makes computing LDP coordinates easy, but it doesn’t duplicate rigorous development of an LDP. While the software generates both forward and inverse coordinates, it may not display scale and elevation factors for the points nor the computed constants associated with the projection.

Nevertheless, here is a .prj for the Limon CBL LDP that works in DNRGPS and, perhaps, some other GIS and surveying systems. The defining constants from Table 1 are easily recognizable.

PROJCS["Lambert_Conformal_Conic",

GEOGCS["GCS_GRS 1980(IUGG, 1980)",

DATUM["D_unknown",

SPHEROID["GRS80",6378137,298.257222101]],

PRIMEM["Greenwich",0],

UNIT["Degree",0.017453292519943295]],

PROJECTION["Lambert_Conformal_Conic"],

PARAMETER["latitude_of_origin",39.268405177778],

PARAMETER["central_meridian",-103.66407758611],

PARAMETER["scale_factor",1.0002526238],

PARAMETER["false_easting",100000],

PARAMETER["false_northing",25000],

UNIT["Meter",1],

PARAMETER["standard_parallel_1",39.268405177778]]

Takeaways

My major takeaway from this project was the importance of the interplay between defining constants and computed constants for Lambert projections. The classifications and the values of the defining constants are design decisions that need careful consideration.

For 2022 secant Lambert state planes, NGS is changing the scale factor from a calculated constant to a defining constant. As a result, the two standard parallels become calculated constants. Knowing their values is neither necessary nor particularly useful and they won’t be nice round numbers, so they won’t be published.

The Northing at the true projection origin might be viewed in the same light. Changing it from a defining constant to a calculated constant would allow the choice of a nice round number for the Northing at the grid origin.

Exercises

Having read this far, perhaps you would consider doing a little work. The first exercise is easy; the last two are for folks who want to learn a bit more.  The reward for correct answers is personal satisfaction. There is no penalty for incorrect ones, but shame on you if you don’t at least do the first one.

1. Determine both the LDP and the geographic coordinates of CBL 150.
2. The Limon CBL is located at the Limon Municipal Airport (KLIC). The geodetic coordinates for the runway ends are 39-16-52.0738N, 103-40-02.5391W and 39-16-06.3973N, 103-39-51.7492W. The slope length of the runway is 4700 feet, rounded or truncated to feet, whichever the FAA requires. What are the Limon CBL LDP coordinates of the runway ends? What is the grid distance from runway end to runway end? Why is the geodetic distance more than a foot different from the grid distance? (Hint: Use the .prj file.)
3. The Limon CBL LDP optimizes grid versus ground near the Limon Municipal Airport. To judge  its performance at some distance from its grid origin, determine Limon CBL LDP coordinates for KJ0231 and KJ0524 and find the LDP grid distance between them. Then use Colorado state plane coordinates and combined factors for the points to compute the ground distance between the two points. (Is the simple mean of the scale factors adequate?)  Is the Limon CBL LDP sufficiently accurate for this computation?
4. Modify the .prj file given in the text to create an LDP that is tangent to the ground at KJ0524. What is the scale factor for the new LDP? the plane coordinates of Limon CBL 0?

More resources:

DNRGPS. A software package originally developed by Minnesota Department of Natural Resources, but now in the public domain. Designed to collect data from Garmin GPS receivers, it works with many other brands, including some survey grade receivers. It can transform coordinates using its extensive library of projection data and it accepts custom projections. Sometimes a bit temperamental, it is nonetheless a great little system. https://www.dnr.state.mn.us/mis/gis/DNRGPS/DNRGPS.html

ESRI Support. Here’s a bit more about .prj files.   https://support.esri.com/en/technical-article/000001897

Krakiwsky, E. J. Conformal Map Projections in Geodesy. http://www2.unb.ca/gge/Pubs/LN37.pdf

Explains the mysterious Qs, Qn, and Qo in Stem’s publication and a host of other stuff.

Stem, James E. State Plane Coordinate System of 1983.. https://www.ngs.noaa.gov/library/pdfs/NOAA_Manual_NOS_NGS_0005.pdf

Note: Two necessary calculated constants for a Lambert projection are K and Ro, the mapping radii at the equator and the central meridian, respectively. Modifying secant state plane projections to single parallel projections can be accomplished by rewriting two equations on page 28 of this publication.

Since k0 is a defining constant for the single parallel projection, the equation that calculates its value (the second equation on the page) can be solved for Ro:

Ro = (k0*a)/(w0*tan(phi0))

Then solve the first equation on the page for K:

K = Ro*exp(Qo*sin(phi0))

With these constants calculated, the forward and inverse formulas can be used as written. These values are determined by GIS software from the information in the .prj file, but they can be calculated in spreadsheet implementations.

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The answers to problem 1 are:

Easting = 99,972.856 meters; Northing = 25,147.523 meters;
Longitude = 103° 39’ 51.81138”; Latitude = 39° 16’ 11.04107”

Any cogo routine will calculate the grid values, but where’s the fun in that? Trig is also effective, but plane geometry is more elegant because it uses only distances and known coordinates to obtain a solution.

The grid length of the CBL is 1,400.2686 meters and we want the coordinates of the point on the CBL that is 150 meters from CBL 0. Thus, the distance from CBL 0 to CBL 150 is 150/1400.2686 of the total length of the CBL. Plane geometry tells us that the x and y coordinates of CBL 150 are each that same fractional distance between the coordinates of CBL 0 and CBL 1400.

So, the plane coordinates of CBL 150 are:

E = (150*99,746.603 + (1.400.2686 – 150)*100,000) / 1,400.2686 = 99,972.856 meters,
N = (150*26,377.150 + (1,400.2686 -150)*25,000) / 1,400.2686 = 25,147.523 meters.

Note that the same process can be used to check the collinearity of CBL 400 with CBL 0 and CBL 1400.
The longitude and latitude of CBL 150 can be computed from the plane coordinates by GIS or surveying software, but that’s not very instructive. Instead, let’s use the NGS forward program and let the GIS software be our check.

The average elevation factor of the CBL is 0.99974681, so a ground distance of 150 meters is an ellipsoid distance of 0.99974681*150, or 149.9620 meters. The geodetic azimuth from CBL 0 to CBL 1400 is 349° 34’ 26.8660”. Entering these parameters, along with the geographic coordinates of CBL 0, into NGS FORWARD gives the following coordinates for CBL 150:

Longitude = 103° 39’ 51.81138”, Latitude = 39° 16’ 11.04107”.

DNRGPS produces the identical result when the plane coordinates are “unprojected.”

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Member
(@john-nolton)
Joined: 9 years ago

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Nice post M.T.  I am not a fan of LDP so I have not gone over this very good. I did notice 2 things;

Under your first post "A shortcut to an LDP" Minnesota uses the word SPHEROID. You might want to

call them and tell them its ELLIPSOID. In your second post calculating "E" you have 1.400.2686

You might want to correct the first decimal point to a comma before somebody jumps on you for that.

See your "N" entry just below it.

JOHN NOLTON

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Spheroid is fine. https://en.oxforddictionaries.com/definition/spheroid

Interesting that you are "not a fan of LDP."  Do you prefer High Distortion Projections?

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(@mathteacher)
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Thanks, John. My old rheumy eyes are getting older and rheumier.

We see one of the objections to LDPs in problem 4, where we create a second, overlapping LDP. But overlapping coordinate systems exist outside the LDP world as well, with state borders and modified ground coordinates being just two examples. Having NGS create wider-area projections may prove very useful in this regard. I can envision one that crosses state lines in the Charlotte, NC - Rock Hill, SC area.

There are no panaceas in modeling the earth's surface, but there are tools. LDPs are just one of them.

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(@mkennedy)
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The usage of spheroid by Minnesota may come from using Esri software. That's what Esri was using when I started in 1994. I believe they got it from DMA as I've found that terminology in their documents from that time.

Melita

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The word spheroid comes from the UK. When the UK say spheroid they mean ellipsoid of revolution. The math as you know,

clearly is an ellipsoid . The governing body that controls the size and shape of the ellipsoid calls it Ellipsoid. Also I would like to

point out that governing body has their international meeting in several months and I do believe we will have some NEW

ellipsoidal constants (small changes but a change). Yes I do think Esri used spheroid and many others. Old habits die hard.

Of course USC&GS used spheroid (Clarke spheroid 1866 and others).

JOHN NOLTON

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I'll have to agree that a great many English words come from England.  Spheroid seems to have essentially the same meaning here and there.  American Heritage Dictionary https://ahdictionary.com/word/search.html?q=spheroid and Defense Mapping Agency, as Melita notes http://earth-info.nga.mil/GandG/publications/tm8358.1/tr83581a.html .  Much like the Oxford definition above.

As you note, the word ellipsoid is now more commonly used.  But spheroid is a technically accurate term for the ellipsoids we commonly use for survey datums.

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This one started as an Iowa Regional Coordinate System .prj, which is an Esri file that uses the word "spheroid." I don't remember which one I used, but here's the one named Spencer:

I substituted the Limon constants for the Iowa ones, including changing the foot conversion to meters, and changed its name. DNRGPS searches its library for an existing projection that duplicates the one it was fed. In this case, it found one, and you can see the differences in the datum definitions. For publication, I used the DNRGPS version instead of going back to my modified IRCS version, to create less confusion for anyone who wanted to use it.

The IRCS .prj files contain both Lambert and Transverse Mercator projections and can be found here: https://iowadot.gov/iarcs/

According to Moritz, the GRS80 ellipsoid is an equipotential ellipsoid of revolution, so, perhaps, spheroid is not a total misnomer. His paper is here: ftp://athena.fsv.cvut.cz/ZFG/grs80-Moritz.pdf

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The solutions to Problem 2 are:

Runway 16 coordinates (meters): 26,413.260N, 99,715.680E
Runway 34 coordinates (meters):25,004.277N, 99,974.346E
Grid distance: 1,432.530 meters, 4,699.89 feet
Ellipsoidal distance from NGS INVERSE: 1,432.1677 meters, 4,698.704 feet

The grid distance is greater than the ellipsoidal distance because, in keeping with its design, the LDP plane is above the ellipsoid.

The published slope length of the runway is 4,700 feet, rounded to the nearest foot. Grid distance is a horizontal distance, so the two are not directly comparable. However, the elevation difference between the runway ends is 25.8 feet, from:

Using that number, the grid distance converts to a slope distance of 4,699.96 feet, but the magnitude of the rounding in the published length is still unknown, so the accuracy of the grid distance can’t be confirmed by this measured distance.

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The answers to Problem 3 are:

Limon CBL LDP Coordinates in meters:

KJ0231: 26,639.255N, 90,083.433E
KJ0524: 17,242.721N, 97,242.883E

Grid distance = 11,829.152 meters

CO State Plane ground distance:

11,829.244 meters, from simple average of scale factors
11,829.246 meters, from Simpson’s Rule average of scale factors

LDP distortion: -8 ppm

The adequacy of both the LDP grid distance and the simple average state plane ground distance computation are matters of professional judgement.

A note on scale factors

Various algorithms and formulas exist for calculating scale factors for points on Lambert planes, but there is one mathematical concept that underlies them all. That inescapable concept is that the scale factor is a periodic function of latitude that is continuous over the domain { 90° < Latitude < 90° }. For every point, even those infinitesimally close to each other, there is a unique scale factor.

For the Limon CBL LDP, that function is:
f(x) = 1.240313883371*(((1+0.081819191042832*sin(x))/(1-0.081819191042832*sin(x)))^0.081819191042832/((1+sin(x))/(1-sin(x))))^(0.632954054800/2)*(1-0.081819191042832^2*sin(x)^2)^(1/2)/cos(x)

where f(x) is the scale factor and x is the latitude of the point. The numbers 1.2403… ,0.6329… , and 0.0818 are constants calculated to simplify the function a bit. The first one is K*sin(phi0)/a where K and phi0 are constants for the projection and a is the semi-major of the GRS80 ellipsoid. The second one is sin(phi0)) and the third one is the eccentricity of the GRS80 ellipsoid.

Pasting the right-hand side of the function into the functions section of this Zweig Media site will let you see the function’s graph and also evaluate the scale factor for any latitude for the Limon CBL LDP: https://www.zweigmedia.com/SVGGrapher/func.php Just remember to enter latitudes in radians.

Going a bit further, the continuous nature of the scale factor function over the domain relevant to a map projection is the source of the use of Simpson’s rule for the most accurate computation of the average scale factor of a line. Simpson’s rule is a numerical approximation to a definite integral of a function. The definite integral calculates the area between the graph of the function and the x-axis over the domain of the integral.

The average value of a continuous function over a given domain is a definite integral divided by the length of the domain. Intuitively, we are dividing an area by a length to get another length. In the case of the average scale factor, we are dividing an area by the difference of two latitudes. Symbolically, for Simpson’s Rule with two intervals, it looks like this, where the x’s are latitudes in radians:

Sometimes Simpson’s Rule is used to calculate an average combined factor, the product of a scale factor and an elevation factor. But the use of this rule for calculating average scale factors comes from the functional nature of the scale factor and integral calculus. Elevation factors follow no such functional rule. Thus, the most accurate combined factor is the product of a Simpson’s Rule average scale factor and a simple average of the elevation factors.

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