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

[mathteacher]

The answers to problem 1 are:

Easting = 99,972.856 meters; Northing = 25,147.523 meters;
Longitude = 103?ø 39ƒ?? 51.81138ƒ?; Latitude = 39?ø 16ƒ?? 11.04107ƒ?

Any cogo routine will calculate the grid values, but whereƒ??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 ƒ?? 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 thatƒ??s not very instructive. Instead, letƒ??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?ø 34ƒ?? 26.8660ƒ?. Entering these parameters, along with the geographic coordinates of CBL 0, into NGS FORWARD gives the following coordinates for CBL 150:

Longitude = 103?ø 39ƒ?? 51.81138ƒ?, Latitude = 39?ø 16ƒ?? 11.04107ƒ?.

DNRGPS produces the identical result when the plane coordinates are ƒ??unprojected.ƒ?

[john-nolton]

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

 

 

[FrozenNorth]

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

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

[mathteacher]

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.

[mkennedy]

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

---

I’m just one of those evil GIS people. Bwah-hah-hah! Seriously, I do coordinate systems and transformations at Esri.

[john-nolton]

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

 

 

[FrozenNorth]

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.

[mathteacher]

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:

PROJCS[“IaRCS_01_Spencer_NAD_1983_2011_LCC_US_Feet”,GEOGCS[“GCS_NAD_1983_2011”,DATUM[“D_NAD_1983_2011”,SPHEROID[“GRS_1980”,6378137.0,298.257222101]],PRIMEM[“Greenwich”,0.0],UNIT[“Degree”,0.0174532925199433]],PROJECTION[“Lambert_Conformal_Conic”],PARAMETER[“False_Easting”,11500000.0],PARAMETER[“False_Northing”,9600000.0],PARAMETER[“Central_Meridian”,-95.25],PARAMETER[“Standard_Parallel_1”,43.2],PARAMETER[“Standard_Parallel_2”,43.2],PARAMETER[“Scale_Factor”,1.000052],PARAMETER[“Latitude_Of_Origin”,43.2],UNIT[“Foot_US”,0.3048006096012192]]

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

 

 

 

[mathteacher]

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:

https://nfdc.faa.gov/nfdcApps/services/ajv5/airportDisplay.jsp?airportId=klic

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 canƒ??t be confirmed by this measured distance.

[mathteacher]

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 Simpsonƒ??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?ø < Latitude < 90?ø }. 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 1.2403ƒ?? ,0.6329ƒ?? , 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 functionƒ??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 Simpsonƒ??s rule for the most accurate computation of the average scale factor of a line. Simpsonƒ??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 Simpsonƒ??s Rule with two intervals, it looks like this, where the xƒ??s are latitudes in radians:

Sometimes Simpsonƒ??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 Simpsonƒ??s Rule average scale factor and a simple average of the elevation factors.

[john-nolton]

MathTeacher; I have printed out your post on LDP and have some things for you to consider.

You say “One way to explore tangent LDPs is to develop one and compare some ground measurements to grid distances”.

I disagree with that. I think you should give the theory with nice drawings FIRST and then give some problems. You did give some problems.

So to help you with that I will give you 3 references to start with ( of course the Internet has a lot).

Ref. #1  “The flat Earth Concept in Local Surveys”  by T. Vincenty (who by the way was a member of the flat earth society, just for laughs).

(Surveying and Mapping, Vol. 49, No.2,1989, pp. 101-102

 

Ref. #2 “A Fresh Look at Tangent Plane Grids” by Joe  F. Dracup  in Surveying and Land Information Systems,

Vol. 58, No. 4, 1998, pp. 205-221

 

Ref. #3 “Design of a Local Coordinate System for Surveying, Engineering, and LIS/GIS  by Earl F. Burkholder

in Surveying and Land Information Systems, Vol. 53, No. 1, 1993, pp. 29-40 and a correction to this paper in the same journal

Vol. 56, No. 2 June 1996, p. 119.

 

Under your paper 11 lines down from “A shortcut to an LDP”

Units[“Degree”, 0.017453292519943295]], this last digit of “5” is correct if it is   TRUNCATED. Do you know

if this is the way Esri does things? This # would be 5769… which would round to 6

 

Look forward to your “NEW Paper”

 

JOHN NOLTON

[mathteacher]

Thanks for the comments, John. The easy one to address is the degree-radian factor, which, according to the calculator in my Pixel 2, is truncated at 16 significant digits. I suspect that the number has proven adequate, but it is an interesting note.

For the most part, the references you cited are inaccessible to me; i don’t have proper credentials. I have read Earl Burkholder’s paper I think, but not the others. The only reference needed for this effort is James Stem’s manual 5 which is freely available on the internet. The Lambert formulas begin on page 26 and definition of the GRS 80 ellipsoid is on page 13. But, if you’re really serious about state plane and low distortion map projections, you need to read the whole thing, cover to cover. The other given references are also freely available on the internet.

You know, learning is a very individual thing. When I first started teaching, I would begin a topic with the theoretical stuff and end with the problems. My best students could learn no matter what I did, but I lost a significant number of the others before I ever got to the problems. Then one day, I began the class with a problem before teaching the math needed to solve it. The change was dramatic and after that I varied the presentation order from topic to topic. My students learned to learn both ways and I learned to teach both ways, so it was win-win.

This one is a hybrid. There’s some math up front, a lot of results, some references, and some more math buried in the references. The idea was to present it in a way that would be informative on its own, but spur questions, comments and discussion to bring about the real learning. The solutions to the problems have introduced some more concepts and more math, again hoping for more discussion and learning from the discussion.

Here’s my first writing on the topic: https://www.xyht.com/surveying/pennsylvania-traverse-low-distortion-projection/ There are errors; one is noted in the replies, but there’s another one in the concept of the elevation factor. It has little effect on the result, but, now, it’s a bit embarrassing. Perhaps you’ll find it an interesting application.

You probably already have the answers to Problem 4, but I’m still working on them. Coming soon, I hope.

 

 

[mathteacher]

Here are the solutions to problem 4.

This modified .prj puts the grid origin at KJ0524 and uses a different false Easting and false Northing to make coordinates more easily distinguishable from those on the CBL LDP.

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”,39.198545625],
PARAMETER[“central_meridian”,-103.695986566667],
PARAMETER[“scale_factor”,1.00026334],
PARAMETER[“false_easting”,200000],
PARAMETER[“false_northing”,75000],
UNIT[“Meter”,1],
PARAMETER[“standard_parallel_1”,39.198545625]]

At KJ0524, the elevation factor is 0.99973673. Its reciprocal, 1.00026334, is used as the scale factor at the grid origin. Note that NGS 2022 specifications limit this scale factor to 6 decimal places; the difference is negligible.

The scale factor function, which can be used to calculate the scale factor for any point, is:

f(x) =1.239459116120*(((1+0.081819191042832*sin(x))/(1-0.081819191042832*sin(x)))^0.081819191042832/((1+sin(x))/(1-sin(x))))^(0.63200963155528/2)*(1-0.081819191042832^2*sin(x)^2)^(1/2)/cos(x)

Calculated Coordinates for all of the points considered are given below on both coordinate systems. The new one is named Limon Southwest LDP. Standard coordinate transformation software should accurately translate, rotate, and scale coordinates from either system to the other. (As a final exercise, determine the coordinates for the true grid origin for the Limon Southwest projection.)

Limon CBL LDP (meters)
Point                  Latitude                           Longitude                Northing              Easting
CBL 0           39.268405177776       -103.664077586110         25,000.000    100,000.000
CBL 400      39.271949486108       -103.664916597220         25,393.591       99,927.580
CBL 1400     39.280806452798      -103.667013661110          26,377.150       99,746.603
Runway 16   39.281131610000        -103.667371970000        26,413.260      99,715.680
Runway 34   39.268443690000      -103.664374783000       25,004.277      99,974.346
True Origin   39.043274846287      -103.664077586110                 0.000    100,000.000
KJ0524          39.198545625000      -103.695986566670         17,242.721       97,242.883
KJ0231           39.283290175000      -103.778983419440        26,659.255       90,083.433
CBL 150*       39.269733630939      -103.664392049992        25,147.523       99,972.856
*Coordinates were calculated

Limon Southwest LDP (meters)
Point                      Latitude                              Longitude               Northing     Easting
CBL 0                 39.268405177776      -103.664077586110     82,758.332   202,754.414
CBL 400            39.271949486108      -103.664916597220    83,151.903   202,681.855
CBL 1400           39.280806452798    -103.667013661110     84,135.410    202,500.529
Runway 16         39.281131610000      -103.667371970000   84,171.509     202,469.593
Runway 34         39.268443690000   -103.664374783000   82,762.600    202,728.758
True Origin*      39.043274846287     -103.664077586110    57,758.107     202,763.213
KJ0524              39.198545625000     -103.695986566670    75,000.000   200,000.000
KJ0231                39.283290175000   -103.778983419440     84,414.107     192,837.147
CBL 150**           39.269733630939   -103.664392049992    82,905.848    202,727.217
*True projection origin for Limon CBL LDP ** Coordinates were calculated

Grid distances on Limon Southwest are slightly different from those on the CBL LDP. For example, the length of the CBL calculated from Limon Southwest coordinates is 1,400.2862 meters, compared to 1,400.2686 meters from the CBL LDP coordinates. Compared to the measured distance of 1,400.2719 meters, the Limon Southwest difference is 10 ppm while that for the CBL LDP is -2ppm.

The situation is different for the distance from KJ0231 to KJ0524. The Limon Southwest distance is 11,829.280 meters compared to a state plane computed ground distance of 11,829.246 meters, about 3ppm different. Recall from Problem 3 that the CBL LDP difference was -8ppm.

Of course, this exercise is illustrative only. Far more work by people far more competent and qualified than I would be required to render either of these projections usable. The purpose here is to provide insight into the development process and to encourage experimentation with near-ground coordinate systems.

Dr. Charles Ghilani, who is indeed both competent and qualified, published a series of articles a few years ago in XYHT magazine that described in some detail the use of state plane coordinates and alternatives. Some of Dr. Ghilaniƒ??s thoughts on LDPs and more conventional methods can be found at the links below. In fact, the series is a valuable reference anyone interested in geodesy and surveying.

https://www.xyht.com/surveying/transformation-observations-2/
https://www.xyht.com/gnsslocation-tech/transformation-of-observations-part-4/