Recent discussions of ground coordinates and LDPs along with @holy-cow calling for more posts prompted me to do a bit of work and offer a bit of math perspective, with Oregon as the focal point. Oregon has an enviable set of LDPs which apparently are used very successfully. In other states, state plane coordinates are adjusted to ground and also used very successfully. I wanted to compare the two methods from a strictly computational viewpoint.
The target is Runway 11/29 at Roberts Field in Redmond OR in the Bend-Redmon-Prineville Zone of the OCRS (Oregon Coordinate Reference System.) The published length of this runway is 7,006 feet. This number is truncated, not rounded, so it's not exact, but it's good enough for checking out the computations.
The spreadsheet below shows the data and computations.
The top half uses OCRS coordinates to compute distances: Grid, Ground, XYZ, and Slope. The bottom half uses Oregon South state plane. The lat/lon and ellipsoid heights come from NGS Aeronautical Data for runway ends.
Runway 29 begins at
44°14'55.5964"N
121°08'18.5266"W
and Runway 11 begins at
44°15'32.1515"N
121°09'40.2406"W.
OCRS points, like State Plane points, have both scale factors and elevation factors and thus combined factors, so both OCRS grid distances and ground distances can be computed. As illogical operationally as it might sound, OCRS coordinates can be adjusted to ground although anyone who did so might suffer unimaginable consequences.
While State Plane has more than 25 times the distortion of OCRS in this example, the difference in their ground distances is only 2/1000 of an International Foot. When rules are followed, good answers result.
Elevation factors are tied to the ellipsoid and don't change unless the ellipsoid changes, so they are the same for both projections. The same is true for XYZ coordinates.
Distances calculated using XYZ coordinates are slope distances because the values of X, Y, and Z all are affected by ellipsoid height. If the height is changed leaving everything else the same, all three coordinates will change.
Regional LDP or ground coordinates? As far as accuracy and computability go, there's really no difference. If it seems better to determine a region-wide combined factor and use it to compute coordinates based on lat/lon measurements, while accepting "minor" differences in measurements, then do it. If it seems better to determine a combined factor for a more limited area with potentially less "minor" differences, then do it.
The math works either way.
@mathteacher The following is old news to you, as it is to the fellow septuagenarians and octogenarians who participate here.
I tend to assume that the vast majority of our younger peers here have no concept of how math computations were carried out in the deep, dark past, say 55 years ago. The first day of university experience for me did not include even a basic pocket calculator that could add, subtract, multiply and divide, but nothing else. It is possible that some professor, somewhere on campus may have had one, but, I doubt it. That was August 1971.
A Brief History: The Busicom LE-120A, known as the HANDY, is the first handheld calculator to use a “calculator on a chip” integrated circuit. According to the Vintage Calculators Web Museum, the calculator featured a 12-digit display in red LED and cost $395 when it first went on sale in January 1971. Because the calculator was so expensive, it came with a wrist strap attached at its base to protect it from being dropped.
Note the price. I was driving a car that cost less than that over 50 miles per day that ran on gas costing 35.9 cents per gallon. Buy a hundred gallons of gasoline today. Imagine paying that amount of money to obtain a little black box that could only add, subtract, multiply and divide, but nothing else.
The HP-65 became available in 1974. It was preceded by the -35 and -45, which added trig functions and storage.
A Brief History: First introduced by Hewlett-Packard as a “Personal Computer,” the calculator allowed users to either buy programs on pre-programmed cards or write programs up to 100 lines long and record them on blank cards. The device featured user-definable keys (with 35 keys controlling more than 80 operations) and was the first HP pocket calculator with base conversions (octal and decimal). The calculator cost $795 when it launched in 1974. In 1975, during the first joint U.S.-Soviet space flight, it became the first handheld calculator in outer space.
Note the gap in time following putting a man on the Moon in July 1969. Again, I was driving a reliable auto that cost less than an HP-65.
In 1979 I was provided a programmable TI handheld calculator to calculate the quantity of dirt required to build earthen dams with the longest being around 2500 feet long and a maximum height of 29 feet. I forget the model number. It came with small magnetic strips to hold programs. We had a hand-drawn topographic map from which we could estimate the ground elevation at stations of 25-foot increments along the proposed centerline of the dam. Determining the cross-sectional area at each station with three different slopes being applied on the front side and a single slope on the backside was a time consuming and super-dull process where errors could be easily made. Averaging paired cross-sections was tedious at best while compiling the long list of incremental volumes to get to the far end of the dam. With my program, we had the answer within seconds of inputting the last piece of data. That allowed charging out something like eight hours of work time for less than an hour of actual time plus the few seconds of magic black box time.
Fascinating. I graduated from college in 1967 with a mathematics degree. The physics department was building a computer which I wasn't allowed to be in the room with given some of my misadventures in chemistry labs.
Do you remember the Bowmar calculators? I was allowed to touch one of those occasionally, but I was programming PCs before I ever owned a calculator. Before them, there were desktop programmable calculators (Monroe 1860) with 512 bytes of program storage that used magnetic cards. Programming those was serious stuff, no high-level language, all 3 digit codes with loop and subroutine capability if you were creative enough to 512 bytes. I learned the difference between branch and jump on those machines.
I think that you and I have lived through the best of times, applied math-wise. We had to learn the theory behind everything we did and be creative in pencil and paper solutions to complex problems. Then computers made it so easy to get answers, but we still knew what they were doing even if we didn't know how they did it. That gave us a tremendous leg up during the transition.
I read half a dozen academic papers a year where the authors seem to search for statistical tests that will make their data say what they want it to say. I've seen similar stuff in surveying where something would not pass at the 95% level so the solution was to either throw out data or go to the 99% level where it would pass.
In very broad terms, an LDP is just another way to select a project combined factor. If it's intended to cover several counties, then it requires careful choices for its origin, error tolerance, and extent, but it's the exact same math that produces state plane projections. I haven't tried it, but I'm certain that the process that @mightymoe uses could be used just as effectively over a wide area.
Thanks for the reminder of the good old days!
In the 1970/71 college year (possibly 71/72?) a couple of us would sneak unto a lab to play with an HP9100, a large table-top unit. It had the functions of a basic scientific calculator with a green CRT maybe 3x5" graphing display.
It cost the price of a new Chevy sedan.
Before I graduated in 72 I saw one student with an HP scientific pocket calculator.
My father was an EE professor. I remember that he had some huge desktop calculator, the size of a large typewriter in his office. I used to go there in the early 70's and manually compute my age in seconds (i.e. how many seconds old I was) on the blackboard in his office, and then he would let me check my work on the calculator. I also remember that in 7th grade (probably 1971) I would come up with complicated math problems that I would solve on the calculator and then take them to my Algebra 1 class to try to embarrass the math teacher. Then he got an HP35 when they first came out in 1972 or so. Game changer. It was not programmable, but it did have trig functions, exponents, etc.
State plane there is fairly benign, 69ppm, the LDP is quite precise at 3ppm, 3ppm at those distances is pushing the limits of modern measurement devices. If a scale of 1.000069 is applied then SP numbers will be similar to the LDP. Basically the system becomes an LDP by scaling it. The main issue I've had with scaling SP is that it doesn't solve the other distortion issue, which is the drift from true north. Depending where the coordinates are in the zone tells just how far it's drifted. Some extreme rotations of +4 degrees make SP so distorted that it becomes a real pain to use, especially doing retracement.
I'd be very happy to see LDP's set up across the states I work in. Rolling your own comes with the problem of sharing, if the zone can't be clicked on with the major programs it becomes only an in-house calculation. We shall see where the industry ends up after the release of 2022.
Yes, extreme convergence angles is one problem that LDPs lessen but don't completely solve because there is no total solution. Coordinate consistency is another problem that is put to bed with an LDP.
In the Oregon Reference manual, three lat/lon coordinate pairs in Zone 4 (this one) are given for checking one's software to see that it produces LDP grid coordinates in meters correct to five decimal places. All three of these points are NGS marks, one on this airport property, but the manual renames them and doesn't give their PIDs. I've always wondered why ODOT would go to the trouble of using NGS marks when coordinate pairs made up would check the software just fine, and then not tell anybody.
It doesn't really matter, it just spurs curiosity.
At the extreme limits of the intended zone coverage for the Portland area LDP- about 50 miles by 50 miles outside of central Portland somewhere in the forests of the Coast Range- the convergence angle grows to about half a degree. The convergence angle for state plane in the same location would be over 2 degrees. In my suburban Portland stomping grounds the LDP's convergence angles max out at 4 minutes. State Plane convergence angles are close to 2 degrees.
A two degree convergence is large. When doing retracements you have to be aware of the convergence and deal with it when necessary. I had a long discussion with a surveyor who was working for an engineering company we were contracted with.
He wanted to double prorate in a missing section corner.
Mean the northings, prorate in the eastings. It was how he was taught.
However, because of the convergence the resulting corner would be approximately 2' from the correct position using "true north". We were just starting to use GPS, I was able to convince him by meaning the latitudes and prorating in the longitudes. Two different positions. The light came on then.
However, because of the convergence the resulting corner would be approximately 2' from the correct position
The question of how to properly double proportion a corner when the as-surveyed data was in state plane was a major question on the Washington State test. At least it was back in 2001.
There is a document by Jerry Wahl that can be had for googling "Double Proportion Made Complex" that, in part, deals with the issue.
That would change the dynamic. It would be the same as retracing an 1880 GLO survey,,,,,,,,, use the bearing basis the original was created with, and with State Plane do it on the surface of the State Plane system. There should be no scale up.