When introducing the concept of state plane coordinates, it may be helpful to be able to say where the origin of the coordinates (that is, northing = 0, easting = 0) is by naming a town or city. This can easily be done of one knows the latitude and longitude of the origin. But I don't know an easy closed-form equation to compute that, given the definition of the zone.
As an example, on an NGS alpha site, we find the definition of the future 2022 Vermont zone to be
Origin latitude = 42° 0' N
Origin longitude east = 287° 24' E [a different meaning of origin]
False easting = 685,800 m
False northing = 0 m
By trial and error I found this to to be within 100 m of N 41° 42' 11.4", W 80° 50' 8.5", that is, Morgan Township, Ohio. But it would be nice to have a formula for this rather than having to iterate.
But I don't know an easy closed-form equation to compute that, given the definition of the zone.
I'm not sure why the alpha version of NCAT isn't working for SPCS northing/easting input...
I get 41 42 13.25405" N, 80 50 09.81705" plugging it into TBC.
For the equations, see here:
https://www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf
See section 3.2 on page 32 (42 of the actual PDF document) for Transverse Mercator calculations. You're looking for section 3.24 for the inverse conversion computations.
@me
No NATRF/SPCS2022 projections available on Earthpoint...
When introducing the concept of state plane coordinates, it may be helpful to be able to say where the origin of the coordinates (that is, northing = 0, easting = 0) is by naming a town or city.
I remember asking this early on in my career and the answer I got was 'somewhere in the pacific ocean'. ♂️
It'd also be interesting to note the map scale factor there and compare it to the value in the middle of the zone. Might even be cool to plot the circle (ellipse?) on a projected map that shows all the 1.000000 scale factor locations.
Oregon and Washington origins for SPC are in the Pacific Ocean.
The 0,0 point of the Oregon North Zone is about 1400 miles west of Fort Bragg, CA
The origin of our state plane coordinate system is codified. Massachusetts General Laws, Chapter 97.
Section 12. For the purpose of more precisely defining said system:
1. The Massachusetts Coordinate System, NAD, 1983, Mainland Zone, Federal Zone Code M2001, is a Lambert conformal conic projection of the North American Datum of 1983, having standard parallels at north latitudes 41° (degrees) 43&sprime; (minutes), and 42° (degrees) 41&sprime; (minutes), along which parallels the scale shall be exact. The origin of coordinates is at the intersection of meridian 71° (degrees) 30&sprime; (minutes), west of Greenwich and the parallel 41° (degrees) 00&sprime; (minutes), north latitude. This origin shall have the coordinates: North (Y) = 750,000 meters and East (X) = 200,000 meters.
2. The Massachusetts Coordinate System, NAD 1983, Island Zone, Federal Zone Code I 2002 is a Lambert conformal conic projection of the North American Datum of 1983, having standard parallels at north latitudes 41° (degrees) 17&sprime; (minutes) and 41° (degrees) 29&sprime; (minutes), along which parallels the scale shall be exact. The origin of coordinates is at the intersection of the meridian, 70° (degrees) 30&sprime; (minutes), west of Greenwich and the parallel 41° (degrees) 00&sprime; (minutes), north latitude. This origin shall have the coordinates North (Y) = 0 meters and East (X) = 500,000 meters.
Historic boundaries and conservation efforts.
I'd also point out that the term "origin" as it applies to map projections is almost never the grid 0,0 point. (edit: @not-my-real-name beat me to it in the above post!)
I don't really care where the 0,0 point is. I scale from the origin only when required to, and even then only grudgingly.
It'd also be interesting to note the map scale factor there and compare it to the value in the middle of the zone. Might even be cool to plot the circle (ellipse?) on a projected map that shows all the 1.000000 scale factor locations.
For Transverse Mercator (and Oblique as well) the scale factor at the center is known.
For Lambert Conformal Conic, it takes a few equations but is pretty easy to figure out.
Plotting the line where grid scale factor is unity is helpful only up to a point, because ellipsoid heights will throw a wrench in there real quick.
If I absolutely have to work in ground and things still need to be on SPCS bearings, I will modify the existing SPCS scale factor and false northings/eastings so as to produce ground values for my project location (with significant truncation in N/E values of course). I set up my TBC project and Civil 3D systems to match, and then I can bring in other georeferenced data, look at my background maps, or continue to survey in that system, without having to worry about grid vs ground.
For LCC projections, it's necessary to reconfigure them to be a single-parallel definition before doing the modifications. But once you have that definition (or a TM definition), it's as simple as applying the specific project scale factor to the base projection scale factor, and then tweaking the false northings/eastings to suit.
In the case of the 2022 Vermont (4400) SPC, there won't be any points with a Scale Factor = 1. This is a result of the Scale Factor at the grid origin being set at 1.000015 which makes all scale factors greater than 1 and all distances on the Plane greater than those on the ellipsoid, bringing many of them closer to ground distances and reducing overall distortion.
In designing a TM projection, the Scale Factor at the origin is assigned first and the meridians with Scale Factor = 1, if there are any, fall wherever that Scale Factor dictates. NGS has adopted the same procedure for Lambert projections. Thus, whereas in NAD83 and NAD27 projections, the standard parallels were selected first and then the Scale Factor at the origin was determined by those parallels, the new scheme lets the standard parallels be determined by the Scale Factor selected for the origin. They aren't published because, without being fundamental constants, they are really just mathematical curiosities.
Every "single parallel" Lambert projection has two standard parallels. These two parallels, one north of the central parallel and one south of the central parallel, have equal scale factors. They may not be equal to 1, but they're equal to each other.
This migration is going to be interesting.
I misunderstood the OP.
In all my years of working with geographic coordinate system designs the only concern I have ever encountered with respect to the 0,0 location is to make sure it is SW of the zone. So I don't understand why it may be helpful to know where 0,0 is other than curiosity. It's certainly not a physical point. As has been noted once the parameters are in any decent software the 0, 0 location can be easily identified.
@me The people I talk to are not surveyors. They are members of search and rescue teams. They sometimes ask what the various numbers around the edges of maps mean. Being able to say "that's an X-Y coordinate system for Vermont; the 0, 0 point is in western New York to make sure all the values are positive" is helpful. Being able to name a specific place makes it less abstract so easier for a beginning student to relate to.
I know that search and rescue teams don't generally use state plane coordinates, they usually use UTM or a related system. But the numbers are on the edge of the map, so people ask about them.