# Coordinate systems and bearings

Ok so I've got a noodle scratcher and want some input before I open my mouth to my higher-ups. I recieved a new boundary description from a surveyor that uses metes and bounds in the form of bearings N XX°XX'XX" W going XX feet and so on. The coordinate system they used was a custom Traverse Mercator projection. However, my state, North Dakota, uses State Plane S Nad83 Int. Feet (a Lambart Conformal Conic projection) as its standard coordinate system; so I believe that all their records would be stored in that system.

My question is what, if any, difference will there be in the bearings if it were transformed into the State Plane system? Both systems use the same units, datum, and spheroid.

I understand their reasoning for using the custom coordinate system as they shifted the false easting and northing to reduce distortion in the location they preformed the survey and using a perfect square coordinate system without converging lines of longitude make sense at this scale. However, because all the records I see for the state never specify a coordinate system I think it is implied that the state's standard coordinate system is used. I don't want this data on file without the custom coordinate system attached and have someone in the future go out and use the description using the wrong coordinate system. Or is the discrepancy between the two, if any, so miniscule that I'm being needlessly nit-picky?

Grid convergence is the angle between true north and the grid north. It varies from point to point except in cylindrical projections, for which all meridians are parallel. For conic projections the meridians all radiate from a central point (off the map to the north in the Northern hemisphere), making it clear the convergence changes as you move in an east-west or north-south direction across the map. For transverse Mercator projections, the grid convergence changes in a more complex fashion, because the meridians are curves joining two fixed points (the projections of the poles).

Tissot indicatrices for ND SP S 83: the red lines point towards true North.

Within the southern part of North Dakota, the grid convergence for the State Plane projection ranges from 2.6 degrees in the southwest corner (meaning that a bearing towards true North would appear to be 2.6 degrees East of North on the map) to -2.9 degrees in the southeast corner. For the UTM projection, the convergence varies from 3.65 degrees to -1.75 degrees in the southeast corner. On average, the UTM convergence is approximately 1.1 degrees greater than the State Plane convergence.

Tissot indicatrices for UTM Zone 14 North: the red lines point towards true North. They tend to point about 1.1 degrees further towards the East (clockwise) than their correspondents in the ND SP S 83 map.

I cannot tell you whether this difference is "minuscule" or not, because that depends on your needs, but given that a 1.1 degree error translates to about 2% of the distance when locations are given as bearings and distances, this seems like an enormous discrepancy for surveying purposes.

• I just wish to add that, on cadastral plans (plats), differences between two "basis of bearings" (grid north), as they're often called, of only a few minutes is usually considered significant. So, yes, do attach appropriate meta data. Commented Nov 8, 2013 at 5:42
• And for future reference (and not wanting to denigrate GIS-SE in any way) a good source of expert advice is surveyorconnect.com Commented Nov 8, 2013 at 5:45
• @martin Thanks for the link. Of course we do not have a monopoly on all things GIS, so it's great to have resources like that to refer to. Commented Nov 8, 2013 at 15:45