Occasionally, I have a need to convert an old magnetic bearing to a NC State Plane Coordinate System NAD83-2011/2010.00 (NC-Grid) bearing. I struggle to understand magnetic to true north conversions when there's an overlap between the declination and the bearing.
In my scenario today, I needed to recreate a boundary line in an extremely poor area where next to no actual surveys exist. There was no evidence of possession. Parol evidence suggest the small one-acre parcel was never occupied. I have no better evidence than a 1947 magnetic bearing of N 2deg 30' E to establish the eastern boundary (historic aerials were reviewed). I'm going to hold that bearing and fix it to an old gum tree that I'm hard pressed to prove isn't the tree called out in the creation deed. Now I need to convert the 1947 magnetic bearing to a NC-Grid bearing so it can be set to a relative precision of 0.12' (Class B NC) with base/rover RTK. I'm not confident I did it correctly but I'd like to fill this gap in my knowledge.
I put the attached notes together in an effort to save myself some time the next time this problem presents itself. If anyone is willing, I'd appreciate a check on my calculations and would love a more concise explanation or graphic to use for future calcs.
Thank you,
Murphy
I look forward to learning from the answers you receive. This is a weak subject for me, as well.
I don't know what GPS program you have, but Trimble will give you the Grid to Geodetic north conversion. Then you can get the mag and historic mag rotation from the NGS.
Doesn't sound like you have a plat, but many of them will mention the mag rotation.
@murphy I cant download the pdf for some reason.
I use TBC Enterprise and didn't think of checking for that option. I'll take a look, thank you.
I removed a graphic, hopefully the attached pdf can now be downloaded.
Back to the OP. First, you need to determine the magnetic declination for you project area in 1947, the NGS web tools are you friend there. Next, applying the declination get you to a geodetic/astronomic azimuth. Finally, you then need to apply the projection convergence angle to determine your grid bearing.
@murphy I don't think it is your file, I think something is up with the site.
Magnetic bearing + magnetic declination = True North (Astronomic North):
Easterly Declination is negative a value; Westerly Declination is a positive a value.
Astronomic North + LaPlace correction = Geodetic North: LaPlace correction is usually small.
Geodetic North - Grid convergence angle (Mapping Angle) + 2nd term = Grid Bearing
Biggest impact of LaPlace correction is in mountainous areas.
2nd term correction is usually only applied in high order surveys.
Example:
Magnetic Bearing = N 2° 30' E
Location: Somewhere in Raleigh, North Carolina
Lat = 35° 46' 52", Long = 78° 38' 30"
Convergence Angle = 0° 12' 24.6": Angle is positive since longitude is east of the central meridian (79°) for the zone
Magnetic declination on January 1, 1947 = 3° 40' west
Astronomic North (Geodetic North) = N 2° 30' E + 3° 40' = N 6° 10' E: LaPlace correction would need to be applied for an accurate Geodetic bearing.
Grid North = N 6° 10' E - 0° 12' 24.6" = 5° 57' 35.4": 2nd term not applied
If the magnetic bearing has already been corrected for declination, then you only need to apply the grid convergence angle.
I'm not sure why the pdf won't view, below is the text in an unfriendly format:
Problem: need to convert a N 2° 30’ E 1947 Magnetic North bearing to NC-Grid (State Plane
NAD83-2011). Solution is as follows:
1. Find 1947 Declination
a. use NGS’s U.S. Historic Declination calculator and set location to nearest
town, Roseboro, NC (See Figure 1) and calculate. Note Longitude
b. 1947 magnetic declination in 1947 in Roseboro, NC = 3° 32’ West of Tn
2. Convert magnetic bearing to True North. Declination is west of Tn (True North)
making it a negative so using the formula on page one (Tn = Mn + Md) the conversion
would be: N 2° 30’ E - 3° 32’ = N 1° 02’ W = Tn bearing for the 1947 N 2° 30’ E
magnetic bearing.
3. Using TBC software or NGS’s NCAT, find the convergence angle.
a. TBC shows it as 18’ 51”, NCAT shows it as 16’ 54”. This example uses TBC’s
value 18’ 51”.
4. Find the central meridian of your state plane coordinate zone. NC has a single zone
and the central meridian is 79° West Longitude. At this longitude grid north and true
or geodetic north are coincident.
a. Because the site longitude 78° 30’ 42” W Longitude it is east of the central
meridian. If east of the CM subtract convergence angle from true north to get
grid. If west of the CM, add convergence angle.
5. Calculate NC-Grid bearing: Grid Bearing = True North Bearing – Convergence
a. Grid Bearing = N 1° 02’ W - 18’ 51” = N 0° 43’ 09” W.
6. See the AI dialogue that follows. Gemini and Grok both made mistakes by not
recognizing the relationship between quadrant bearings and azimuths. It is very easy
to add or subtract incorrectly.
AI Dialogue:
Below is a concise, technical explanation you can reuse to show exactly where Gemini
AI went wrong, using your specific numbers.
The core pitfall: mixing quadrant bearings with azimuth math
Your correct intermediate result
After applying 1947 declination:
• True bearing: N 1°02′00″ W
This is a quadrant bearing, which carries direction by letters, not by sign.
What Gemini did wrong
Gemini implicitly treated:
N 1°02′00″ W
as if it were the signed value
−1°02′00″
That is an azimuth-style signed angle, not a quadrant bearing.
Once that silent conversion happened, Gemini then subtracted convergence
algebraically:
−1°02′00″ − (+0°18′51″) = − 1°20′51″
and converted back to a bearing as:
N 1°20′51″ W
Why that result is wrong
1. Quadrant bearings are not signed angles
• N 1°02′ W means “1°02′ west of north”
• The “west” is already encoded
• Writing −1°02′ double-counts direction unless everything is converted to a signed
coordinate system
Gemini mixed:
• quadrant bearing input
• signed-angle arithmetic
That is mathematically invalid.
2. The geometry contradicts Gemini’s result
At your site in Roseboro, NC:
• Grid north is 18′51″ east of true north
• Your line is west of true north
When the reference north rotates toward the line, the line must appear closer to north,
not farther away.
So the grid bearing must be smaller than 1°02′, not larger.
Gemini’s answer (1°20′51″ W) violates this physical geometry.
Correct handling (what you did)
Staying consistently in quadrant bearings:
• True: N 1°02′00″ W
• Convergence: 0°18′51″ E
1°02′00″ − 0°18′51″ = 0°43′09″
NC-Grid (NAD83-2011) bearing:
N 0°43′09″ W
One-sentence takeaway (use this verbatim if you like)
Gemini’s error was treating a quadrant bearing as a signed azimuth and then
subtracting convergence, which double-counted direction and produced a
geometrically impossible result.
This is the classic bearings vs. azimuths trap—and you caught it correctly.
Astronomic North (Geodetic North) = N 2° 30' E + 3° 40' = N 6° 10' E
This is my main holdup and I can't conceptualize it well. I calculated N 6° 10' E my first go around but when I draw it on a compass rose, it doesn't make sense. If you draw a rose with Mn at 0°, that puts Tn at N 3° 32’ E (Roseboro, NC declination is 3° 32’,West of Tn), the Mn bearing of N 2° 30' E would fall between lines representing Mn and Tn and the result would seem be N 2° 30’ E_Mnorth - 3° 32’_Mdeclination = N 1° 02’ W_Tn.
It is way easier to visualize this in azimuths. The formula for magnetic to grid north is azimuth + declination + LaPlace + convergence angle. LaPlace is relatively small and probably lost in the accuracy of a compass. Both declination and convergence are positive east and negative west. In your case the sum of the declination and convergence is roughly 3d58'51" plus the LaPlace if you choose. You just need to add that value to any magnetic azimuth, no pesky quadrants to worry about.
Maybe I'm misunderstanding declination. NGS lists it as 3° 32’ West, so wouldn't that be a negative declination making it -3° 32’ declination + 18’ 51” convergence = -3° 13' 9"?
@john-putnam had it close to right, but Convergence Angle should be subtracted, not added. Part of the confusion stems from trying to remember add or subtract instead of working with signed numbers. Both convergence angles and declinations can be positive or negative and it's much easier to remember a formula and plug in the correct signed number than it is to constantly ask, "Should I add or subtract?"
Here are the two formulas to remember:
(1) True Azimuth = Magnetic Azimuth + Declination
(2) True Azimuth = Grid Azimuth + Convergence Angle
Things equal to the same thing are equal to each other, so
(3) Grid Azimuth + Convergence Angle = Magnetic Azimuth + Declination, and
(4) Grid Azimuth = Magnetic Azimuth + Declination - Convergence Angle
John is absolutely correct in saying that using azimuths instead of bearings makes this much easier.
Magnetic bearing = N 2deg 30min E = Azimuth 2deg 30min
Declination = 3deg 32min W = Azimuth 356deg 28min
Convergence angle = 18min 51sec
Grid Azimuth = 2deg 30min + 356deg 28min - 18min 51sec
Grid Azimuth = 358deg 39min 09sec = N 1deg 20min 51sec W.
The problem with @murphy 's calculation is that his N 1deg 02sec W has to be treated as a negative number, which is one problem encountered when using bearings; they're unsigned when sometimes signed numbers are needed.
The AI correction to Gemini is laughable because N 1deg 02sec W does not "encode" anything arithmetically, it's just textual description. Changing that to a negative so that it can be used in a computation is exactly what should be done. In Ai speak, this is a hallucination.
Thank you. That makes sense and seems to align more with some references I had from NGS. Key points: convert bearings to azimuths, subtract convergence angle if site is east of central meridian and add if west of central meridian.
Just be careful with those rules for adding and subtracting. Convergence angles and declinations are signed numbers for a reason and formulas take advantage of that.
In the formula True Azimuth = Grid Azimuth + Convergence Angle, suppose the Grid Azimuth is -15 minutes. Then
True Azimuth = Grid Azimuth +(-15 minutes) = Grid Azimuth - 15 minutes.
If we know the True Azimuth and need the Grid Azimuth, then Grid Azimuth = True Azimuth - Convergence Angle.
Grid Azimuth = True Azimuth - (-15 minutes) = True Azimuth + 15 minutes.
We didn't have to concern ourselves with east or west of anything nor whether we were converting from or to True Azimuth. The formulas remain constant, the Convergence Angle can be either positive or negative and arithmetic rules take care of the rest.
No thinking errors and no worries.
As you move east from the Central Meridian, true north drifts to the left. Point your instrument to 0 grid, and the north pole will be to the left. If you're far enough east of the CM to make 15' of convergence then grid north of 0 is 0-15-00 true north. This means to convert from grid to true 15 minutes needs to be added to grid. Reverse it when west of the Central Meridian. Grid north of 0 is true north of 359d45', because true north is drifting to the right.
Well, yes. But, if you're east of the CM, convergence Angle will be positive, if you're west of the CM, It will be negative.
Using the formula True North = Grid North + Convergence, along with the signed number that represents Convergence, eliminates the continual analysis.
West of the CM with convergence Angle = -15 minutes:
True North = Grid North + (-15 minutes) = Grid North - 15 minutes.
For the Lambert projection that NC uses for state plane,
Convergence Angle = (Lon of CM - Lon of Point) * sin(Central Parallel), angles in radians.
That formulation forces points West of the CM to have negative Convergence Angles.
The signs are your friend. Using them helps to reduce blunders.
Formulas work well for those who find it easier to memorize them than to remember or understand the concept. I personally would much rather understand the concept and figure out the answer than memorize formulas to figure out the answer.