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3. Grid Direction

To convert the geodetic azimuth of line Jerry-A1 to grid, we need to know:

  • where the two meridians are with respect to each other, and
  • if the arc-to-chord correction should be applied

a. Meridians

Convergence is the angle from Geodetic N to Grid N. Jerry's SPC convergence is +0°11'03.9". Because it is positive, Geodetic N is west of Grid N, Figure K-4.

 

Figure K-4
Geodetic and Grid North


Another way to determine the meridians' relationship is by comparing longitudes of Jerry and the CM. This requires knowledge of the zone parameters. The CMs of all three Wisconsin SPC zones are at 90°00'00"W longitude. Jerry's longitude is 89°43'53.76413"W which places it east of the CM. Because Jerry is east of the CM, Geodetic N west of Grid N, Figure K-5.
  

Figure K-5
Longitudes

 

b. Arc-to-Chord

The arc-to-chord correction for line Jerry-A1 can be computed using either Equation H-9 or H-10. We'll use both to see if there's a difference.

Equation H-9
Equation H-10


Both equations require No which is the north coordinate of the CP at the CM. Even though a constant, it is not given as a zone parameter and must be computed using the projection equations. NCAT does not provide this information but the NAD 83 Coordinate Conversion workbook does.

Wis South SPC Zone No = 510,708.62 ft = 155,664.30 m
(1) Equation H-9

Because Equation H-9 is set up for metric coordinates,we'll need to convert point A1.

Equation H-9 in terms of point identifiers:

Coordinate differences:

Substituting into the correction equation:

(2) Equation H-10

Equation H-10 in terms of point identifiers is:

This coordinates may be metric or Imperial. We'll stick to feet.

This equation requires ro which, like No, is constant for for a zone but not given as a defining parameter. NCAT does not provide this information but the NAD 83 Coordinate Conversion workbook does.

For Wis SPC South zone, ro = 20,920,156.06 ft.

Coordinate differences:

Substituting into the correction equation:

(3) Application

Both equations yield the same result (surprise, surprise).

Is the arc-to-chord correction significant? That's up to the surveyor and the type of project.  Even though it's less than a second, we'll apply it as it may affect rounding.

To compute the Grid Azimuth from Geodetic, convergence, and arc-to-chord correction, use Equation H-7.

Equation H-7


A sketch can also be used to determine how to apply the correction. Because the North coordinates of Jerry and A1 are less than No, the line is south of the CP which makes it concave northerly toward the CP, Figure K-6.

Figure K-6
Line Jerry-A1 Concavity


The arc-to-chord correction computed with Equations H-9 and H-10 is negative making it an angle to to the left which Figure K-6 supports. Figure K-7 shows all the elements to convert the Geodetic Azimuth to Grid.

Figure K-7
Grid Azimuth


Using Jerry's convergence and a negative arc-to-chord correction, the Grid Az of Jerry-A1 is: