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3. Grid Direction

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

  • where the two meridians are with respect to each other, and
  • if the arc-to-chord correction is significant

a. Meridians

Convergence is the angle from Geodetic N to Grid N. Jerry's UTM convergence is -1°51'37.6". Because it is negative, Geodetic N is east of Grid N, Figure L-4.

Figure L-4
Geodetic and Grid North

We can also determine the relationships by comparing longitudes. Jerry's coordinates are in UTM Zone 16 whose CM is at 87° W longitude. Jerry is west of the CM at 89°43'53.76403" W longitude, Figure L-5, placing Geodetic N east of Grid N

Figure L-5

b. Arc-to-Chord

The arc-to-chord correction for line Jerry-A1 can be computed using either Equation H-11 or H-12. We'll use both to see if there's a difference.

Equation H-11
Equation H-12

Both equations require Eo which is a defining zone parameter: it is the east coordinate of the CM (false easting). Eo is 500,000.00 m for all UTM zones.

(1) Using Equation H-11

Equation H-11 is set up for metric coordinates so we don't need to convert any points. Set up in terms of point identifiers:

Coordinate differences:

Substituting into the correction equation:

(2) Using Equation H-12

Equation H-12 in terms of point identifiers is:

This equation requires ro which 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 UTM Zone 16, ro = 6,354,209.61 m

Substituting coordinate differences into the correction equation:

(3) Application

As expected (with fingers crossed) both equations yield the same result.

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

Another way to determine the correction's behavior is by examining the line's position relative to the CM. In a transverse cylindric projection the projected line is concave toward the CM.  Line Jerry-A1 is west of the CM so it is concave to the east, Figure L-6.

Figure K-6
Line Jerry-A1 Concavity

Putting the pieces together, Figure K-7, it's apparent that the correction's value must be added in Equation H-7.

Figure K-7
Grid Azimuth

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