### 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

Longitudes

#### 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 E_{o} which is a defining zone parameter: it is the east coordinate of the CM (false easting). E_{o} 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 r_{o} 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, r_{o}= 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

LineJerry-A1Concavity

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: