### 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 N_{o} 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 N_{o}= 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 r_{o} which, like N_{o}, 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, r_{o}= 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 N_{o}, 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: