
Occupy Reilly with a total station and backsight to Bromilow. State plane coordinates are known for both stations; the grid azimuth from Reilly to Bromilow can be calculated.

Establish a temporary station called Temp in a position that's visible from both Reilly and Wakeman.

With the total station at Reilly, backsight to Bromilow, foresight to Temp and record horizontal angle, zenith angle and slope distance to Temp. Observe the horizontal and zenith angles in both direct and reverse positions.

Move the total station to Temp, backsight to Reilly, foresight to Wakeman and record horizontal angle, zenith angle and slope distance. As before, observe all angles direct and reverse.
Station Name  Latitude (North)  Longitude (West) 
Bromilow  32 16 52.33969  106 45 15.77636 
Reilly  32 16 55.93458  106 45 15.16429 
Wakeman  32 17 0.10142  106 45 29.49809 

Calculate the grid azimuth from Reilly to Bromilow using state plane coordinates determined earlier:
Name  Northing (meters) 
Easting (meters) 
(2) Bromilow  142158.262  452489.852 
(1) Reilly  142268.912  452506.387 
1  DN(21) = 110.650m  DE(21) = 16.535m 
Figure 2, below shows the quadrant of the azimuth from Reilly to Bromilow. As can be seen, the azimuth from is greater than 180Â° but less than 270Â°. The equation for azimuth is
a = tan^{1} (E2  E1) = tan^{1} DE(N_{2}  N_{1})DN
Solving this equation gives
a = 8Â° 29' 56.8"
We know the azimuth is in the third quadrant (greater than 180Â°), so we must add 180Â° to a to get grid azimuth.
Azimuth Reilly to Bromilow = 188Â° 29' 56.8"
= 188Â° 29' 57"

Calculate the azimuth of other lines in the traverse.
First, let me give the horizontal angles (angles to the right) from Reilly to Temp and from Temp to Wakeman.
Horiz. angle at Reilly from Bromilow to Temp = 68Â° 02' 24"
Horiz. angle at Temp from Reilly to Wakeman = 271Â° 15' 42"
Referring to sketch in Figure 1,
Az Reilly to BromilowÂ Â Â Â Â Â Â 188Â° 29' 57"
+ angle right at ReillyÂ Â Â Â +Â Â Â Â 68Â° 02' 24"
Â Â Â Â Â Â Â
Az Reilly to TempÂ Â Â Â Â Â Â 256Â° 32' 21"
Â Â Â  180Â°
Az Temp to ReillyÂ Â Â Â Â Â Â 76Â° 32' 21"
+ angle right at Temp +Â Â Â Â 271Â° 15' 42"
Â Â Â Â Â Â Â
Az Temp to WakemanÂ Â Â Â Â Â Â 347Â° 48' 03" 
We need to calculate horizontal distances from Reilly to Temp and from Temp to Wakeman.
Reilly to Temp
Zenith angle = 91Â° 11' 36"
Slope distance = 1111.45 ft (338.697m)
Horizontal distance
= slope distance x sin (zenith angle)
Horiz. dist Reilly to Temp = (1111.45 ft) sin (91Â° 11' 36")
= 1111.21 ft (338.697m)
Temp to Wakeman
Zenith angle = 89Â° 55' 45"
Slope distance = 701.75 ft (213.894m)
Horiz. dist Temp to Wakeman = (701.75) sin (89Â° 55' 45")
= 701.75 ft (213.894m)
Note: State plane coordinates are horizontal coordinates; height calculations are not needed.

Reduce horizontal distances to grid distances.
This is a twostep process. First, horizontal distances at the surface must be reduced to the ellipsoid (geodetic distances) by using the elevation factor, then reduced to the grid (grid distances) by using the scale factor.
I'm going to borrow two figures from the NGS state plane coordinate manual1 (Figures 4.1b and 4.1c on pages 47 and 48). These will be combined as Figure 3 in this column. As can be seen from the equation in the figure,
S = D ( R/ (R + N + H) )
WhereÂ Â Â Â S = Geodetic Distance
Â Â Â D = Horizontal Distance
Â Â Â H = Mean Elevation
Â Â Â N = Mean Geoid Height
Â Â Â R = Mean Radius of Earth
NGS recommends using R = 6,372,000m. In Las Cruces, N.M.,
N = 25m and H = 1188.720m (3,900 ft).
Putting these numbers into the above equation gives
Geodetic Distance = Horizontal Distance (0.99982)
Elevation factor = 0.99982
The scale factors are given in Figure 4 of this column. We can average the scale factor. Since the traverse is short, let's use 0.9999278 = 0.99993. Multiplying the scale factor times the elevation factor gives the "combined factor."
Combined Factor = (0.99982) (0.99993) = 0.99975
Because state plane coordinates are given in meters, I'm going to use the horizontal distances in meters to calculate grid distances, also in meters.
Grid Distance =
Horizontal Distance x Combined Factor.
Grid Distance Reilly to Temp = (338.697m) (0.99975)
= 338.612m
Grid Distance Temp to Wakeman = (213.894m) (0.99975)
= 213.840m
STATION NAME 
LATITUDE (NORTH) 
LONGITUDE (WEST) 
NORTHING (Y) METER 
EASTING (X) METER 
ZONE  CONVERGENCE  SCALE FACTOR 
ELEV (M) 
GEOID HT (M) 

D  M  S  
Bromilow  32 16 52.33969  106 45 15.77636  142158.262  452489.852  NM C  0  16  9.78  0.99992783  1 
1 Â 
Reilly  32 16 55.93458  106 45 15.16429  142268.912  452506.387  NM C  0  16  9.48  0.99992781  1  1 
Wakeman  32 17 0.10142  106 45 29.49809  142399.023  452131.948  NM C  0  16  17.17  0.99992825  1  1 

Calculate state plane coordinates of stations Temp and Wakeman.
The following equations are needed to calculate coordinates of station Temp:
E Temp = E Reilly + (Grid Distance Reilly to Temp ) sin (Azimuth Reilly to Temp )
N Temp = N Reilly + (Grid Distance Reilly to Temp ) cos (Azimuth Reilly to Temp )
E Temp = 452506.387m + (338.612m) sin (256Â° 32' 21") = 452177.077m
N Temp = 142268.912m + (338.612m) cos (256Â° 32' 21") = 142190.090m
Repeating the process going from Temp to Wakeman,
E Wakeman = 452177.077m + (213.840m) sin (347Â° 48' 03") = 452131.890m
N Wakeman = 142190.090m + (213.840m) cos (347Â° 48' 03") = 142399.101m
How close did I get to the published coordinates?
Using a COGO package, you may find some minor errors in my calculations; I used an HP 48 calculator (Hewlett Packard, Palo Alto, Calif.).
1  Northing (meters)  Easting (meters) 
Published Values  142399.023  452131.948 
Calculated Values  142399.101  452131.890 
1  0.078m  0.058m 
Now I have to tell what instrument I used. When I went to my office to get the instrument for this survey, only an old manual total station was available. Even with that, the error of closure is about 1:5000. With modern instruments available in your company, you should get much better results. The point of this series of articles was to show how easy and convenient it is to work with state plane coordinates.
Long article, and I havenÂ¿t started on surface coordinates. I will have to continue this in the next column to make a fair comparison of state plane coordinates vs. surface coordinates. I'll also explain convergence of the meridian.