State Plane Coordinates
vs. Surface Coordinates, Part 8.
by James P. Reilly, Ph.D.

In the last column I gave a sketch of the sample problem, (Figure 2, page 18, Oct. POB), that will be used to demonstrate traversing using state plane coordinates. There is a change, as shown in Figure 1 of this column, because now a building is under construction that obstructs Wakeman from being observed from Reilly. Do the following:

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

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

3. With the total station at Reilly, back-sight to Bromilow, fore-sight 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.

4. Move the total station to Temp, back-sight to Reilly, fore-sight to Wakeman and record horizontal angle, zenith angle and slope distance. As before, observe all angles direct and reverse.

Figure 1. Sketch of sample problem.

As we said in earlier columns, state plane coordinates are based on conformal map projections. This means all horizontal angles as measured are correct, but all measured distances must be reduced to grid distances. Since the observed angles are correct, let’s start by calculating the grid azimuths from Reilly to Temp and from Temp to Wakeman.

1. Calculate the grid azimuth from Reilly to Bromilow using state plane coordinates determined earlier:

Figure 2, below shows the quadrant of the azimuth from Reilly to Bromilow. As can be seen, the azimuth from is greater than 180° but less than 270°. The equation for azimuth is

a = tan¯¹ (E2 - E1) = tan¯¹ DE
               
(N2 - N1) DN

Solving this equation gives

a = 8° 29' 56.8"

We know the azimuth is in the third quadrant (greater than 180°), so we must add 180° to a to get grid azimuth.

Azimuth Reilly to Bromilow = 188° 29' 56.8"
= 188° 29' 57"

Figure 2. Relative location of Bromilow from Reilly.

2. 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 = 68° 02' 24"
   Horiz. angle at Temp from Reilly to Wakeman = 271° 15' 42"
   
   Referring to sketch in Figure 1,
   
   Az Reilly to Bromilow        188° 29' 57"
   + angle right at Reilly     +     68° 02' 24"
          
   Az Reilly to Temp        256° 32' 21"
       - 180°
          
   Az Temp to Reilly        76° 32' 21"
   + angle right at Temp +     271° 15' 42"
          
   Az Temp to Wakeman        347° 48' 03"

3. We need to calculate horizontal distances from Reilly to Temp and from Temp to Wakeman.
   
   Reilly to Temp
   Zenith angle = 91° 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 (91° 11' 36")
   = 1111.21 ft (338.697m)
   
   Temp to Wakeman
   Zenith angle = 89° 55' 45"
   Slope distance = 701.75 ft (213.894m)
   Horiz. dist Temp to Wakeman = (701.75) sin (89° 55' 45")
   = 701.75 ft (213.894m)
   Note: State plane coordinates are horizontal coordinates; height calculations are not needed.

Figure 3. Reduction to the ellipsoid.

4. Reduce horizontal distances to grid distances.
   This is a two-step 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) )
   Where     S = Geodetic Distance
       D = Horizontal Distance
       H = Mean Elevation
       N = Mean Geoid Height
       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

4. 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 (256° 32' 21") = 452177.077m
   
   N Temp = 142268.912m + (338.612m) cos (256° 32' 21") = 142190.090m
   
   Repeating the process going from Temp to Wakeman,
   
   E Wakeman = 452177.077m + (213.840m) sin (347° 48' 03") = 452131.890m
   
   N Wakeman = 142190.090m + (213.840m) cos (347° 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.).

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 haven’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.

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James P. Reilly serves as head of the Department of Surveying at New Mexico State University, College of Engineering, in Las Cruces.