State Plane Coordinates
vs. Surface Coordinates, Part 9.
by James P. Reilly, Ph.D.

This is the final article of this series. I know not everything was covered, but it was a good review. I’m interested in comments because we may need to have an additional article or two blended in with the GPS topics that will begin with the next column.

To summarize, we converted the latitude and longitude coordinates, determined by GPS, of stations BROMILOW, REILLY and WAKEMAN to state plane coordinates in the New Mexico Central zone, using program SPCS83.

As can be seen from previous articles, it’s a trivial task to work in the state plane coordinate system. All slope distances are reduced to horizontal distances, and these are multiplied by a combined factor to get grid distances. This is the only thing different from working in a plane surface coordinate system. There is one exception: If the traverse lines are long, a correction must be made to the azimuth. Before I define ‘long line,’ let’s look at Figure 1 (Figure 2.5 from NOAA Manual NOS NGS 5, State Plane Coordinate System of 1983 by James E. Stem).

t=a—g+d

Figure 1. Azimuths.

In Figure 1, the grid azimuths we calculated in our small traverse are labeled t. The following is a quote from the NOAA manual:

The projection of the geodetic azimuth a onto the grid is not the grid azimuth, but the projected geodetic azimuth symbolized as T. Convergence y is defined as the difference between the geodetic and projected geodetic azimuths. Hence by definition, a = T + y, and the sign of y should be applied accordingly. The angle obtained from two projected geodetic azimuths is a true representation of an observed angle.

    When an azimuth is computed from two plane coordinate pairs, the resulting quantity is the grid azimuth symbolized as t. The relationship between projected geodetic azimuth T and grid azimuth t is subtle and may be more clearly understood in [Figure 1]. The difference between these azimuths is a computable quantity symbolized as d or more often as (t-T). For the purpose of sign convention it is defined as d= t-T. For reasons apparent in [Figure 1], this term is also identified as the ‘arc-to-chord’ correction.

Only in precise traverses is the (t-T) correction significant. An approximate equation is: (t-T)=25.4(DN)(DE)10-10 where DN and DE are in meters. To solve this equation you will need more information. For the Lambert projection, DN is the distance from the central axis of the zone, and DE is the difference of eastings of the endpoints of the line. For the Transverse Mercator projection, DE is the distance of the central axis of the projection, and DN is the difference of northings of endpoints of the line. If you would like to see an example of a traverse with long lines showing these calculations, let me know.

Referring again to Figure 1, the angle y is the convergence angle (also called the mapping angle). As can be seen, adding y to t gives the geodetic azimuth. Referring to the last two columns, y (called convergence in the articles) is calculated every time the program SPCS83 converts from geodetic coordinates to state plane coordinates and from state plane coordinates to geodetic coordinates. If, after completing a grid traverse, you want the geodetic azimuth, you add (algebraically) the convergence angle to the grid azimuth. This is necessary because the meridians converge toward the poles while the north-south grid lines are parallel to the central meridian. On the central meridian of each zone, the convergence is zero.

Surface Coordinates
It was common to take each line on the grid and divide the length by the combined factor. This gave the horizontal distance on the earth’s surface. Doing this allowed conventional surveying crews to retrace grid lines. However, the use of computers in all surveying offices has changed that. Today, surveyors want to convert the state plane coordinates to surface coordinates (sometimes called project coordinates). I can tell you that it’s bad practice, but let’s show how it is done.

There can be only one combined factor for a project if working in surface coordinates. On anything other than short traverses, the scale factor changes for each line, and the elevation factor may change. That’s why state plane coordinates work.

So that there is no confusion, it’s a common practice to drop the first digit or two from the station’s state plane coordinate. In our case, the state plane coordinates of the three known stations were as follows:

A person wanting surface coordinates might do the following:

(But in most cases the coordinates would be in feet.)

Let me tell of two different situations where surface coordinates were used and problems detected. Two civil engineering firms were given five miles of highway design by a state department of transportation; the projects were end-to-end. Both firms used surface coordinates. When the first firm finished its work, its ending coordinates did not match the beginning coordinates of the second firm’s project.

In another incident, I received an E-mail asking if I could help a surveyor having problems with his state plane coordinates. When I called him, he said the problem was not the state plane coordinates, but his surface coordinates, which were only good to about 10 miles. He was looking for a way to extend beyond 10 miles.

I need to talk to more people who use surface coordinates. With only one combined factor per project, this can create havoc. I hope you can see from this series of articles that state plane coordinates are the way to go.

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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.