Point of Beginning

State Plane Coordinates vs. Surface Coordinates, Part 2.

July 7, 2000
When the state plane coordinate system was established, the authors described the system in simple terms, easily understood by users. Figure 1 shows a two-dimensional coordinate system familiar to just about everybody. Today we would call this an x, y rectangular coordinate system or a two-dimensional right-handed Cartesian coordinate system. The authors of the state plane coordinate system called it a grid. The following quote from Coast and Geodetic Survey Special Publication No. 235, "The State Coordinate Systems," shows how they described it.

As with any plane-rectangular coordinate system, a projection employed in establishing a State coordinate system may be represented by two sets of parallel lines, intersecting at right angles. The network thus formed is termed a grid. One set of these lines is parallel to the plane of the meridian passing approximately through the center of the area shown on the grid, and the grid line corresponding to that meridian is the Axis of Y of the grid. It is also termed the central meridian of the grid. Forming right angles with the Axis of Y and to the south of the area shown on the grid is the Axis of X. The point of intersection of these axes is the origin of coordinates. The position of a point represented on the grid can be defined by stating two distances, termed coordinates. One of these distances, known as the x-coordinate, gives the position in an east-and-west direction. The other distance, known as the y-coordinate, gives the position in a north-and-south direction; this coordinate is always positive. The x-coordinates increase in size, numerically, from west to east; the y-coordinates increase in size from south to north. All x-coordinates in an area represented on a State grid are made positive by assigning the origin the coordinates: x = 0 plus a large constant. For any point, then, the x-coordinate equals the value of x adopted for the origin, plus or minus the distance (x') of the point east or west from the central meridian (Axis of Y); and the y-coordinate equals the perpendicular distance to the point from the Axis of X. The linear unit of the State coordinate systems is the foot of 12 inches defined by the equivalence: 1 international meter = 39.37 inches exactly.

The linear distance between two points on a State coordinate system, as obtained by computation or scaled from the grid, is termed the grid length of the line connecting those points. The angle between a line on the grid and the Axis of Y, reckoned clockwise from the south through 360°, is the grid azimuth of the line. The computations involved in obtaining a grid length and a grid azimuth from grid coordinates are performed by means of the formulas of plane trigonometry.

The state coordinate system was developed so there would be a direct relationship between geodetic coordinates, latitude and longitude, and grid coordinates, x and y. This is explained as follows:

For more than a century the United States Coast and Geodetic Survey has engaged in geodetic operations which determined the geodetic positions - the latitudes and longitudes -of thousands of monumented points distributed through the country. These latitudes and longitudes are on an ideal figure - a spheroid of reference which closely approaches the sea-level surface of the Earth. By mathematical processes, the positions of the grid lines of a State coordinate system are determined with respect to the meridians and parallels of the spheroid of reference. A point that is defined by stating its latitude and longitude on the spheroid of reference may also be defined by stating its x- and y-coordinate on a state grid. If either position is known, the other can be derived by formal mathematical computation. So too with lengths and azimuths: the geodetic length and azimuth between two positions can be transformed into a grid length and azimuth by mathematical operations. Or the process may be reversed when grid values are known and geodetic values are desired.

In general, any survey computations involving the use of geodetic position data can also be accomplished with the corresponding grid data; but with this difference: results obtained with geodetic data are exact, but they require the use of involved and tedious spherical formulas and of special tables. On the other hand, results obtained with grid data are not exact, since they involve certain allowances that must be made in the transfer of survey data from the curved surface of the Earth (spheroid) to the plane surface of a State coordinate system; but the computations with the grid data are quite simple, being made with the ordinary formulas of plane surveying; and with the State coordinate systems, exact correlation of grid values and grid values and geodetic values is readily obtained by simple mathematical procedures.

In modern geodesy the expression "ellipsoid of revolution" has replaced "spheroid." Notice the statements about the direct relationship between geodetic coordinates and state plane grid coordinates. That relationship doesn't exist if one uses surface coordinates.

Some people are confused when the expression "map projections" is used. The state coordinate systems put an ellipsoidal-shaped Earth on a flat plane at an accuracy acceptable for surveying, and in order to do this the U. S. Coast and Geodetic Survey selected map projections that cartographers use to put a round earth on flat paper.

By using a conformal map projection as the base for a state coordinate system and limiting one dimension of the area which is to be covered by a single grid, two things are accomplished [this is a repeat from Part 1, but worded differently].

On a conformal map projection, angles are preserved. This means that, at a given point, the difference between geodetic and grid azimuths of very short lines is a constant, and angles on the earth formed by such lines are truly represented on the map. For practical purposes of land surveying, this condition holds for distances up to about ten miles. For longer lines, the difference varies, and the correction to be applied to an observed (geodetic) angle to obtain a corresponding grid angle is the difference of the corrections to the azimuths of the lines, separately derived.

"The limitation in the width of the projection or grid permits a control of deviations of grid lengths from geodetic lengths. When the width of an area covered by a single grid is 158 statue miles, the extreme difference between geodetic and grid lengths will be 1/10,000 of the length of a line, which is quite satisfactory for most land surveys.

"Deviations of grid lengths from geodetic lengths will be a maximum along the margins of the longer dimension of the grid and midway between those margins. Along the margins, the grid length of a line will be greater than its geodetic length; along the center line, the geodetic length will be the greater. Between these limits are two lines along which grid and geodetic lengths are equal: these are lines of exact scale. The quantity by which a geodetic length is multiplied to obtain the corresponding grid length is termed a scale factor. It is greater than unity [one] outside the lines of exact scale; decreases to unity along those lines; and continues to decrease to a minimum about midway between them. The magnitude of the scale factor at any point depends upon the position of the point with respect to the lines of exact scale. It is constant along a line - any line - which is parallel with the lines of exact scale. These lines of equal scale correction are grid lines on the transverse Mercator grid, being parallel with the central meridian or Axis of Y; on the Lambert grid they are curved lines, being parallels of latitude. Scale factors [are simple to calculate]. For any given survey line, the scale factor may be readily ascertained and applied to the geodetic length of the line to obtain its grid length; or in an inverse operation, employed in obtaining a geodetic from a grid length. Where the exact length of a line is desired, it is thus easily obtained. It must be remembered that a geodetic length is a distance on the spheroid (sea level surface of the Earth), whose relationship to the corresponding ground-level distance may be expressed by very simple formulas and accurately illustrated by a geometrical figure. On the other hand, a grid length is a distance on a plane which is mathematically related to the spheroid, so that the relationship between corresponding lengths on the two surfaces can be expressed by mathematical formulas, but [cannot be graphically demonstrated]. The commonly used examples of a developed cone for the Lambert grid and a developed cylinder for the transverse Mercator grid are inexact and serve only as illustrations.

While a width of 158 statue miles was adopted as a standard in devising the State coordinate systems, departures from that width have been made where geographic conditions permitted or surveying requirements justified the change. If the width of a State is less than 158 miles, the width of the grid was decreased and the effect of the scale factor thereby also decreased. The narrower the strip on the Earth's surface which it is desired to portray on a plane, the smaller will be the distortion involved in the process. The north-south dimension of Connecticut is less than 80 miles: the maximum scale factor for the Connecticut coordinate system, (Figure 2 on p. 18) along the northern and southern boundaries of the State, expressed as a ratio, is around 1:40,000. Midway between the lines of exact scale it is 1:79,000. Where a state is too wide to be covered by a single grid, it is divided into belts, called zones, for each of which a separate grid is adopted. The boundary lines between zones follow county lines. The limiting scale factors for the various zones of a State coordinate system need not be the same. For example, the Illinois Coordinate System, (Figure 3 on p. 18) comprises two zones. The eastern zone, in which Chicago is located, has much smaller scale factors than the western zone. One thing sought in devising the State coordinate system was to keep the number of zones in any State to a minimum, consistent with the requirements of scale accuracy. For example, by allowing the scale ratio to go slightly above 1:10,000, the entire State of Texas was divided into five zones.

Long article. Because of the length, I eliminated at least two sketches that might have made the description clearer; these will be included in the next column. Remember, the state coordinate systems discussed refer to the NAD 27, the North American Datum of 1927. Changes were made for the North American Datum of 1983.