State Plane Coordinates
vs. Surface Coordinates, Part 4.
by James P. Reilly, Ph.D.

The three basic projections, discussed in the last column, are shown in Figure 1. Projection surfaces used for the State Plane Coordinate systems are modifications, also discussed in the last column and shown in Figure 2. These are called secant projections: a secant cone in Lambert’s Projection and secant cylinder in Mercator’s Projection. In the Mercator Projection, the secant cylinder has been rotated 90o so the axis of the cylinder is perpendicular to the axis of rotation of the datum surface. Occasionally the cylinder is rotated into a predetermined azimuth, creating an oblique Mercator Projection; there is one state plane coordinate zone in Alaska that uses this concept.

Please note that these projection surfaces intersect the ellipsoid, not the earth’s surface. The secant cone intersects the surface of the ellipsoid along two parallels of latitude called standard parallels. Specifying these two parallels defines the cone; specifying a central meridian orients the cone with respect to the ellipsoid. The transverse secant cylinder intersects the surface of the ellipsoid along two small ellipses equidistant from the meridian through the center of the zone. The secant cylinder is defined by specifying this central meridian plus a desired grid scale factor on the central meridian. The ellipses of intersection are standard lines; their location is a function of the central meridian scale factor.

The specification of the latitude and longitude of the grid origin and grid values assigned to that origin is needed to uniquely define a zone of either the Lambert or transverse Mercator Projection. Figure 3, taken from State Plane Coordinate System of 1983 by James E. Stem, shows how the Lambert and Transverse Mercator Systems are defined.

Before we get into defining zones and zone constants, let’s look again at Figure 2 and ask, “When does one use the Lambert Conformal Conic Projection?” and “When does one use the transverse Mercator Projection?” (Note: Although the word “conformal” is not used in naming the transverse Mercator Projection, the projection is conformal). The Lambert Projection provides the closest approximation to the datum surface for a rectangular zone longest in the east-west direction. The transverse Mercator Projection provides the closest approximation to a rectangular zone longest in a north-south direction. The narrower the strip of the earth’s surface desired to be portrayed onto a plane, the smaller the scale distortion on the projection. As mentioned in an earlier column, “when the width of an area covered by a single grid is 158 statute miles, the extreme differences between geodetic and grid length will be 1/10,000 of the length of a line.” For a state like Connecticut that is somewhat longer in the east-west direction, the Lambert Projection is ideal. The north-south distance across Connecticut is less than 158 statute miles; one zone can and does cover the entire state. New Hampshire, New Jersey and Rhode Island are somewhat longer in the north-south direction; all three states use the transverse Mercator Projection and, as with Connecticut, one zone covers each state.

What about the larger states? If a state is large, it doesn’t matter which of the two projections is used; you just have to divide the state into two or more zones. I’m sure a lot of thought was given to the selection of projection and number of zones for each state. Even though California is much longer in the north-south direction, the non-rectangular shape made it more practical to use the Lambert Projection with seven zones. Table 1, a large table for the State Plane Coordinate System of 1927, summarizes everything we have discussed up to this point in time. For each state it identifies the projection(s) used, names the zones, gives the latitude and longitude and scale factor selected for the central meridian or parallels, and gives the latitude, longitude and x and y coordinates selected for the origin. The origin of every zone was far enough south so that all rectangular y-coordinates will be positive numbers. With few exceptions, the x-coordinate of the zone central meridian was 500,000 feet or 2,000,000 feet.

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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.