State Plane Coordinates
vs. Surface Coordinates, Part 3.
by James P. Reilly, Ph.D.
As I stated earlier in this series, state plane coordinates are based on conformal map projections. Since we’re surveyors, we can’t think of a map projection as used only for paper maps—but this concept may be hard for some people to grasp.
There are many definitions of map
projections. One reference states, a map projection is a systematic representation of all
or part of a surface of a round body, especially the earth, onto a plane (Snyder). Another
reference says, a projection is a means of transferring points on one surface to
corresponding points on another surface (Buckner). When surveying or mapping a large area,
a projection is required. No matter what projection is used, there will be distortions. If
the survey or map covers a small area—like a town—distortions may not be
visible, but they do exist. Determine what distortion is the least objectionable, and
select that projection for the survey or map.
With few exceptions, there are three developable surfaces which are the basis of most map projections: the cylinder, cone and plane. A developable surface can be “cut” and unrolled to form a plane. This is shown in Figure 1. For illustrative purposes, let’s describe these surfaces on a global basis.
Cylinder
Surface touches the equator throughout its circumference.
The meridians of longitude will be projected onto the cylinder as equidistant straight lines perpendicular to the equator.
The parallels of latitude are projected as lines parallel to the equator, and mathematically spaced for certain characteristics.
The Mercator Projection is the best known example, and its parallels must be mathematically spaced (see Figure 2).
Cone
If a cone is placed over the globe, with its peak along the polar axis of the
earth and with the surface of the cone touching the globe along some particular parallel
of latitude, a conic projection can be produced (see Figure 3).The meridians are projected onto the cone as equidistant straight lines radiating from the peak.
The parallels are projected as lines around the circumference of the cone in planes perpendicular to the earth’s polar axis, spaced for the desired characteristics.
Plane
A plane tangent to one of the earth’s poles is the basis for polar azimuthal projections. An azimuthal projection is one on which the directions or azimuths of all points are shown correctly with respect to the center.
The group of projections is named for the function, not the plane, since all tangent-plane projections on a sphere are azimuthal.
The meridians are projected as straight lines radiating from a point, but they are spaced at their true angles instead of the smaller angles of the conic projections. One example is shown in Figure 4.
The parallels of latitude are complete circles, centered on the pole.
We are interested in modifications to the characteristics described above.
The cylinder or cone may be secant to or cut the globe at two parallels instead of being tangent to just one. This provides two standard parallels.
The plane may cut through the globe at any parallel instead of touching a pole.
The axis of the cylinder or cone can have a direction different from that of the polar axis, while the plane may be tangent to a point other than a pole. This type of modification leads to important oblique, transverse and equatorial projections, in which most meridians and parallels are no longer straight lines or arcs of circles.
This will be the topic of the next column, because the modifications are used in the state plane coordinate system.
References:
1 “Map Projections—A Working Manual,” by John P. Snyder, U.S. Geological
Survey Professional Paper 1395, 1987.
2 State Plane Coordinates in Modern Surveying Practice, by R. B. Buckner, 1993.
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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.