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FIGURE 5.4 Concepts underlying creation of a State Plane Coordinate System grid from conic projection.

Angles are correctly represented on this projection. The scale is exact only along the standard parallels, and is continuously changing along the meridians. Since more than one cone may be fitted to the ellipsoid to create projection zones, the projection concerned is also known as the Lambert conformal polyconic projection.

With respect to projections, the term conformal has a special meaning. Since it is impossible to develop a spherical surface onto a plane, all maps will contain distortions according to the projection used in their compilation. The distortions may relate to scale; area; angles; or to the shape of figures. The unique properties of a conformal projection include the following:

• All figures on the surface of the earth retain their shape on the map.

• Angles measured on the map approximate their true values on the surface of the earth.

• The map scale at any point is uniform in all directions.

These properties are important to all who use grid coordinates in their work, including surveyors, engineers, and planners. The Lambert conformal conic projection is typically used as the basis for the State Plane Coordinate Systems in states having the greatest dimension in the east-west direction.

### Mercator Projection

The Mercator projection is a projection that conceptually uses a cylinder tangent to the ellipsoid at the equator to develop the spherical surface of the earth into a plane surface. The equator is represented as a straight line true to scale; meridians of longitude are represented as straight lines perpendicular to the equator; and parallels of latitude are represented by straight lines parallel to the equator. The scale is exact only along the equator and expands along the meridians north and south of the equator. A line of constant bearing—called a rhumb line—appears as a straight line on this projection. Therefore, the course between two points can be scaled from the map, a useful characteristic for mariners. This will not, however, be the shortest distance between the two points concerned.

### Transverse Mercator Projection

The transverse Mercator projection is equivalent to a Mercator projection rotated 90 degrees. Neither meridians of longitude—except for the central meridian—nor parallels of latitude appear as straight lines. This projection is used as a basis for the State Plane Coordinate Systems in states having the greatest dimension in a north-south direction.

### Other Projections

There are a number of other projections in use for various mapping purposes, including the Universal Transverse Mercator (UMT) projection system. This is a system devised for military purposes, but is used for some civil purposes. It divides the entire earth into 62 zones, much larger than the zones used for the State Plane Coordinate System. The later systems are to be preferred for use in most public planning, engineering, surveying, and large-scale mapping efforts. Unlike the Transverse Mercator projection used as a basis for the state plane coordinate systems, which is a tangent projection, the transverse Mercator projection used as a basis for the UMT projection is a secant projection.

### The State Plane Coordinate System

The State Plane Coordinate System converts the spherical coordinates—latitude and longitude—of the projection, as measured in degrees, minutes and seconds, to rectangular grid coordinates measured in feet. The x axis values are known as eastings, the y axis values are known as northings. See Figure 5.4. The State Plane Coordinate Systems permit the conduct of surveys using plane surveying methods, while accounting for the curvature of the earth, and maintaining distortions within specified limits.

In the use of the State Plane Coordinate System, it is important to determine whether the system concerned is the older system based on the North American Datum of 1927, in which the grid coordinate values and related distances are expressed in U.S. survey feet; or is the newer system based on the North American Datum of 1983, in which the grid coordinate values and related distances are expressed in meters. It is also important to understand the relationship of ground level distances to State Plane Coordinate distances; see Figure 5.5. The following definitions may be helpful in examining Figure 5.4, and are essential to the proper understanding and use of the State Plane Coordinate Systems. The term scale factor is defined as the state plane coordinate distances divided by the ellipsoidal, or sea level, distances. The term sea level reduction factor is defined as the ellipsoidal, or sea level, distances, divided by the horizontal ground level distances. The combination scale and sea level reduction factor is defined as the product of the scale factor and the sea level reduction factor so that State Plane Coordinate grid distances are equal the horizontal ground level distances multiplied by the combination factor.

### Survey Control Networks

The third fundamental element required for the creation of a map is a system of survey control that makes manifest on the surface of the earth the map projection used, and makes it possible to accurately relate the measurements involved in surveying and mapping to the map projection. A survey control network consists of a framework of monumented points, or stations, the locations of which, on the surface of the earth, are accurately known—either relatively or absolutely. The control survey stations are used to locate, orient, and adjust local surveys, to provide a means of verification and check for such surveys, and in the use of surveys in the preparation of maps. Survey control networks are of two types:

horizontal, in which the locations of the monumented stations are given either in terms of latitude and longitude, or in terms of State Plane Coordinates; and vertical in which the orthometric heights—elevations—of the monumented stations (called bench marks) are given in feet above the geoid—mean sea level configuration—of the earth.

There are, in fact, two horizontal survey control networks in place and used within much of the United States: (1) the geodetic control survey network created by the federal government through the National Geodetic Survey, formerly known as the U.S. Coast and Geodetic Survey; and (2) the U.S. Public Land Survey System, also created by the federal government through the Bureau of Land Management and its predecessor agencies. The National Geodetic Survey also provides a vertical control network complimentary to the horizontal control network. The geodetic control survey system provides the basis for all federal topographic mapping efforts, and for the preparation of all nautical and aeronautical charts within the continental United States, including Alaska.

The U.S. Public Land Survey System provides the basis for all real property boundary surveys and mapping in much, but not all, of the continental United States. The states in which the Public Land Survey System has been established include the 30 states that were created out of the public domain—that is, out of the lands originally ceded to, and owned by, the federal government. These states are Alabama, Alaska, Arizona, Arkansas, California, Colorado, Florida, Idaho, Illinois, Indiana, Iowa, Kansas, Louisiana, Michigan, Minnesota, Mississippi, Missouri, Montana, Nebraska, Nevada, New Mexico, North Dakota, Oklahoma, Ohio, Oregon, South Dakota, Utah, Washington, Wisconsin, and Wyoming.

### The National Geodetic Survey Control System

As already noted, the national geodetic survey control system actually consists of two networks: a horizontal survey control network and a complimentary vertical survey control network. The system is a scientific system designed to provide, as already noted, the basic control for all federal topographic mapping and hydro-graphic and aeronautical charting operations.

The horizontal survey control network consists of thousands of monumented points whose positions on the surface of the earth—expressed in terms of latitude and longitude—were established to known high orders of accuracy by the U.S. Coast and Geodetic Survey (USC and GS), now National Geodetic Survey. The positions were established by high-order triangulation and traverse surveys. The stations are relatively widespread, being typically located on topographic high points, and are often relatively inaccessible. The use of the stations requires knowledge and skill in the use of geodetic survey techniques and equipment, knowledge, and skills, not generally historically available to public works engineers and land surveyors.

In order to make the national horizontal survey control network more readily available to and usable by local surveyors, the USC and GS in 1933 devised the State Plane Coordinate System. As already noted, the State Plane Coordinate

System, as originally developed, is based on the Clark Spheroid of 1866 and the attendant North American Datum of 1927 (NAD-27). The State Plane Coordinate grid values are expressed in U.S. survey feet. Also as noted, the system translates the spherical coordinates—latitude and longitude—of the primary federal survey control stations into rectangular coordinates on a plane surface mathematically related to the ellipsoid on which the spherical coordinates have been determined. The State Plane Coordinate System is designed so that the effect of the distortion inherent in the projection of the curved surface of the ellipsoid onto the plane used for the rectangular grid is not more than one part in 10,000. The State Plane Coordinate System, thus, permits local engineers and surveyors to connect surveys by simple, well established plane surveying techniques tied to the extensive network of precisely located triangulation and traverse stations of the national geodetic control survey network.

If the location of a point is defined by stated coordinates on the State Plane Coordinate System grid, then the location of that point is also known by its corresponding latitude and longitude. Thus, the precise location on the surface of the earth of all survey stations and landmarks established in local engineering and land surveys can be accurately described by stating their coordinates referred to the common origin of the State Plane Coordinate System grid. Once plane coordinates are established for any survey station or landmark, these coordinates may become the best available evidence for the original positions concerned, should physical monuments be lost. Computations relating to the lengths and bearings of lines and to related coordinate values using the State Plane Coordinate System are simple, being made with the well-established formulas of plane surveying.

The National Geodetic Survey (NGS) readjusted the horizontal survey control network in 1983, creating the North American Datum of 1983 (NAD-83). This datum has been further refined in some areas of the United States, such refined datums being identified by a suffix attached to the notation NAD-83. For example, in Wisconsin the refined datum is indicated as NAD-83(91). As already noted, NAD-83 is based on the Geodetic Reference System ellipsoid of 1980. The attendant State Plane Coordinate values are given in meters. The shifts in latitude and longitude values between the two datums can be significant. For example, in Southeastern Wisconsin, the maximum shift in latitude between NAD-27 and NAD-83(91) approximates 11 feet; while the maximum shift in longitude approximates 39 feet. Although these shifts are important globally, affecting courses and distances between intercontinental locations, the shifts do not significantly affect the relative bearings and distances between monumented points and control stations within local areas.

The vertical survey control network of the national geodetic control system consists of a network of monumented benchmarks, the elevations—orthometric heights—of which have been determined by the USC and GS through differential spirit level surveys. The original vertical datum is known as National Geodetic Vertical Datum of 1929 (NGVD-29). This datum was also known as Mean Sea Level Datum. The national level net concerned was based on 26 tide stations located along the coasts of the United States and Canada, interconnected by high-order-accuracy differential spirit level surveys.

The NGS readjusted the level network in 1988 to produce a new vertical datum known as the North American Vertical Datum of 1988 (NAVD-88). The differences in elevations on the two datums can be significant. For example, in Southeastern Wisconsin, the difference between NGVD-29 and NAVD-88 elevations ranges from 0.08 to 0.32 foot.

Some local vertical control survey datums are still in use. These can present significant problems when engineering efforts requiring an areawide approach—such as the hydrologic and hydraulic studies that are required to be conducted on a watershed bases—are involved. The use of the national datums and the conversion of local datums to the national datums is to be encouraged.

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