## Info

FIGURE 5.2 Relationships between the topographic surface of the earth, the geoid, and a reference ellipsoid.

sea level if the seas extended under the continents. The geoid is the figure to which all survey measurements are referenced and to which surveying instruments are oriented through the use of spirit level vials or plumb lines. The geoid is, however, an irregular, undulating surface.

Since it is impossible to make survey computations on the irregular surface of the geoid, a mathematical surface is substituted that closely approximates the geoid. This surface is created by rotating a two-dimensional ellipse about its semiminor axis to create a three-dimensional, mathematically defined surface, known as the ellipsoid.

Geographic locations and relationships—distances and directions—are expressed as though they were located on the ellipsoid. Historically, a number of ellipsoids have been used as a basis for surveying and mapping operations. Within the continental United States, the historically most widely used ellipsoid is know as the Clark spheroid of 1866. This ellipsoid was adopted for use in geodetic surveying and in hydrographic mapping operations in the continental United States by the U.S. Coast and Geodetic Survey—now known as the National Geodetic Survey—and for use in topographic mapping operations by the U.S. Geological Survey in the later part of the nineteenth century. This ellipsoid is the basis for the North American Datum of 1927 (NAD-27), and for the older State Plane Coordinate Systems still in use within the United States. The grid coordinate values under these older systems are expressed in U.S. survey feet. The newer State Plane Coordinate Systems in use within the United States are based on the Geodetic Reference System of 1980. This spheroid is the basis for the North American Datum of 1983 (NAD-83), and for the revised State Plane Coordinate Systems. The grid coordinate values under these newer systems are expressed in meters.

### Map Projections

The second of the foundational elements required for the creation of a map is a map projection. A map projection typically consists of a set of mathematical equations for converting the spherical surface of the earth to a flat surface upon which maps may be constructed. Map projections thus serve to convert the spherical geometry of the mapping ellipsoid to the plane geometry of the flat mapping surface. A number of projection systems are in use for surveying and mapping.

### Tangent Plane Projection

The tangent plane projection was the most common form of projection once used by land surveyors and civil engineers. It is the basis for "plane" surveying procedures, the procedures usually taught in introductory courses in surveying. This projection is applicable only to the surveying and mapping of small areas and is being replaced by more sophisticated projections and attendant and mapping surveying procedures. Many land surveyors and civil engineers engaged in the application of plane surveying may not even be aware that they are using this type of projection in the preparation of maps and plats attendant to their work. In application, the surveyor selects a point in, or near, the area to be surveyed and mapped, at which point the survey is oriented to some form of directional control that is recoverable. The directional control may be provided by magnetic observation; celestial observation; or by the direction of a line defined by two monumented U.S. Public Land Survey system one-quarter section corners. The surveyor then measures the angles formed by the lines of the survey and the central direction. This is very different from independently measuring a magnetic or astronomic direction for each line as is the case in projection-less mapping. The angles so measured may then be drawn to scale on the map. Bearings shown on tangent projection maps do not represent the astronomic or geodetic bearings of the survey lines. The curvature of the earth and convergence of the meridians are ignored. The distances are measured as, or reduced to, horizontal distances. The distances are assumed to be measured at the mean elevation of the area surveyed, and are horizontal and not level distances. No adjustments are made for differences between distances as measured on the surface of the earth and these same distances on the flat plane of the projection. The map—often termed a plat—derived from the measured angles and distances is, in effect, a projection of the curved surface of the earth onto a flat plane.

The principal advantage of this system is its simplicity. Straight lines are considered to have a constant bearing; parallel straight lines are considered to have the same bearing; level surfaces are considered to be flat planes; and plumb lines are considered to be parallel. The errors introduced by these assumptions become noticeable when the areas concerned exceed about 75 square miles, and then approximate 0.05 foot and 0.1 second of arc. Individually compiled maps cannot be coordinated and become diagrams rather than true maps. Other surveys conducted in the same manner will disagree in the lengths and directions of common lines, and directions between identical points on adjacent parcels will have different values. This means that discrepancies, gaps, and overlaps will not be apparent from mere review of plats of survey of adjacent parcels. Resurveys are entirely dependent on recovery of survey markers or monuments set during the original work.

Tangent plane projection surveys are of limited use to comprehensive planners and to civil and environmental engineers concerned with areawide projects. The lack of a common reference system makes the task of relating individual parcels to each other difficult or impossible. Indeed, existing municipal maps compiled from plats of survey are often no more than representations of the compilations of paper records, so poorly done as to make their use in planning and engineering difficult and costly, and the use of such plan implementation devices as official mapping legally questionable.

The need to identify real property boundaries permanently and precisely, and the need for large-scale, areawide planning and engineering has led to the use of projection systems that eliminate the disadvantages of the tangent plane system.

### Lambert Projection

The Lambert conformal conic projection conceptually uses a cone passed through two parallels of the ellipsoid—known as standard parallels—to develop the spherical surface into a plane surface. See Figures 5.3, 5.4, and 5.5. Meridians of longitude are represented on the projection by converging straight lines; and parallels of latitude are represented as arcs of circles with a common center.

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