# Coordinate Definitions for Land Surveyors

**coordinate–**^{1}adj. Equal in rank, quality, or significance; similar in order or nature; not subordinate**. **^{2}Composed of things of equal rank or order; coordinated. ^{3} n. Any one of a set of numbers used in specifying the location of a point on a line, in space, or on a given plane or other surface (latitudes and longitudes are coordinates of a point on the Earth’s surface).** ^{4} **

*pl.*Linear or angular quantities, or both, which designate the position of a point in relation to a given reference frame. There are two general divisions of coordinates used in surveying—polar coordinates and rectangular coordinates. These can each be subdivided into three classes: plane coordinates, spherical coordinates, and space coordinates.

**Coordinate method, variation of—**See *variation of coordinate method. *

**Coordinate system—**A reference system for defining points in space or on a particular surface by means of distances or angles, or both, with relation to designated axes, planes, or surfaces. Three general types of reference systems are commonly used in surveying and mapping:** ^{1} **Plane-pol in which points in a plane are defined by distance from a specified point along a ray with a known direction with respect to a specified base line;

^{2}Rectangular, in which points are defined by linear distances from two perpendicular axes or from three mutually perpendicular planes;

^{3}Spherical, in which points on a spherical or ellipsoidal surface are defined by the angles between a normal or radius through the point and two selected perpendicular diametrical planes. Examples are geographic, astronomic, and azimuth-altitude systems. To obtain the advantages of rectangular coordinates for defining points on the Earth’s surface, a number of special plane-rectangular coordinate systems have been developed and adapted to specific areas. The usual procedure is to project the geographic coordinate system to a plane by mathematical transformation and to superimpose a rectangular grid on the plane projection. The plane-coordinate (grid) system then takes the name of the projection. The most common projections for this purpose are the Lambert conformal conic and the transverse Mercator. Such systems have been developed for each U.S. state and Puerto Rico, with the larger states subdivided into two or more zones. A similar system, on the transverse Mercator projection, was developed for world-wide use in zones of 6° longitudinal width; the universal transverse Mercator (UTM) grid. See also

*map projection, Lambert conformal conic projection; map projection, transverse Mercator; map projection, Universal Transverse Mercator; coordinates, state plane; grid.*

**coordinate system, curvilinear—**Any coordinate system in which at least one of the geometric elements (lines or surfaces) used for reference is curved. Coordinates are determined by the intersection of curved lines or surfaces rather than by intersections of straight lines or planes only.

**coordinate system, state plane—**See *coordinate system; coordinates, state plane.*

**coordinates, assumed plane—**A local plane-coordinate system set up at the convenience of the surveyor. The reference axes are usually assumed so that all coordinates are in the first quadrant. The Y-axis may be in the direction of astronomic (true) North, magnetic North, or an assumed North.

**coordinates, astronomic—**Quantities which define the position of a point on the geoid with reference to the planes of the celestial equator and of a selected celestial meridian. See also *latitude, astronomic; longitude, astronomic.*

**coordinates, Cartesian—**Values representing the location of a point in a plane in relation to two intersecting straight lines, called axes. The point is located by measuring its distance from each axis along a parallel to the other axis. If the axes are perpendicular to each other the coordinates are rectangular; if not perpendicular, they are oblique coordinates. This system is extended to represent the location of points in three-dimensional space by referencing to three mutually perpendicular coordinate axes which intersect at a common point of origin.

**coordinates, geocentric—**Coordinates that define the position of a point with respect to the center of the Earth. Geocentric (or terrestrial) coordinates can be either Cartesian (x, y, z) or spherical (latitude, longitude, and radial distance); geocentric coordinate system; geocentric position.

**coordinates, geodetic—**The quantities of latitude and longitude which define the position of a point on the surface of the Earth with respect to the reference ellipsoid. Also called \”geographic coordinates.\”

**coordinates, geographic—**An inclusive term, used to designate both geodetic coordinates and astronomic coordinates. Also called “terrestrial coordinates.”

**coordinates, grid—**Two distances which fix the position of a point on a grid. The perpendicular distance to the point from the axis of Y, termed the abscissa or x-coordinate; and the perpendicular distance from the axis of X, termed the ordinate or y-coordinate. In surveying operations, the nominal origin at the intersection of the axes is usually given large numerical coordinates to avoid the inconvenience of using negative coordinates. Geodetic coordinates (latitudes and longitudes) may be transformed into grid coordinates, and all survey computations relating to them may then be made by the methods and formulas of plane surveying. See also *coordinates, state plane; easting, false; meridian, central; northing, false.*

**coordinates, military—**The U.S. Army formerly used fire-control coordinates based on a polyconic projection. Universal transverse Mercator (UTM) coordinates and universal polar stereographic (UPS) coordinates are currently used for military purposes along with a military grid reference system.

**coordinates, origin of—**A point in a system of coordinates which serves as an initial point in computing its elements or in prescribing its use. The term, origin of coordinates, has several definitions, each so well established that a single definition cannot be prescribed to the exclusion of others. The following are given in the order of preferred use; to avoid misunderstanding, the use should be defined by stating the position of the origin in the system and giving the numerical coordinates assigned it. ^{1} The origin of coordinates is the point of intersection of the coordinate axes, from which the coordinates are reckoned. In mathematical treatises this origin is usually given the coordinates (0,0); in surveying, however, it is standard practice to give this origin coordinates having large positive numerical values, thereby avoiding the use of negative coordinates. See also *coordinates, state plane. *^{2} The origin of coordinates is the point to which the coordinate values (0,0) are assigned, irrespective of its position with reference to the axes. ^{3} The origin of coordinates is the point from which the computation of the elements of the coordinate system (projection) proceeds.

**coordinates, plane rectangular—**The perpendicular distances (coordinates) of a point from a pair of axes which intersect at right angles, reckoned in the plane defined by those axes. Plane rectangular coordinates are usually computed from data which are in the form of polar coordinates-that is, distance and direction (bearing or azimuth) from a previously determined point; for example, the computation of latitudes and coordinates, in land surveying. The methods used are based on plane trigonometry and geometry. The position of a point on the earth can be defined by plane rectangular coordinates on a tangent plane (local system of plane coordinates), or on a so-called conic or cylindrical map projection, such as are used in the state plane coordinate systems.

**coordinates, polar—**The distance and direction from a central point of reference to a point whose position is being defined. The point of reference is called the “pole or origin,” the line (distance) connecting the origin with the point whose position is being defined is the “radius vector,” and the angle between the fixed line to which the direction is referred and the radius vector is the “vectorial angle.” In surveying operations, observations are usually put in the form of polar coordinates as a first step in the computation of plane or spherical coordinates. For example, computations of geodetic positions (latitudes and longitudes) are based on azimuths and distances from known positions.

**coordinates, rectangular**—Coordinates on any system in which the axes of reference intersect at right angles.

**coordinates, rectangular space**—The perpendicular distances of a point from planes defined by each pair of a set of three axes which are mutually perpendicular to each other at a common point of origin. In photogrammetry, space coordinates are also called “survey coordinates,” and are the x-coordinates and y-coordinates which define the horizontal position of a point on a ground system, and the z-coordinate, which is the elevation of the point with reference to the ground system. These coordinates are also useful in airport zoning.

**coordinates, rectilinear—**See *coordinates, rectangular.*

**coordinates, spherical—**A system of coordinates defining a point on d sphere or ellipsoid by its angular distances from a primary great circle and from a reference secondary great circle, as latitude and longitude.

**coordinates, state plane—**The plane-rectangular coordinate systems established by the U.S. Coast and Geodetic Survey (predecessor of the National Geodetic Survey), one for each state in the United States, for use in defining positions of geodetic stations in terms of plane-rectangular (X and Y) coordinates. Each state is covered by one or more zones, over each of which is placed a grid imposed upon a conformal map projection. The relationship between the grid and the map projection is established by mathematical analysis. Zones of limited east-west dimension and indefinite north-south extent have the transverse Mercator map projection as the base for the state coordinate system, whereas zones for which the above order of magnitude is reversed have the Lambert conformal conic map projection with two standard parallels. For a zone having a width of 158 statute miles, the greatest departure from exact scale (scale error) is 1 part in 10,000. Only adjusted positions on NAD 27 or NAD 83 may be properly transformed into plane coordinates on a state system. All such geodetic positions, validated by the National Geodetic Survey, are transformed into state plane-rectangular coordinates on the proper grid, and are distributed by that bureau with the geodetic positions. State plane coordinates are extensively used in recording land surveys, and in most states such use has received approval by legislative enactment. See also *coordinate system; coordinates, grid.*

**coordinates, Universal Polar Stereographic—**The Universal Polar Stereographic (UPS) grid consists of two polar stereographic projections, one for the north polar area and one for the south polar area. Its specifications are: (a) Polar stereographic projection; (b) International Spheroid; (c) North zone—polar area north of 84° N; and South zone—polar area south of 80° S; (d) The unit of length is the meter; (e) The longitude of origin is 0° 180° E-W; (f) The latitude of origin is 90° N and 90° S; (g) The false northing and false easting are both 2,000,000 m; (h) The scale factor at origin is 0.994; (i) The UPS grids extend to 80° 30′ N and 79° 30′ S providing a 30-minute overlap with the UTM grid; (j) The approximate secant line of unity scale factor is 81° 06′ 52″. The UPS grids and the UTM grids make use of the Military Grid Reference System.

**coordinates, Universal Transverse Mercator—**The Universal Transverse Mercator (UTM) grid has the following specifications: (a) Transverse Mercator projection (Gauss-Kruger type) in zones 6° wide in longitude; (b) The GRS80 Ellipsoid if using NAD 83, and for North America, Clarke\’s Ellipsoid of 1866 if using NAD 27. For other areas one should consult the publications of the Defense Mapping Agency (now NGA) of the U.S. Department of Defense; (c) For North America, either NAD 27 or NAD 83 is correct, although NAD 83 is now the datum generally preferred; (d) The longitude of the origin is the central meridian of each zone; (e) The latitude of the origin is 0° (the equator); (f) The unit of length is the meter; (g) The false northing is 0 m for the northern hemisphere and 10,000,000 m for the southern hemisphere; (h) The false easting is 500,000 m; (i) The scale factor at the central meridian is 0.9996; (j) The zones are numbered by beginning with 1 on the zone from 180° W to 174° W, and increasing eastward to 60 on the zone from 174° E to 180° E. All grid zones are identical except for the displacement in longitude; (k) The limits of latitude are 84° N and 80° S; (1) The zones are bounded by meridians whose longitudes are multiples of 6° west or east of Greenwich. On large-scale maps and in tables, an overlap of approximately 25 miles on either side of the junction is provided for surveyors and for artillery survey and firing. This overlap is never used, however, in giving a grid reference. The UTM system was designed for worldwide use between the latitude limits given herein and has been particularly useful for work in countries that have no other coordinate system. The polar areas are covered by the Universal Polar Stereographic (UPS) grid. The UPS grids and the UTM grids make use of the Military Grid Reference System.

**coordinates, vertical—**The vertical distance (elevation) of a point above or below a surface of reference (datum). The vertical coordinate of a point may be plus or minus, according to whether the point is above or below the datum; the datum may be assigned a large positive elevation so that all elevations referred to it will be plus.

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Source: NSPS “Definitions of Surveying and Related Terms“, used with permission.

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