Coordinate Systems Overview

These materials were developed by Peter H. Dana, Department of Geography, University of Texas at Austin, 1995.  These materials may be used for study, research, and education in not-for-profit applications.  If you link to or cite these materials, please credit the author, Peter H. Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder.  These materials may not be copied to or issued from another Web server without the author's express permission.  Copyright © 1999 Peter H. Dana.  All commercial rights are reserved.  If you have comments or suggestions, please contact the author or Kenneth E. Foote at k.foote@colorado.edu.

This page is available in a framed version.  For convenience, a Full Table of Contents is provided.

Associated Overviews

Introduction

• This overview of coordinate systems for georeferencing provides a brief description of local and global systems for use in precise positioning, navigation, and geographic information systems for the location of points in space.
• There are many different coordinate systems, based on a variety of geodetic datums, units, projections, and reference systems in use today.
• As an example, this overview often uses the position of one of the thousands of geodetic control points in the United States, the star in the hand of the Goddess of Liberty atop the Capitol building in Austin, Texas.
• The Texas Capitol Building
• The Star in the Hand of The Goddess of Liberty
• One Location Described by a Variety of Systems

Basic Coordinate Systems

• There are many basic coordinate systems familiar to students of geometry and trigonometry.
• These systems can represent points in two-dimensional or three-dimensional space.
• René Descartes (1596-1650) introduced systems of coordinates based on orthogonal (right angle) coordinates.
• These two and three-dimensional systems used in analytic geometry are often referred to as Cartesian systems.
• Similar systems based on angles from baselines are often referred to as polar systems.
• Plane Coordinate Systems
• Two-dimensional coordinate systems are defined with respect to a single plane.
• A Point Described by Cartesian Coordinates in a Plane
• A Line Defined by Two Points in a Plane
• Distance Between Two points or Line Length from the formula of Pythagoras
• A Point Described by Polar Coordinates in a Plane
• Conversion of Polar to Cartesian Coordinates in a Plane
• Three-Dimensional Systems
• Three-dimensional coordinate systems can be defined with respect to two orthogonal planes.
• A Point Described by Three-Dimensional Cartesian Coordinates
• Distance Between Two Points Described by Three-Dimensional Cartesian Coordinates
• A Point Described by Three-Dimensional Polar Coordinates
• Conversion of Three-Dimensional Polar to Three Dimensional Cartesian Coordinates
• Table of Contents

Reference Ellipsoids

• Ellipsoidal earth models are required for accurate range and bearing calculations over long distances. Loran-C, and GPS navigation receivers use ellipsoidal earth models to compute position and waypoint information. Ellipsoidal models define an ellipsoid with an equatorial radius and a polar radius. The best of these models can represent the shape of the earth over the smoothed, averaged sea-surface to within about one-hundred meters.
• Reference ellipsoids are defined by semi-major (equatorial radius) and semi-minor (polar radius) axes.
• Other reference ellipsoid parameters such as flattening, and eccentricity are computed from these two terms.
• Reference Ellipsoid Parameters
• Many reference ellipsoids are in use by different nations and agencies.
• Selected Reference Ellipsoids
• Table of Contents

Geodetic Datums

• Geodetic datums define the reference systems that describe the size and shape of the earth. Hundreds of different datums have been used to frame position descriptions since the first estimates of the earth's size were made by Aristotle. Datums have evolved from those describing a spherical earth to ellipsoidal models derived from years of satellite measurements.
• Modern geodetic datums range from flat-earth models used for plane surveying to complex models used for international applications which completely describe the size, shape, orientation, gravity field, and angular velocity of the earth. While cartography, surveying, navigation, and astronomy all make use of geodetic datums, the science of geodesy is the central discipline for the topic.
• Referencing geodetic coordinates to the wrong datum can result in position errors of hundreds of meters. Different nations and agencies use different datums as the basis for coordinate systems used to identify positions in geographic information systems, precise positioning systems, and navigation systems. The diversity of datums in use today and the technological advancements that have made possible global positioning measurements with sub-meter accuracies requires careful datum selection and careful conversion between coordinates in different datums.
• Geodetic Datum Overview, Department of Geography, University of Texas at Austin
• Table of Contents

Coordinate Systems

Global Systems

• Latitude, Longitude, Height
• The most commonly used coordinate system today is the latitude, longitude, and height system.
• The Prime Meridian and the Equator are the reference planes used to define latitude and longitude.
• Equator and Prime Meridian
• The geodetic latitude (there are many other defined latitudes) of a point is the angle from the equatorial plane to the vertical direction of a line normal to the reference ellipsoid.
• The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane.
• The geodetic height at a point is the distance from the reference ellipsoid to the point in a direction normal to the ellipsoid.
• Geodetic Latitude, Longitude, and Height
• Table of Contents

• ECEF X, Y, Z
• Earth Centered, Earth Fixed Cartesian coordinates are also used to define three dimensional positions.
• Earth centered, earth-fixed, X, Y, and Z, Cartesian coordinates (XYZ) define three dimensional positions with respect to the center of mass of the reference ellipsoid.
• The Z-axis points toward the North Pole.
• The X-axis is defined by the intersection of the plane define by the prime meridian and the equatorial plane.
• The Y-axis completes a right handed orthogonal system by a plane 90 degrees east of the X-axis and its intersection with the equator.
• ECEF X, Y, and Z
• ECEF X, Y, Z Coordinate Example
• Table of Contents

• Universal Transverse Mercator (UTM)
• Universal Transverse Mercator (UTM) coordinates define two dimensional, horizontal, positions.
• UTM zone numbers designate 6 degree longitudinal strips extending from 80 degrees South latitude to 84 degrees North latitude.
• UTM zone characters designate 8 degree zones extending north and south from the equator.
• There are special UTM zones between 0 degrees and 36 degrees longitude above 72 degrees latitude and a special zone 32 between 56 degrees and 64 degrees north latitude.
• UTM Zones
• Each zone has a central meridian. Zone 14, for example, has a central meridian of 99 degrees west longitude. The zone extends from 96 to 102 degrees west longitude.
• UTM Zone 14
• Eastings are measured from the central meridian (with a 500km false easting to insure positive coordinates).
• Northings are measured from the equator (with a 10,000km false northing for positions south of the equator).
• UTM Zone 14 Example Detail
• UTM Coordinate Example
• Table of Contents

• Military Grid Reference System (MGRS)
• The Military Grid Reference System (MGRS) is an extension of the UTM system. UTM zone number and zone character are used to identify an area 6 degrees in east-west extent and 8 degrees in north-south extent.
• UTM zone number and designator are followed by 100 km square easting and northing identifiers.
• The system uses a set of alphabetic characters for the 100 km grid squares.
• Starting at the 180 degree meridian the characters A to Z (omitting I and O) are used for 18 degrees before starting over.
• From the equator north the characters A to V (omitting I and O) are used for 100 km squares, repeating every 2,000 km.
• Northing designators normally begin with 'A' at the equator for odd numbered UTM easting zones.
• For even numbered easting zones the northing designators are offset by five characters, starting at the equator with 'F'.
• South of the equator, the characters continue the pattern set north of the equator.
• Complicating the system, ellipsoid junctions (spheroid junctions in the terminology of MGRS) require a shift of 10 characters in the northing 100 km grid square designators. Different geodetic datums using different reference ellipsoids use different starting row offset numbers to accomplish this.
• Military Grid Reference System
• UTM zone number, UTM zone, and the two 100 km square characters are followed by an even number of numeric characters representing easting and northing values.
• If 10 numeric characters are used, a precision of 1 meter is assumed.
• 2 characters imply a precision of 10 km.
• From 2 to 10 numeric characters the precision changes from 10 km, 1 km, 100 m 10 m, to 1 m.
• MGRS Example
• Table of Contents

• World Geographic Reference System (GEOREF)
• The World Geographic Reference System is used for aircraft navigation.
• GEOREF is based on latitude and longitude.
• The globe is divided into twelve bands of latitude and twenty-four zones of longitude, each 15 degrees in extent.
• World Geographic Reference System Index
• These 15 degree areas are further divided into one degree units identified by 15 characters.
• GEOREF 1 Degree Grid
• Two numeric characters designate the integer number of minutes of longitude east of the one degree quadrangle boundary longitude.
• Two additional numeric characters designate the number of minutes of latitude north of the one degree quadrangle boundary latitude.
• GEOREF Example
• The World Geographic Reference System can be extended to refer to larger areas of operation.
• A larger East-West area can be designated by adding an "S" and the number of of nautical miles to the east and west sides of the referenced point.
• A larger north-south area can be designated by adding an "X" and the number of nautical miles to the north and south.
• A circular area can be designated by adding an "R" and the radius of the circle in nautical miles.
• An altitude zone can be defined by adding an "H" and a value of altitude. The number of digits indicates the precision of the value. Five digits implies units in feet. Four digits implies tens of feet, three digits, hundreds of feet, and two digits, thousands of feet.
• Table of Contents

• Local systems
• Universal Polar Stereographic (UPS)
• The Universal Polar Stereographic projection (UPS) is defined above 84 degrees north latitude and south of 80 degrees south latitude.
• The eastings and northings are computed using a polar aspect stereographic projection.
• Zones are computed using a different character set for south and north Polar regions.
• North Polar Area UPS Grid
• North Polar UPS Example
• South Polar Area UPS Grid
• South Polar UPS Example
• Table of Contents

• National Grid Systems
• Many nations have defined grid systems based on coordinates that cover their territory. Australia, Belgium, Great Britain, Finland , Ireland, Italy, The Netherlands, New Zealand, and Sweden are a examples of nations that have defined a National Grid System.
• British National Grid (BNG)
• The British National Grid (BNG) is based on the National Grid System of England, administered by the British Ordnance Survey.
• The BNG has been based on a Transverse Mercator projection since the 1920s.
• The modern BNG is based on the Ordnance Survey of Great Britain Datum 1936 (Airy Ellipsoid).
• The true origin of the system is at 49 degrees north latitude and 2 degrees west longitude.
• The false origin is 400 km west and 100 km north.
• Scale at the central meridian is 0.9996012717
• The first BNG designator defines a 500 km square.
• The second designator defines a 100 km square.
• British National Grid 100 km Squares
• The remaining numeric characters define 10 km, 1 km, 100 m, 10 m, or 1 m eastings and northings.
• British National Grid Example
• Table of Contents

• Irish National Grid
• The Irish National Grid (ING) is administered by the Irish Ordnance Survey.
• The ING has been based on a Transverse Mercator projection since the 1920s.
• The ING is based on the Ordnance Survey of Great Britain Datum 1936 or the Ireland Datum 1965.
• The true origin of the system is at 53 degrees, 30 minutes north latitude and 8 degrees west longitude.
• The false origin is 200 km west and 250 km south of the true origin.
• Scale at the central meridian is 1.000035.
• The first ING designator defines a 100 km square.
• Irish National Grid 100 km Squares
• The remaining numeric characters define 10 km, 1 km, 100 m, 10 m, or 1 m eastings and northings.
• Irish National Grid Example
• Table of Contents

• State Plane Coordinates
• In the United States, the State Plane System was developed in the 1930s and was based on the North American Datum 1927 (NAD27).
• NAD 27 coordinates are based on the foot.
• While the NAD-27 State Plane System has been superseded by the NAD-83 System, maps in NAD-27 coordinates (in feet) are still in use.
• NAD-27 State Plane Coordinate Example
• Most USGS 7.5 Minute Quadrangles use several coordinate system grids including latitude and longitude, UTM kilometer tic marks, and applicable State Plane coordinates.
• Three Coordinate Systems on the Austin, East USGS 7.5' Quadrangle
• The State Plane System 1983 is based on the North American Datum 1983 (NAD83).
• NAD 83 coordinates are based on the meter.
• State plane systems were developed in order to provide local reference systems that were tied to a national datum.
• Some smaller states use a single state plane zone.
• Larger states are divided into several zones.
• State plane zone boundaries often follow county boundaries.
• Lambert Conformal Conic projections are used for rectangular zones with a larger east-west than north- south extent.
• State Plane Zone Example
• Texas State Plane Zones and Parameters
• NAD-83 State Plane Coordinate Example
• Transverse Mercator projections are used to define zones with a larger north-south extent.
• One State Plane zone in Alaska uses an oblique Mercator projection for a thin diagonal area.
• Alaska State Plane Zone 5001
• Table of Contents

• Public Land Rectangular Surveys
• Public Land Rectangular Surveys have been used since the 1790s to identify public lands in the United States.
• The system is based on principal meridians and baselines.
• Townships, approximately six miles square, are numbered with reference to baseline and principal meridian.
• Ranges are the distances and directions from baseline and meridian expressed in numbers of townships.
• Every four townships a new baseline is established so that orthogonal meridians can remain north oriented.
• U.S. Rectangular Survey
• Sections, approximately one mile square, are numbered from 1 to 36 within a township.
• Township Sections
• Sections are divided into quarter sections.
• Quarter sections are divided into 40-acre, quarter-quarter sections.
• Quarter-quarter sections are sometimes divided into 10-acre areas.
• Subdivided Section
• Fractional units of section quarters, designated as numbered lots, often result from irregular claim boundaries, rivers, lakes, etc.
• Abbreviations are used for Township (T or Tps), Ranges (R or Rs), Sections(sec or secs), and directions (N, E, S, W, NE, etc.).
• A Township and Range Property Description
• Table of Contents

• Metes and Bounds
• Metes and Bounds identify the boundaries of land parcels by describing lengths and directions of lines.
• Lines are described with respect to natural or artificial monuments and baselines defined by these monuments.
• The metes and bounds survey is based on a point of beginning, an established monument.
• Line lengths are measured along a horizontal level plane.
• Directions are bearing angles measured with respect to a previous line in the survey.
• Metes and Bounds Example
• Table of Contents

• Miscellaneous Systems
• Postal Codes
• Postal codes such as the United States ZIP code can be used to identify areas.
• Three digit codes identify large areas.
• Texas 3-Digit Postal Zip Codes
• Five-digit ZIP codes identify smaller areas.
• Maidenhead Grid Squares
• The Maidenhead Grid Square system was designed to facilitate the designation of geographical positions for use within the amateur radio community. The Maidenhead Grid identifies "Fields" consisting of an area twenty degrees of longitude by ten degrees of latitude with two alphabetic characters. An additional set of two numeric digits locates a specific two-degrees of longitude by one-degree of latitude" grid square" area within the Field. Two additional alphabetic characters can be used to refer to a 5.0 minutes of longitude by 2.5 minutes of latitude "Sub-Square" within the Grid Square. In each case the longitude character precedes the latitude designator.
• Variations and extensions to the Maidenhead system are in use. Some Global Positioning System (GPS) receivers display positions in an extended Maidenhead system that appends one or two additional sets of numeric and alphabetic pairs, increasing the precision with which a location can be specified.
• Some amateur radio operators use other terms for the Maidenhead system such as World Wide Locator (WWL) squares or QTH locator squares.
• Ham operators use these grid designators to communicate transmitter positions to each other. Several utility programs are available to convert between latitude and longitude and the Maidenhead Grid Square system. Some of these also allow computation of distance and azimuth between stations.
• Maidenhead Grid Square Fields
• Maidenhead Grid Squares
• Maidenhead Grid Sub-Squares
• AT&T V and H Coordinate System
• The AT&T V and H (Vertical and Horizontal) coordinate system was devised in 1957 by Jay K. Donald for the easy computation of distances between telephone switching centers. The system is based on the Donald Elliptic Projection, a two-point equidistant projection covering the land masses of the continental United States and Canada. The system is based on units of the square-root of one-tenth of a mile.
• AT&T V and H Coordinates - Donald Elliptic Projection
• Once the coordinates of switching sites are known, distances between sites can be simply found by calculating the square root of the sum of the squares of the differences in the vertical and horizontal coordinates divided by ten. Designed for simple distance calculations that could be accomplished in the field with a slide rule, the system is still found imbedded in some telephone rate computation software.
• V & H Coordinate Example
• Navigation System Coordinates
• Navigation systems can define locations by referencing measurements of electronic signals.
• Loran-C time-differences can identify positions with an accuracy of one-quarter of a mile.
• Loran-C Time Differences
• Omega phase-differences can identify positions with an accuracy of 1-5 kms.
• VOR-DME (Very high frequency Omni Range - Distance Measuring) measurements from an aircraft can identify locations with an accuracy of 0.5-3 kms.
• VOR-DME Chart Detail
• VOR - DME Coordinates
• Navigational buoys, and other aids to navigation can be used as visual reference points, bearings to visual references can identify locations with varying accuracies.
• Navigational Buoy
• Table of Contents

• References
• Defense Mapping Agency. 1977. The American Practical Navigator Publication No. 9, Defense Mapping Agency Hydrographic Center.
• Laurila, Simo H. 1976. Electronic Surveying and Navigation. New York: John Wiley & Sons
• Muehrcke, P.C and Juliana O. Muehrcke. 1992. Map Use. Madison, WI: JP Publications.
• National Imagery and Mapping Agency. 1997. World Geodetic System 1984 (WGS 84) - Its Definition and Relationships with Local Geodetic Systems, 3rd Edition. Washington, DC: National Imagery and Mapping Agency.
• Snyder, John P. 1987. Map Projections, A Working Manual. Washington, DC: US Govt. Printing Office.
• Thomas, P. D. 1970. Spheroidal Geodesics, Reference Systems and Local Geometry. Washington, DC: U. S. Naval Oceanographic Office.
• U. S. Air Force and Navy. 1983. Air Navigation, AFM 51-40. Washington, DC: Department of the Air Force.
• U. S. Army. 1967. TM 5-241-1 Grids and Grid References. Washington, DC: Department of the Army.
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