This section was edited by Kenneth Foote, Department of Geography, University of Texas Austin.

Your comments on these materials are welcome. A link to an evaluation form is provided at the end of this document.

Topics covered in this unit

• This unit provides an overview of coordinate systems used for georeferencing, including:
  • a description of basic coordinate systems
  • a description of the shape of the Earth
  • some examples of global and regional systems used for precise positioning, navigation, and geographic information systems
• The overview discusses the rationale behind these systems and how they are used

Learning Outcomes

• After learning the material covered in this unit, students should be able to:
  • List the major global georeferencing systems
  • Explain how the UTM georeferencing system is organized
  • Locate a given local landmark in three or more global or regional coordinate systems
  • Express the location of the student's home in UTM and one regional coordinate system
  • Identify the georeferencing system used most widely in the student's locality.
  • Differentiate between different global systems of georeferencing including UTM, MGRS, and GEOREF systems.
  • Collect examples of regional and local coordinate systems employed by nations or states not mentioned in the text.
  • Find and plot one example of a metes-and-bounds survey
  • Plan a regional GIS project that spans two or more UTM or SPC zones.

Full Table of Contents

Instructors' Notes

Metadata and Revision History

1. 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 Decartes (1596-1650) introduced systems of coordinates based on orthogonal (right angle) axes.
  • 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.

1.1. Plane Coordinate Systems

• Two-dimensional coordinate systems are defined with respect to a single plane, as demonstrated in the following figures:
  • Figure 1. A Point Described by Cartesian Coordinates in a Plane
  • Figure 2. A Line Defined by Two Points in a Plane
  • Figure 3. Distance Between Two points (Line Length) from the formula of Pythagoras
  • Figure 4. A Point Described by Polar Coordinates in a Plane
  • Figure 5. Conversion of Polar to Cartesian Coordinates in a Plane

1.2. Three-Dimensional Systems

• Three-dimensional coordinate systems can be defined with respect to two orthogonal planes.
  • Figure 6. A Point Described by Three-Dimensional Cartesian Coordinates
  • Figure 7. A Point Described by Three-Dimensional Polar Coordinates
  • Figure 8. Conversion of Three-Dimensional Polar to Three Dimensional Cartesian Coordinates

2. Earth-Based Locational Reference Systems

• Reference systems and map projections extend the ideas of Cartesian and polar coordinate systems over all or part of the earth.
  • Map projections portray the nearly spherical earth in a two-dimensional representation.
• Earth-based reference systems are based on various models for the size and shape of the earth.
  • Earth shapes are represented in many systems by a sphere
  • However, precise positioning reference systems are based on an ellipsoidal earth and complex gravity models.

2.1. Reference Ellipsoids

• Ellipsoidal earth models are required for precise distance and direction measurement over long distances.
  • Ellipsoidal models account for the slight flattening of the earth at the poles. This flattening of the earth's surface results at the poles in about a twenty kilometer difference between an average spherical radius and the measured polar radius of the earth.
  • The best ellipsoidal 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 either:
  • semi-major (equatorial radius) and semi-minor (polar radius) axes, or
  • the relationship between the semi-major axis and the flattening of the ellipsoid (expressed as its eccentricity).
  • Figure 9. Reference Ellipsoid Parameters
• Many reference ellipsoids are in use by different nations and agencies.
  • Table 1. Selected Reference Ellipsoids
• Reference ellipsoids are identified by a name and often by a year
  • for example, the Clarke 1866 ellipsoid is different from the Clarke 1858 and the Clarke 1880 ellipsoids.

2.2. Geodetic Datums

• Precise positioning must also account for irregularities in the earth's surface due to factors in addition to polar flattening.
• Topographic and sea-level models attempt to model the physical variations of the surface:
  • The topographic surface of the earth is the actual surface of the land and sea at some moment in time.
    • Aircraft navigators have a special interest in maintaining a positive height vector above this surface.
  • Sea level can be thought of as the average surface of the oceans, though its true definition is far more complex.
    • Specific methods for determining sea level and the temporal spans used in these calculations vary considerably.
    • Tidal forces and gravity differences from location to location cause even this smoothed surface to vary over the globe by hundreds of meters.
• Gravity models and geoids are used to represent local variations in gravity that change the local definition of a level surface
  • Gravity models attempt to describe in detail the variations in the gravity field.
    • The importance of this effort is related to the idea of leveling. Plane and geodetic surveying uses the idea of a plane perpendicular to the gravity surface of the earth which is the direction perpendicular to a plumb bob pointing toward the center of mass of the earth.
    • Local variations in gravity, caused by variations in the earth's core and surface materials, cause this gravity surface to be irregular.
  • Geoid models attempt to represent the surface of the entire earth over both land and ocean as though the surface resulted from gravity alone.
• Geodetic datums define reference systems that describe the size and shape of the earth based on these various models.
  • While cartography, surveying, navigation, and astronomy all make use of geodetic datums, they are the central concern of the science of geodesy.
• Hundreds of different datums have been used to frame position descriptions since the first estimates of the earth's size were made by the ancient greeks.
  • 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.
• Different nations and international agencies use different datums as the basis for coordinate systems in geographic information systems, precise positioning systems, and navigation systems.
  • In the United States, this work is the responsibility of the National Geodetic Survey (http://www.ngs.noaa.gov/).
  • Links to some of the NGS's counterparts in other nations are listed below in Section 7.2 (Web References).
• Linking geodetic coordinates to the wrong datum can result in position errors of hundreds of meters.
  • 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.
• For the purposes of this unit, reference system can be divided into two groups:.
  • Global systems can refer to positions over much of the Earth.
  • Regional systems have been defined for many specific areas, often covering national, state, or provincial areas.

3. Global Systems

3.1. 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.
  • Figure 10. Equator and Prime Meridian
• There are several ways to define these terms precisely. From the geodetic perspective these are:
  • The geodetic latitude of a point is the angle between the equatorial plane and 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.
  • Figure 11. Geodetic Latitude, Longitude, and Height

3.2. ECEF X, Y, Z

• Earth Centered, Earth Fixed (ECEF) Cartesian coordinates can also be used to define three dimensional positions.
• ECEF X, Y, and Z Cartesian coordinates define three dimensional positions with respect to the center of mass of the reference ellipsoid.
  • The Z-axis points from the center toward the North Pole.
  • The X-axis is the line at the intersection of the plane defined by the prime meridian and the equatorial plane.
  • The Y-axis is defined by the intersection of a plane rotated 90° east of the prime meridian and the equatorial plane.
  • Figure 12. ECEF X, Y, and Z
  • Table 2. ECEF X, Y, Z Coordinates Example

3.3. Universal Transverse Mercator (UTM)

• Universal Transverse Mercator (UTM) coordinates define two dimensional, horizontal, positions.
• Each UTM zone is identified by a number
  • UTM zone numbers designate individual 6° wide longitudinal strips extending from 80° South latitude to 84° North latitude.
  • (Military UTM coordinate systems also use a character to designate 8° zones extending north and south from the equator, see below).
  • Figure 13. UTM Zones
• Each zone has a central meridian.
  • For example, Zone 14 has a central meridian of 99° west longitude.
    • The zone extends from 96 to 102° west longitude.
  • Figure 14. UTM Zone 14
• Locations within a zone are measured in meters eastward from the central meridian and northward from the equator. However,
  • Eastings increase eastward from the central meridian which is given a false easting of 500 km so that only positive eastings are measured anywhere in the zone.
  • Northings increase northward from the equator with the equator's value differing in each hemisphere
    • in the Northern Hemisphere, the Equator has a northing of 0
    • for Southern Hemisphere locations, the Equator is given a false northing of 10,000 km
  • Figure 15. UTM Zone 14 Example Detail
  • Table 3. UTM Coordinate Example

3.4. Military Grid Reference System (MGRS)

• The Military Grid Reference System (MGRS) is an extension of the UTM system.
• A UTM zone number and an additional zone character are used to identify areas 6° in east-west extent and 8° in north-south extent.
  • A few special UTM zones do not match the standard configuration (see Figure 13)
    • between 0° and 42° east longitude, above 72° north latitude in the area of the Greenland and Barents Seas, and the Arctic Ocean.
    • in zones 31 and 32 between 56° and 64° north latitude including portions of the North Sea and Norway.
• UTM zone number and character are followed by two characters designating the eastings and northings of 100 km square grid cells.
  • Starting eastward from the 180° meridian, the characters A to Z are assigned consecutively to up to 24 strips covering 18° of longitude (characters I and O are omitted to eliminate the possibility of confusion with the numerals 1 and 0).  The sequence begins again every 18°.
  • From the equator northward, the characters A to V (omitting characters I and O) are used to sequentially identify 100 km squares, repeating the sequence every 2,000 km.
    • for odd numbered UTM easting zones, northing designators normally begin with 'A' at the equator
    • for even numbered UTM 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.
  • Figure 16. Military Grid Reference System
• For a full MGRS location, UTM zone number and character and the two grid square designators are followed by an even number of digits representing more precise easting and northing values.
  • 2 digits give a coordinate precision of 10 km.
  • 10 digits give a coordinate precision of 1 m.
  • Table 4. MGRS Example
• MGRS and UTM systems are often employed in products produced by the US National Imagery and Mapping Agency (http://www.nima.mil/), formerly the Defense Mapping Agency.

3.5. 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° in extent.
  • Figure 17. World Geographic Reference System Index
• These 15° areas are further divided into one degree units identified by 15 characters.
  • Figure 18. GEOREF 1° Grid
  • Table 5. GEOREF Example

4. Regional Systems

• Several different systems are used regionally to identify geographic location
• Some of these are true coordinate systems, such as those based on UTM and UPS systems
• Others, such as the metes and bounds and Public Land Survey systems describe below, simply partition space

4.1. Transverse Mercator Grid Systems

• Many nations have defined grid systems based on Transverse Mercator coordinates that cover their territory.

4.1.1. An example - the British National Grid (BNG)

• The British National Grid (BNG) is based on the National Grid System of England, administered by the British Ordnance Survey (http://www.ordsvy.gov.uk/)
• 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.
• The true origin of the system is at 49° north latitude and 2 degrees west longitude.
  • The false origin is 400 km west and 100 km north.
• Scale factor at the central meridian is 0.9996012717.
• The first BNG designator defines a 500 km square.
• The second designator defines a 100 km square.
  • Figure 19. British National Grid 100 km Squares
• The remaining digits define 10 km, 1 km, 100 m, 10 m, and 1 m eastings and northings.
  • Table 6. British National Grid Example

4.2. Universal Polar Stereographic (UPS)

• The Universal Polar Stereographic (UPS) projection is defined above 84° north latitude and south of 80° 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.
• Figure 20. North Polar Area UPS Grid
  • Table 7. North Polar UPS Example
• Figure 21. South Polar Area UPS Grid
  • Table 8. South Polar UPS Example

4.3. State Plane Coordinates (SPC)

• State plane systems were developed in order to provide local reference systems that were tied to a national datum.
• In the United States, the State Plane System 1927 was developed in the 1930s and was based on the North American Datum 1927 (NAD-27).
  • NAD-27 coordinates are in English units (feet).
  • Figure 22. NAD-27 State Plane Coordinate Example
• The State Plane System 1983 is based on the North American Datum 1983 (NAD-83).
  • NAD-83 coordinates are metric.
  • Table 9. NAD-83 State Plane Coordinate Example
  • While the NAD-27 State Plane System has been superceded by the NAD-83 System, maps in NAD-27 coordinates are still in use.
• Most USGS 7.5 Minute Quadrangles show several coordinate system grids including latitude and longitude, UTM kilometer tic marks, and applicable State Plane coordinates.
  • Figure 23. Three Coordinate Systems on the Austin, East USGS 7.5' Quadrangle
• Each state has its own State Plane system with specific parameters and projections.
  • Software is available for easy conversion to and from latitude and longitude.
  • A popular public domain software package, CORPSCON is maintained by the US Army Corps of Engineers
• Some smaller states use a single state plane zone while larger states are divided into several zones.
  • State plane zone boundaries often follow county boundaries.
  • Figure 24. State Plane Zone Example
• Two projections are used in all State Plane systems, with one exception:
  • Lambert Conformal Conic projections are used for regions with a larger east-west than north-south extent.
    • examples are Nebraska and North Carolina
  • Transverse Mercator projections are used for regions with a larger north-south extent.
    • examples are New Hampshire and Illinois
  • Some states use both projections
    • in Florida, the Lambert Conformal Conic projection is used for the North zone while the Transverse Mercator projection is used for the East and West zones.
  • The exception is one State Plane zone in Alaska which uses an Oblique Mercator projection for a thin diagonal area.
    • Figure 25. Alaska State Plane Zone 5001

4.4. Public Land Rectangular Surveys (USPLS)

• Public Land Rectangular Surveys have been used since the 1790s to identify public lands in the United States.(USPLS = US Public Land Survey)
  • The system is based on principal meridians and baselines.
• Townships, square with six miles on each side, are numbered with reference to a baseline and principal meridian.
  • actually, few townships are truly square due to convergence of the meridians.
• 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.
  • Figure 26. U.S. Rectangular Survey
• Sections, approximately one mile square, are numbered from 1 to 36 within a township.
  • Figure 27. 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.
  • Figure 28. 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.).
  • Table 10. A Township and Range Property Description

4.5. Metes and Bounds

• Metes and Bounds identify the boundaries of land parcels by describing lengths and directions of a sequence of lines forming the property boundary.
  • Lines are described with respect to natural or artificial monuments and to baselines drawn from 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 the previous line in the survey.
  • Table 11. Metes and Bounds Example
  • Figure 28a.  Metes and Bounds graphic
• Metes and bounds descriptions are also referred to as COGO (Coordinate Geometry) when used in GIS and CAD systems

5. Summary

• This overview has introduced a number of global and regional coordinate systems. A single point on the Earth can be described in a variety of systems. Each GIS project may require the use of a specific locational reference system. It is important to be aware of the variety of systems in use.
• As an example of a point that could be referred to by a number of different system, one of the horizontal control monuments used in the survey network maintained by the National Geodetic Survey (the star in the hand of the statue of the Goddess of Liberty on top of the state capitol building in Austin, Texas) has been used throughout this overview.
  • Figure 29. The Texas Capitol Building
  • Figure 30. The Star in the Hand of The Goddess of Liberty
• This horizontal control monument can be described by many different locational reference systems.
  • Table 12. One Location Described by a Variety of Systems

6. Review and Study Questions

6.1. Essay and Short Answer Questions

• In what ways does the long and widespread use of SPC, UTM, COGO, and USPLS reference systems limit the possibility of building regional and state-wide GIS?
• What is metes and bounds surveying and how is it used to measure and record land records?
• The US Public Land Survey is a method of cadastral partitioning. How has it influenced the appearance of the American landscape and why?
• What is the rationale behind both the State Plane Coordinate and Universal Transverse Mercator coordinate systems?
• From the standpoint of locational reference systems (SPC and UTM) and methods of cadastral partitioning (USPLS, metes and bounds, etc.), why is Texas such an unusual state?
• Describe the Township and Range land surveying system. Use a diagram.
• What is a false origin? In practice, why are they always placed outside of the map zone being used?
• In a state of Texas's size, why can't SPC or UTM coordinates be used for mapping and GIS projects that span the entire state?
• Why was the State Plane Coordinate System such an important advance for mapping in the US?

6.2. Multiple-choice questions

Choose the best or most appropriate answer(s) to the question.

• Which of the following statements are true of both SPC and UTM coordinate systems?
  1. both SPC and UTM are for mapping in the US
  2. both SPC and UTM employ conformal, equidistant projections
  3. within the US both systems yield horizontal coordinates of equal precision
  4. UTM zones correspond to state boundaries, whereas SPC zone are aligned with county boundaries

• Which system incorporates a false origin to measure position within a Cartesian grid?
  1. metes and bounds
  2. State Plane Coordinate System (SPC)
  3. Universal Transverse Mercator (UTM)
  4. Township and Range
  5. long lots

• Which of the following are true about the SPC?
  1. accuracy is 1 part in 10000
  2. the system is best used in regional and statewide GIS projects
  3. city governments resist using the SPC because of cost
  4. both A and B
  5. none of the above

7. Reference Materials

7.1. Print References

Bugayevskiy, Lev M. and John P. Snyder. 1995. Map Projections: A Reference Manual. London: Taylor and Francis.
This book contains a general exposition on map projection theory followed by sections on particular types of projection. Projections are classified by those whose parallels are straight, in the shape of concentric circles, or in the shape on non-concentric circles. Other types map projections and current map projection research are discussed. This is an excellent resource especially when paired with Snyder's Map Projections 1987.

Clarke, Keith C. 1995. Analytical and Computer Cartography, 2nd ed. Englewood Cliffs, NJ: Prentice Hall.
This book contains descriptions of most coordinate systems used in GIS along with enough technical details (including source code examples and a diskette) to work out many coordinate system conversions including computer raster graphic transformations not included in many other books on map projections.

Defense Mapping Agency. 1977. The American Practical Navigator: Publication No. 9, Defense Mapping Agency Hydrographic Center.
A venerable reference work containing many practical details for using maps and navigation systems. While primarily useful for working with nautical charts, the book contains sections on numerous navigation aids, from sextants to GPS.

Defense Mapping Agency. 1991. World Geodetic System 1984 (WGS 84) - Its Definition and Relationships with Local Geodetic Systems, 2nd Edition. Washington, DC: Defense Mapping Agency (DoD).
The primary source for WGS-84 information, including lists of reference ellipsoids, geodetic datums, and the simple three-parameter datum shift values required for datum transformation approximations.

Laurila, Simo H. 1976. Electronic Surveying and Navigation. New York: John Wiley & Sons.
An excellent source for geodetic formulas, including details on latitude, longitude, height systems, rectangular coordinate systems and ellipsoidal geodesics. The book, while somewhat dated now, provides a good background on many surveying and navigation systems in use today.

Muehrcke, P.C and Juliana O. Muehrcke. 1992. Map Use. Madison, WI: JP Publications.
While not a technical manual for mapping transformations, the book has very clear descriptions of most coordinate systems as well as discussions of many more detailed GIS issues relating to terrain surfaces and statistical evaluations.

Maling, D.H. 1992. Coordinate systems and map projections.  2nd ed. New York: Pergamon Press.
A reference manual containing algorithms and formulas for conversion between different coordinate systems and map projections.

Robinson, Arthur H., Joel L. Morrison, Phillip C. Muehrcke, A. Jon Kimerling, and Stephen C. Guptill. 1995. Elements of Cartography. 6th ed. New York: John Wiley and Sons, 41-58, 91-111.
A book that has served as the basis for cartography courses for more than 40 years. An indispensable reference book covering all phases of map making and map reading.

Snyder, John P. 1987. Map Projections: A Working Manual. Washington, DC: US Government Printing Office.
The best single reference for details on map projection methods, the book includes numerical examples for help in producing map projection code.

US Army. 1967. TM 5-241-1 Grids and Grid References. Washington, DC: Department of the Army.
A complete description of MGRS and UTM, including maps of the world with the MGRS preferred "spheroids" and MGRS row offsets. This old edition is out of print and does not contain WGS-84-based MGRS details.

7.2. Web References

7.2.1. US Federal Agencies

• US Army Corps of Engineers, http://www.tec.army.mil/
  • maintains the CORPSCON program which is described at http://crunch.tec.army.mil/software/corpscon/corpscon.html
• US Geological Survey, http://www.usgs.gov/
  • EROS Data Center, http://edcwww.cr.usgs.gov/
  • National Mapping Division, http://www-nmd.usgs.gov/
  • Global Land Information System, http://edcwww.cr.usgs.gov/webglis/
• US National Geodetic Survey, http://www.ngs.noaa.gov/
  • Geodetic Control Subcommittee, http://www.ngs.noaa.gov/FGCS/index.htm
  • a set of links to programs for geodetic adjustments and coordinate system translations is maintained by the NGS  at http://www.ngs.noaa.gov/PC_PROD/pc_prod.html
• US National Imagery and Mapping Agency (NIMA), http://www.nima.mil/

7.2.2. Non-US Federal Agencies

• Australian Surveying and Land Information Group (AUSLIG), http://www.auslig.gov.au/index.htm
  • AUSLIG Geodesy Division, http://www.auslig.gov.au/geodesy/newhome.htm
• British Geological Survey, http://www.bgs.ac.uk/
• Geomatics Canada/Géomatique Canada
  • National Atlas Information Service, http://www-nais.ccm.emr.ca/
  • Geodetic Survey of Canada/Division des levés géodésiques, http://www.geod.nrcan.gc.ca/
• Geographical Survey Institute of Japan, http://www.gsi-mc.go.jp/ENGLISH/index.html
• Institut Géographique National (France), http://www.ign.fr/
• National Survey and Cadastre (Denmark), http://www.kms.dk/index_en.html
• Ordnance Survey (United Kingdom), http://www.ordsvy.gov.uk/
• Ordnance Survey of Northern Ireland, http://www.osni.gov.uk/
• Sistema Nactional de Informação Geográphica (Portugal), http://www.cnig.pt/index_I.html
• Statens Kartverk (Norway), http://www.statkart.no/

7.2.3. Other relevant webpages

• The following are related sections by the same author in The Geographer's Craft Project at the University of Texas Austin:
  • Dana, Peter H. 1995. Geodetic Datum Overview, http://www.colorado.edu/geography/gcraft/notes/datum/datum_f.html
  • Dana, Peter H. 1995. Coordinate Systems Overview, http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html
  • Dana, Peter H. 1995. Map Projections Overview, http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html

Evaluation

Citation

Dana, Peter H. (1997) Coordinate Systems Overview, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u013/u013.html, posted (today).