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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

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  Refer Diagram page No.4

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.

Refer Diagram page No.5
ECEF X, Y, Z Coordinate Example
NAD-83 Lattitude N 30:16:28.82  Longitude W 97:44:25.19
X = -742507.1             Y = -5462738.5           Z = 3196706.5

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
NAD-83 Lattitude N 30:16:28.82  Longitude W 97:44:25.19
Zone 14R  Easting 621160.98 meters  Northing 3349893.53 meters

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
NAD-83 Lattitude N 30:16:28.82  Longitude W 97:44:25.19
NAD-83 Military Grid Reference 14RPU2116149894

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