Coordinate Systems and Projections

Coordinate Systems and Projections Class 5 GISG 110

Objectives • Coordinate systems overview • The Earth • Geodetic Datum • Projections

Coordinate systems overview • A reference system consisting of points, lines, and/or surfaces and a set of rules, used to define the positions of points in space in either two or three dimensions • Coordinates are used to identify locations on the earth's surface

Coordinate systems Two types • Geographic • Also known as Global reference system • Not uniform across the earth’s surface • Latitude/Longitude • Projected (Planar) • Measures of length and angle are uniform • Cartesian • Polar

Geographic coordinate system • A reference system that uses latitude and longitude to define the locations of points on the surface of a sphere or spheroid (earth) • Features store lat-long values • Defined by a datum, prime meridian, and angular unit • angular unit – the unit of measurement on a sphere or a spheroid, usually degrees

Geographic coordinates – Latitude and Longitude Latitude • The angular distance along a meridian north or south of the equator • Usually measured in degrees • Also called parallels (except equator) Longitude • The angular distance of a point on the earth’s surface east or west of the prime meridian (Greenwich) • Intersects the equator and pass through the north and south poles, also called meridians • Expressed in degrees, minutes, and seconds

Equator The parallel of reference that is equidistant from the poles and defines the origin of latitude values as 0 degrees north or south Prime Meridian Greenwich meridian Any line of longitude designated as 0 degrees east and west, to which all other meridians are referenced Equator and Prime Meridian

Point P Latitude of 30 degrees North Longitude 20 degrees West Most commonly used reference system for locating positions on the earth Latitude and Longitude

Parallels of latitude • Are all small circles, except for the equator • True east-west lines • Always parallel • Any two are always equal distances apart • An infinite number can be created • Parallels are related to the horizontal x-axes of the Cartesian coordinate system • Measured relative to equatorranges from -90° at South Pole to + 90° at North Pole

Meridians of longitude • Are halves of great circles, connecting one pole to the other • All run in a true north-south direction • Spaced farthest apart at the equator and converge to a point at the poles • An infinite number can be created on a globe • Meridians are similar to the vertical y-axes of the Cartesian coordinate system • Measured relative to prime meridianranges from -180° when traveling west and +180° traveling east

Degrees, minutes, seconds • Angular measurement must be used to specify location on the earth's surface • This is based on a sexagesimal scale • A circle has 360 degrees, 60 minutes per degree, and 60 seconds per minute • A point is expressed as 45° 33' 22" (45 degrees, 33 minutes, 22 seconds) • The earth rotates on its axis once every 24 hours, therefore • Any point moves through 360° a day, or 15° per hour

Great circles, small circles Great circle • A circle formed by passing a plane though the exact center of a sphere • Largest circle that can be drawn on a sphere's surface • An infinite number of great circles can be drawn on a sphere • Used to calculate distance between two points on a sphere Small circle • Produced by passing a plane through any part of the sphere other than the center

Great circle The plane forming great circles must pass through the exact center of the sphere Small Circle The plane forming small circles can be anywhere else Great circles, small circles

Figure of the Earth • “Figure of the Earth" was refined from flat-earth models to spherical models • Used for global exploration, navigation, mapping, and mathematical calculations • After late 1700s, measurements showed the earth was ellipsoidal in shape

Ellipsoid (spheroid) • The shape and size of a geographic coordinate systems surface is defined by a sphere or spheroid • Formed by rotating an ellipse about an axis • Required for precise distance measurement over long distances • Accounts for flattening of earth at poles

Reference ellipsoids defined by either Semi-major (equatorial radius) and semi-minor (polar radius) axes or Relationship between semi-major axis and flattening of the ellipsoid (eccentricity) Flattening – a measure of how much a spheroid differs from a sphere F = (a-b)/a Ellipsoid and flattening

Geodetic Datum • Spheroid approximates shape of earth • Datum defines position of spheroid relative to the center of earth • Provides a frame of reference for measuring locations on the surface of the earth • Defines the origin and orientation of latitude and longitude lines • Simplified definition: the way a coordinate system is linked to the physical Earth • Two types of datum • Earth-centered • Local

Earth-centered Has origin placed at earth’s currently known center of mass More accurate overall Used for remote sensing WGS84 (World Geodetic System 1984) datum Local Aligned so that it closely corresponds to earth’s surface for a particular area and can be more accurate for that particular area Used by local government agencies NAD27 (North American Datum 1927) Earth-centered and local datum

Earth-centered and local datum

Changes in datum/coordinate system • Whenever you change the datum, the coordinate values of your data will change • NAD83 • 117 12 57.75961 34 01 43.77884 • NAD27 • -117 12 54.61539 34 01 43.72995 • Difference • Longitude: three seconds • Latitude: 0.05 seconds

Referencing locations • Locations on the earth are referenced to the datum • Different datum have different coordinate values for the same location

Using wrong datum when interpreting latitude, longitude, and height values can result in coordinate values having position errors in three dimensions of up to one kilometer Shifts in Datums

Datums in use • Hundreds in use around the world • Two horizontal used almost exclusively in North America are NAD27 and NAD83 • NAD83 is used for United States marine, aviation, and topographic maps • San Diego datum of choice • Accuracy update of NAD83: High Accuracy Reference Network (HARN) or High Precision GPS Network (HPGN) • Washington State Plane North, NAD83 HARN, feet • GPS is based on the World Geodetic System 1984 (WGS-84)

NAD27 • Uses Clarke 1866 spheroid to represent shape of earth • Origin is a point on earth referred to as Meades Ranch Kansas • Many control points were calculated from observations taken in the 1800s • Calculations were done manually and in sections over many years • Errors vary from station to station

NAD83 • The origin for datum is the earth’s center of mass • Based on both earth and satellite observations, using GRS 1980 spheroid (almost identical to WGS 1984 spheroid) • WGS 1984 and NAD 1983 coordinate systems both earth centered • Compatible with GPS data • New data in the US is referenced to NAD83

Geographic coordinate system review • Datum (e.g., NAD83) • Prime meridian • Angular unit (measurement on a spheroid)

Projected Coordinate Systems

Projected coordinate system Also known as planar • A reference system used to locate x,y, and z positions of point, line, and area features in two or three dimensions • Features store x,y values • Defined by a geographic coordinate system, a map projection, and parameters needed by the map projection, and a linear unit of measurement • Two types • Cartesian coordinate system • Polar coordinate system

Cartesian coordinate system • A two-dimensional, planar coordinate system in which horizontal distance is measured along an x-axis and vertical distance is measured along a y-axis Each point on the plane is defined by an x,y coordinate Relative measures of distance, area, and direction are constant throughout the Cartesian coordinate plane

Two axes, at right angles to each other Coordinates identified as x and y x is horizontal and y vertical x is east, y is north (x,y) = (7,6) Cartesian coordinates simplified

Use distance from origin (r) and angle from fixed direction (q) Fixed direction is north and angle is measured clockwise from it Useful for measuring from fixed point (e.g., center of a city) Polar coordinates

Map projections • Map projections convert the Earth’s curved surface to a flat surface (paper) through a mathematical transformation • Three projection types (or developable surfaces) • Cylindrical Projection surface (Cylinder) • Secant Conic Projection (Cone) • Secant Planar Projection (Plane) • Each case, a sheet of paper is wrapped around the Earth and positions of objects are projected onto the paper

Planar (Planes) Eccentric circles formed Conic (Cones) Concentric circles formed Cylindrical (Cylinders) Parallels are straight Projection types (surfaces)

Cylindrical projection surface example Goal: Convert the earth’s curved surface to a flat piece of paper

Projection distortion Distortions make geographers SADD • Shape • Area • Distance • Direction

Projections designed to minimize distortion Conformal projections (shape) • Preserve local shape • Meridians (longitude) and parallels (latitude) must intersect at right angles • No map projection can preserve shapes of larger regions Equal area projections (area) • Preserve area of displayed features (as result, shape, angle, and scale are distorted) • Meridians and parallels may not intersect at right angles • In smaller regions, shapes are not obviously distorted so distinguishing equal area from conformal is difficult

Projections designed to minimize distortion Equidistant projections (distance) • Preserve distances between certain points • Most have one or more lines on map with same length (at map scale) as same line on the globe • No projection is equidistant to and from all points on a map True-direction (azimuthal) projections (direction) • Maintains some of the great circle arcs, giving directions of all points on the map correctly with respect to center • Some true-direction projections are also conformal, equal area, or equidistant

Types of projections and spatial attribute preserved *Type classified by spatial attribute they preserve

Supported map projections 65 total map projections • Examples from previous page: • Albers Equal Area Conic • Equidistant Conic • Lambert Azimuthal Equal Area • Lambert Conformal Conic • UTM (global)* • State Plane Coordinates (regional)* * Also a projected coordinate system, uses Cartesian coordinate system to specify locations

UTM • A grid-based method of specifying locations on the surface of the earth • Not a single map projection • Employs a series of sixty zones, each of which is based on a specifically defined Transverse Mercator projection (cylindrical and conformal)

UTM Universal Transverse Mercator (UTM) • Commonly used projected coordinate system that divides the globe into 60 zones, starting at -180 degrees longitude (each spanning 6 degrees of longitude) • Coordinates in meters • Has own central meridian

UTM • Provides georeferencing at high levels of precision for the entire globe • Established in 1936 by the International Union of Geodesy and Geophysics • Adopted by the US Army in 1947 • Adopted by many national and international mapping agencies, including NATO • Commonly used in topographic and thematic mapping, for referencing satellite imagery and as a basis for widely distributed spatial databases

UTM Advantages • UTM is frequently used • Consistent for the globe • Is a universal approach to accurate georeferencing Disadvantages • Rectangular grid superimposed on zones defined by meridians causes axes on adjacent zones to be skewed with respect to each other • Problems arise in working across zone boundaries • No simple mathematical relationship exists between coordinates of one zone and an adjacent zone • Full georeference requires the zone number, easting and northing (FYI) • For more information, visit wikipedia.org and look up UTM

State Plane Coordinates (SPCS) • Set of more than 100 geographic coordinate systems designed for specific regions of the United States • Each state contains one or more state plane zones, the boundaries of which usually follow county lines • Units are generally in feet

State Plane Coordinates (SPCS) • The system is widely used for geographic data by state and local governments • Each state's shape determines which projection is chosen to represent that state • Projections are chosen to minimize distortion over the state • A state may have 2 or more overlapping zones, each with its own projection system and grid • Each zone is based on either a Transverse Mercator projection or a Lambert conformal conic projection • The choice between the two map projections is based on the shape of the state and its zones

State Plane Coordinates Zones divided north-south and east-west based state’s shape Smaller states may be 1 zone, larger states have 6 or more

State Plane Coordinates • Lambert conformal conic projection is used for states that extent mostly east-west • Transverse Mercator projection for those that extend north-south • Oblique Mercator projection used for panhandle of Alaska • Each zone uses either NAD27 or NAD83 horizontal datum

State Plane Coordinates Advantages • May give a better representation than the UTM system for a state's area • Coordinates may be simpler than those of UTM Disadvantages • Not universal from state to state • Problems may arise at the boundaries of projections • For more information, visit wikipedia and look up State Plane Coordinate System

Spatial reference (putting it all together) Geographic Coordinate System (lat/long) Planar or Projected Coordinate System (x,y) Ellipsoid modelDatum references ellipsoid Units (feet or meters), zones, projection, & parameters Projected Data GIS ready

Lat/Long data Villages with number of boats Projected data (mapped) Coordinate system components X,Y data

Coordinate Systems and Projections