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1 Map Projection and Coordinates The shape of the earth Models Sphere with radius of ~6378 km Ellipsoid (or Spheroid) with equatorial radius (semi-major axis) of ~6378 km and polar radius (semi-minor axis) of ~6357 km The Shape of the Earth Difference of ~21 km usually expressed as flattening (f ) ratio of the ellipsoid: f = difference/major axis = ~1/300 for earth

2 The Shape of the Earth Spheroid is the result of rotating an ellipse around an axis a = Semimajor axis b = Semiminor axis X, Y, Z = Reference frame The Shape of the Earth Ellipsoid The Earth revolves easterly on its axis, generating centrifugal force, causing a bulge in the middle and flattening at the poles Geoid Representation of the Earth as an equigravitational surface Paul Bolstad, GIS Fundamentals Due to variations in gravity, the geoid does not follow the ellipsoid exactly (100 m deviation) Geoid height difference between ellipsoid and geoid (geoid undulation)

3 The Shape of the Earth Many attempts to measure the size and shape of the earth s spheroid. An estimate of ellipsoid allows calculation of the elevation of every point on earth including the sea level. A base reference for the adjustment of geodetic measurements is called datum. The Shape of the Earth Why different Ellipsoids Better methods for approximating the surface of the Earth Developed from different areas of the Earth Just an approximation

4 The Shape of the Earth We have to know datum and ellipsoid when using a GPS (Global Positioning System) receiver. In Thailand, a topographic map is based on Indian datum, and Everest Spheroid. Datum The ellipsoid is a mathematical model that describes the shape of the earth. A datum defines the position of the ellipsoid relative to the center of the earth Provides a frame of reference for measuring locations on the surface of the earth Aligns its spheroid to closely fit the earth s surface in a particular area ESRI, ARC/INFO Help

5 Datum Horizontal Datum - A base reference for a coordinate system. It includes the latitude and longitude and orientation of an initial point of origin and an ellipsoid that models the surface of the earth in the region of interest. Prime Meridian Z Y X Traditional Horizontal Datums Limitations to the Traditional Approach NAD 27 (Clarke Ellipsoid ) ED 50 (International Ellipsoid)

6 Datums Regional vs. Global Approach Global replaces regional datums with a common, accurate standard One system for maps of the entire planet Datums North American Datums NAD27 (North American Datum of 1927) NAD83 (North American Datum of 1983) World Datums WGS84 (World Geodetic System of 1984) Used for GPS Other Datums European Datum of 1979 Ordnance Survey Datum of Great Britain, 1936 Indian Datum 1975

7 Graticule: the pattern of meridians and parallels on the earth based on Equally spaced parallels between the equator and the poles Parallels are always parallel to one another, so they are always the same distance apart The Graticule Parallels and Meridians Meridians are also known as lines of longitude Example is the prime meridian Parallels are also known as lines of latitude Example is the equator Paul Bolstad, GIS Fundamentals

8 The Graticule Meridians are spaced farthest apart on the equator and converge to a single point at the poles Parallels and meridians cross one another at right (90 o ) angles Concept of a Projection A systematic rendering of a graticule Projection Surfaces Plane Developable Surfaces Cone Cylinder Different projections create different distortions ESRI, ArcView Help Paul Bolstad, GIS Fundamentals

9 Projections Conformal Projections Conformal projections preserve local shape. They show the perpendicular graticule lines intersecting at 90 degree angles However, an area may become more distorted to maintain the angles Equal Area Projections preserve the area of features. Other properties such as shape, angle, and scale are distorted. Equidistant projections preserve distances between certain points. Scale is not maintained correctly, however, typically one or more lines has its scale maintained. ESRI, Understanding Map Projections 9

10 Planar Projections Project map data onto a flat surface touching the globe Usually tangent to the globe, but may be secant Tangent point is focus of the projection Conic Projections Paul Bolstad, GIS Fundamentals Conic Projection - tangent to the globe along a line of latitude distortion increases away from the standard parallel Conic projections are used for mid-latitude zones that have an eastto-west orientation Somewhat more complex conic projections contact the global surface at two locations called secant conic projections

11 Cylindrical Projections Paul Bolstad, GIS Fundamentals Mercator is most common cylindrical projection Equator is typically the line of tangency Meridians are of equal space, lines of latitude increases toward poles Displays true direction along straight lines Transverse projections use meridian lines as tangent point, therefore, North/South lines are preserved Azimuthal Projections Gnomonic projection views the surface data from the center of the Earth, Stereographic projection views it from pole to pole. Orthographic projection views the Earth from an infinite point, as if viewed from deep space.

12 Major Properties of Projections Conformality retention of correct angles. Preserves shape, but not size. Also allows for accurate directions Equivalence retains unit area. Good for measuring area phenomena (amount of arable land). There are scale changes however, and changes in shape Projection Effects Conformal projections Mercator, Lambert conformal Equivalence Sinusoidal Equidistance Azimuthal

13 Compromise Projections Miller Cylindrical similar to Mercator, but distorts size somewhat less However, its not conformal, and lack the directional integrity Robinson minimizes visually disturbing distortions Different Projections Mercator projection conformality Sinusoidal equivalence Millers Cylindrical compromise Robinson - compromise

14 Basic Coordinate-System Concepts: The fundamental spherical coordinate system used to locate positions on the earth s surface. A way of representing the earth s spherical shape as a flat surface. A smooth approximation of the earth s irregular surface. What is a Coordinate System A coordinate system defines the location of a point on a planar or spherical surface A geographic coordinate system is a three dimensional reference system that locates points on the Earth s surface. The unit of measure is usually decimal degrees. A point has two coordinate values: latitude and longitude. Latitude and longitude measure angles.

15 Cartesian VS. Geographic Coordinate Systems Paul Bolstad, GIS Fundamentals Facts About Latitude Lines of latitude (parallels) are evenly spaced (small circles) from 0 o at equator (a great circle) to 90 o at poles. 60 nautical miles (~ 110 km)/1 o, ~1.8 km/minute and ~ 30 m/second of latitude. N. latitudes are positive (+φ) S. latitudes are negative (- φ).

16 Longitude (λ) Longitude is the angle (λ) between the plane of the prime meridian and the meridianal plane containing the point of interest (P). Facts About Longitude (λ) Lines of longitude (meridians) converge at the poles. The distance of a degree of longitude varies with latitude. Zero longitude is the Prime (Greenwich) Meridian (PM); longitude is measured from o east and west of the PM. East longitudes are positive (+λ), west longitudes are negative ( λ).

17 Example: Given the following DMS coordinate, 45 o convert it to DD. M S DD = D DD = = Going the other way DD trunc(dd) * 3600 = total seconds / 60 = total minutes. Total minutes - Trunc(total minutes) * 60 = seconds Coordinate Systems cont. A projected coordinate system is a twodimensional planar surface. However, the Earth s surface is three-dimensional. Transforming three-dimensional space onto a two-dimensional surface is called projection.

18 Coordinate Systems (cont.) Projection formulas are mathematical expressions that convert data from a geographical location (latitude and longitude) on a sphere or spheroid to a corresponding location (x and y) on a flat, two dimensional surface The projected coordinate system uses two axes: the x-axis, representing east-west, and the y-axis, representing north-south. They intersect at the origin, (0,0) Universal Transverse Mercator (UTM) Based on transverse Mercator projection Covers Earth surface between 80 o South and 84 o North 60 north-south zones 6 degrees of longitude wide

19 Universal Tranverse Mercator The UTM graticule coverage Each belt is 6 O in longitude wide 84 o N Equator 0 meters 10,000,000m 80 o S 180 o 0 o 180 o Projections UTM is an international metric coordinate system that covers the entire earth. It has the advantage of being mathematically consistent and well defined for the entire earth. Local coordinate systems are often used to fit mapping needs for a particular region.

20 Universal Transverse Mercator (UTM) Each zone overlaps ½ degree into the adjoining zones False origin of 500,000 meters west of the central meridian of each UTM zone Universal Transverse Mercator (UTM) Ex: UTM Zone 48N Projection :Transverse Mercator Ellipsoid : Everest Central Meridian: 105E Reference Latitude: 0 False Easting: 500,000 False Northing: 0 Scale Factor:.9996

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