Map projections and datums

Maps are flat flat Earth is curved

Map Distortion No map is as good as a globe. A map can show some of these features True direction or azimuth True angle True distance True area True shape But not all of them!

common frame of reference Coordinate System common frame of reference for all data on a map

GIS needs Coordinate Systems to: perform calculations relate one feature to another specify position in terms of distances and directions from fixed points, lines, and surfaces

Coordinate Systems Cartesian coordinate systems: perpendicular distances and directions from fixed axes define positions Polar coordinate systems: distance from a point of origin and an angle define positions

Each coordinate system uses a different model to map the Earth’s surface to a plane

GCS Geographic Coordinate Systems Degrees of latitude and longitude Spherical polar coordinate system “Real” distance varies

Spherical Coordinates Any point uniquely defined by angles passing through the center of the sphere Meridian  Equator 

The Graticule Map grid (lines of latitude and longitude) A transformation of Earth’s surface to a plane, cylinder or cone that is unfolded to a flat surface

Decimal Degrees 30º 30' 0" = 30.5º 42º 49' 50" = 42.83º 35º 45' 15" = ? 35.7541

Standard Geographic Features Parallels of latitude are drawn relative to the Equator and meridians of longitude are drawn relative to the central meridian.

Parallels of Latitude Equator Slicing the Earth into pieces

Measuring Parallels Give the slices values

Lines of Longitude Antimeridian A Meridian A Meridians of longitude slice the Earth relative the pole. Establish a way of slicing the Earth from pole to pole

Prime Meridian Establishes an orthogonal way of slicing the earth

Longitude North America Values of pole-to-pole slices

Earth Grid Comparing the parallels and the lines In some GISs, the longitude west of the Prime Meridian and the latitudes South of the Equator are negative. Comparing the parallels and the lines

Latitude and Longitude Combining the parallels and the lines

Grid for US What is wrong with this map? Parallels and Lines for US

Sphere vs. Ellipsoid Globes versus Earth The earth is almost a perfect sphere. The ellipticity of a sphere is 0.0, whereas that of the earth is 0.003353. Flattening of the earth occurs at the poles. And bulging occurs at the equator. The south pole is closer to the equator than the north pole.

Shape of the Earth Approximated by an ellipsoid Rotate an ellipse about its minor axis = earth’s axis of rotation Semi-major axis a = 6378 km Semi-minor axis b = 6356 km NP b a SP

Ellipsoids and Geoids The rotation of the earth generates a centrifugal force that causes the surface of the oceans to protrude more at the equator than at the poles. This causes the shape of the earth to be an ellipsoid or a spheroid, and not a sphere. The nonuniformity of the earth’s shape is described by the term geoid. The geoid is essentially an ellipsoid with a highly irregular surface; a geoid resembles a potato or pear.

The Ellipsoid The ellipsoid is an approximation of the Earth’s shape that does not account for variations caused by non-uniform density of the Earth. Examples Clarke 1866 Clarke 1880 GRS80 WGS60 WGS66 WGS72 WGS84 Danish

Satellite measurements have led to the use of geodetic datums WGS-84 (World Geodetic System) and GRS-1980 (Geodetic Reference System) as the best ellipsoids for the entire geoid.

The Geoid The maximum discrepancy between the geoid and the WGS-84 ellipsoid is 60 meters above and 100 meters below. Because the Earth’s radius is about 6,000,000 meters (~6350 km), the maximum error is one part in 100,000.

Geodetic Datums

Geodetic Datum Defined by the reference ellipsoid to which the geographic coordinate system is linked The degree of flattening f (or ellipticity, ablateness, or compression, or squashedness) f = (a - b)/a f = 1/294 to 1/300

Geodetic Datums A datum is a mathematical model Provide a smooth approximation of the Earth’s surface. Some Geodetic Datums WGS60 WGS66 Puerto Rico Indian 1975 Potsdam South American 1956 Tokyo Old Hawaiian European 1979 Bermuda 1957

Common U S Datums NAD27 North American Datum 1927 WGS84 World Geodetic System 1984 (based on NAD83)

Map Projections A map projection is a systematic arrangement of grids on a flat surface (map). Map projections are required because the earth is spheroid. One easy way to understand what map projections are is to visualize a light shining through the earth onto a surface, called a projection surface. Any representation of the earth will have some distortion, in shape, area, distance, or direction. The four basic properties of map projections are: Shape Area Distance Direction Maps are flat. Different projections produce different distortions. The characteristics of each projection make them useful for some purpose.

Making a Map Concept of the Light Source The light source can be taken as the center of the earth, producing a gnomonic projection. The point of tangency is a single point on the surface of the earth. Shape distortion increases with distance from the center. Directional measurements are accurate from the center. No line distances are accurate. Distortion of area measurements increase with distance from the center. The stereographic projection views the surface from one surface to the opposite surface. Local shape is accurate. True scale is at the center and so distortion of areas increases with distance from the center. From the center, directions are accurate. Local angles are accurate everywhere. Distance errors increase with distance from the center. The orthographic projection views the earth from an infinite point in space. Shapes are minimally distorted near the center, maximum distortion occurs near the edge. Area get smaller away from the center Directions are only true from the central point. Distances decrease away from the center. Concept of the Light Source

Projection Families Although there are many projections, they can be grouped into one of three families according to the projection surface. Universal Transverse Mercator is a system in which the earth is divided into 60 zones, each spanning six degrees of longitude. Each zone has its own central meridian. Therefore each zone spans 3 degrees west and 3 degrees east of its central meridian. The origin for each zone is the Equator and its central meridian. To eliminate negative coordinates, the projection alters the coordinate system at the origin. The value given to the central meridian has a false easting of 500,000, and a false northing of 0. For locations in the southern hemisphere, the origin is assigned a false easting of 500,000 and false northing of 10,000,000. The UTM system has accurate representation for small shapes and minimum distortion of larger shapes within each zone. Areas have minimal distortion within each zone. Directional angles in UTM are locally true. Distances along central meridians in UTM are constant, but at a scale factor of 0.9996 to reduce lateral distortion. Errors and distortion increase beyond one UTM zone.

Types of Projection Families An infinite number of projections can be produced by changing the orientation of the projection surface. There three basic types of projections: azimuthal, cylindrical, and conic. The normal orientation for cylindrical projections places the cylinder parallel to the polar axis. The cylinder is upright and tangent to the globe at the equator The normal orientation for conic projections has the cone tangent to parallel of interest and the axis of the cone is in line with the axis of the globe. Transverse projections have the projection surfaces at 90 degree angles to the normal projections. The surfaces onto which the earth’s surface can be projected can have an infinite number of orientations to the earth itself. Any surface at non-polar or non-transverse positions is called an oblique projection.

Standard Point/Line for Projection For regular or normal azimuthal projections, the orientation is tangent to the poles of the earth. A projection has either a point of tangency or a line of tangency. Distortion increases as you move away from the point or line of tangency. The only place where the map is true to scale is at the point or line of tangency.

Regular Azimuthal Azimuthal projections are the easiest to understand. In them, the grid network is projected onto a plane

Azimuthal Projections In a the azimuthal projection the scale changes dramatically across the United States. This is not a good choice for a map of the US.

Azimuthal Projections Shapes are distorted everywhere except at the center Distortion increases from center True directions can be plotted from the center outward Distances are accurate from the center point

Polyconic Projections A series of conic projections stacked together Have curved rather than straight meridians Not good choice for tiles across large areas

Albers Conic Equal Area Projections Good choice for mid-latitude regions of greater east-west than north-south extent Scale factor along two standard parallels is 1.0000 Scale is reduced between the two standard parallels and increased north or south of the two standard parallels

Equal Area Projections Projections that preserve area are called equivalent or equal area. Equal area projections are good for small scale maps (large areas) Examples: Mollweide and Goode Equal-area projections distort the shape of objects

Conformal Map Projections Projections that maintain local angles are called conformal. Conformal maps preserve angles Conformal maps show small features accurately but distort the shapes and areas of large regions Examples: Mercator, Lambert Conformal Conic

Conformal Map Projections The area of Greenland is approximately 1/8 that of South America. However on a Mercator map, Greenland and South America appear to have the same area. Greenland’s shape is distorted.

Map Projections For a tall area, extended in north-south direction, such as Idaho, you want longitude lines to show the least distortion. You may want to use a coordinate system based on the Transverse Mercator projection.

Map Projections For wide areas, extending in the east-west direction, such as Nebraska, you want latitude lines to show the least distortion. Use a coordinate system based on the Lambert Conformal Conic projection.

Map Projections For a large area that includes both hemispheres, such as North and South America, choose a projection like Mercator. For an area that is circular, use a normal planar (azimuthal) projection

The UTM System

Universal Transverse Mercator 1940s, US Army 120 zones (coordinate systems) to cover the whole world Based on the Transverse Mercator Projection Sixty zones, each six degrees wide

UTM Zones Zone 1 Longitude Start and End 180 W to 174 W Linear Units Meter False Easting 500,000 False Northing 0 Central Meridian 177 W Latitude of Origin Equator Scale of Central Meridian 0.9996

UTM Zones Zone 2 Longitude Start and End 174 W to 168 W Linear Unit Meter False Easting 500,000 False Northing 0 Central Meridian 171 W Latitude of Origin Equator Scale of Central Meridian 0.9996

UTM Zones Zone 13, Colorado, Nebraska Panhandle, etc. Longitude Start and End 108 W to 102 W Linear Unit Meter False Easting 500,000 False Northing 0 Central Meridian 105 W Latitude of Origin Equator