Map projections Rüdiger Gens Coordinate systems Geographic coordinates f a: semi-major axis b: semi-minor axis f: flattening = (a-b)/a b Expresses as a fraction 1/f = about 300 a Map projections Geographic Geodetic latitude latitude – geographical coordinates imply spherical Earth model – geodetic coordinates imply ellipsoidal Earth model GEOS 639 – InSAR and its applications (Fall 2006) 2 Sphere versus spheroid Source: ArcGIS help file y assumption that the earth is a sphere is possible for small-scale maps (smaller than Map projections 1:5000000) y to maintain accuracy for larger-scale maps (scales of 1: 1000000 or larger) a spheroid is necessary GEOS 639 – InSAR and its applications (Fall 2006) 3 Common Spheroids y Bessel 1841 y Clarke 1866, Clarke 1880 y GEM 6, GEM 10C y GRS 1967, GRS 1980 y International 1924, International 1967 Map projections y WGS 72, WGS 84 GEOS 639 – InSAR and its applications (Fall 2006) 4 Reference surfaces Source: R. Gens, UAF Map projections y three reference surfaces x topography x geoid x ellipsoid GEOS 639 – InSAR and its applications (Fall 2006) 5 Reference surfaces Source: R. Gens, UAF Map projections y topography represents the physical surface of the Earth GEOS 639 – InSAR and its applications (Fall 2006) 6 Reference surfaces Source: R. Gens, UAF Map projections y geoid defined as level surface of gravity field with best fit to mean sea level x maximum difference between geoid and mean sea level about 1 m GEOS 639 – InSAR and its applications (Fall 2006) 7 Reference surfaces Source: R. Gens, UAF Map projections y ellipsoid defines mathematical surface approximating the physical reality while simplifying the geometry GEOS 639 – InSAR and its applications (Fall 2006) 8 Datum y describes the relationship between a particular local ellipsoid and a global geodetic reference system (e.g. WGS84) y local datum defines the best fit to the Earth's surface for particular area (e.g. NAD27) Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 9 Common Datums y World Geodetic System 1972 (WGS 72) y World Geodetic System 1984 (WGS 84) y North American Datum 1927 (NAD 27) y North American Datum 1983 (NAD 83) y European Datum 1950 (ED 50) Map projections y South American Datum 1969 (SAD 69) GEOS 639 – InSAR and its applications (Fall 2006) 10 Datum y datum x describes the relationship between a particular local ellipsoid and a global geodetic reference system y coordinate system x shape and size given by the ellipsoid x position given by the fixing of the origin Map projections x fixing of the origin defines a datum GEOS 639 – InSAR and its applications (Fall 2006) 11 Geographic coordinate system y A point is referenced by its longitude and latitude values y Longitude and latitude are angles measured from the earth’s center to a point on the earth’s surface Source: ESRI Map projections Source: ESRI GEOS 639 – InSAR and its applications (Fall 2006) 12 Coordinate systems Cartesian coordinates y geodetic coordinates inappropriate for satellite y z imagery → cartesian P coordinates Map projections x GEOS 639 – InSAR and its applications (Fall 2006) 13 Map projections y problem of mapping three-dimensional coordinates related to a particular datum on a flat surface x maps are two-dimensional x impossible to convert spheroid into flat plane without distortions of shape, area, distance, or direction Map projections → map projections GEOS 639 – InSAR and its applications (Fall 2006) 14 Cylindrical projections y cylinder that has its entire circumference tangent to the Earth’s surface along a great circle (e.g. equator) Map projections GEOS 639 – InSAR and its applications (Fall 2006) 15 Cylindrical projections y The map projection has distorted the graticule (data near the poles is stretched) Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 16 So.. y Where would you use a cylindrical projection? y And why? At the equatorial regions Map projections As the projection is true at the equator and distortion increases towards the poles GEOS 639 – InSAR and its applications (Fall 2006) 17 Cylindrical: Examples y Mercator projection y Transverse Mercator projection y Oblique Mercator projection Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 18 Conic projections y Simplest conic projection is tangent to the surface along a small circle (called standard parallel) y The meridians are projected onto the conical surface, meeting at the apex y Parallel lines of latitude are Map projections projected onto the cone as rings. GEOS 639 – InSAR and its applications (Fall 2006) 19 Conic projections y further you get from the standard parallel, the more distortion increases. Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 20 Conic projections y secant projection: a more complex conic projection y contact the global surface at two locations y defined by two standard parallels y less overall distortion than a tangent projection Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 21 So.. y Where would you use a conic projection? y And why? In mid-latitudes Map projections Projection is true along some parallel between the poles and equator GEOS 639 – InSAR and its applications (Fall 2006) 22 Conic: Examples y Conic projection with two standard parallels y Lambert Conformal Conic projection (preserves angles) y Albers Conic Equal-Area projection (preserves areas) Map projections GEOS 639 – InSAR and its applications (Fall 2006) 23 Azimuthal (Planar) projections Map projections x projecting positions directly to a plane tangent to the Earth’s surface GEOS 639 – InSAR and its applications (Fall 2006) 24 Azimuthal (Planar) projections y Point of contact specifies the aspect and is the focus of the projection. y The focus is identified by a central longitude and a central latitude. y Possible aspects are polar, equatorial, and oblique Source: ESRI Map projections GEOS 639 – InSAR and its applications (Fall 2006) 25 So.. y Where would you use a azimuthal projection? y And why? In polar regions Map projections Projection is best at their center point and distortion is generally worst at the edges GEOS 639 – InSAR and its applications (Fall 2006) 26 Azimuthal (Planar) projections y Lambert Azimuthal Equal-Area projection y Stereographic (conformal) projection Map projections GEOS 639 – InSAR and its applications (Fall 2006) 27 Equidistant projections 1 kp 1 km = 1 Map projections Sphere Projection x scale factor along a meridian (line of equal longitude) is equal to 1 x shape and area of square are distorted GEOS 639 – InSAR and its applications (Fall 2006) 28 Equal-area projections 1 1 kp km Map projections Sphere Projection x equal areas are represented by the same map area regardless of where they occur GEOS 639 – InSAR and its applications (Fall 2006) 29 Conformal projections kp 1 km 1 Sphere Projection Map projections x angles on a conformal map are the same as measured on the Earth’s surface x meridians intersect parallels at right angles GEOS 639 – InSAR and its applications (Fall 2006) 30 Summary y map purpose x for distribution maps: equal area x for navigation: projections that show azimuths or angles properly y size of area x some projections are better suited for East-West Map projections extent, others for North-South x for small areas the projection is relatively unimportant x for large areas the projection is very important GEOS 639 – InSAR and its applications (Fall 2006) 31.