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Geoide Ssii-109 GEOIDE SSII-109 MAP DISTORTIONS MAP DISTORTION • No map projection maintains correct scale throughout • The intersection of any two lines on the Earth is represented on the map at the same or different angle • Map projections aim to either minimize scale, angular or area distortion TISSOT'S THEOREM At every point there are two orthogonal principal directions which are perpendicular on both the Earth (u) and the map (u') Figure (A) displays a point on the Earth and (B) a point on the projection TISSOT'S INDICATRIX • An infinitely small circle on the Earth will project as an infinitely small ellipse on any map projection (except conformal projections where it is a circle) • Major and minor axis of ellipse are directly related to the scale distortion and the maximum angular deformation SCALE DISTORTION • The ratio of an infinitesimal linear distance in any direction at any point on the projection and the corresponding infinitesimal linear distance on the Earth • h – scale distortion along the meridians • k – scale distortion along the parallels MAXIMUM ANGULAR DEFORMATION AND AREAL SCALE FACTOR • Maximum angular deformation, w is the biggest deviation from principle directions define on the Earth to the principle directions define on the map • Occurs in each of the four quadrants define by the principle directions of Tissot's indicatrix • Areal scale factor is the exaggeration of an infinitesimally small area at a particular point NOTES • Three tangent points are taken for a number of different azimuthal projection: (φ, λ)= (0°, 0°), (φ, λ)= (45° N, 0°), (φ, λ)= (80° N, 0°) • Each projection covers a ‘square’ on the Earth, with latitude and longitude increasing in increments of 1° • Ellipsoidal (WGS84) equations are used for the projections AZIMUTHAL PROJECTIONS • Direction or azimuth from the centre of the projection to every other point on the map is show correctly • Scale and distortions change only with the distance from the centre • One standard point (centre of projection) LAMBERT AZIMUTHAL EQUAL AREA • Equal area • Central meridian is a straight line on all aspects • All other meridians and parallels are complex curves • Not a perspective projection • Scale increases radially as the distance increases from the tangent point • Scale increases in the direction perpendicular to radii as the distance increases from the tangent point LAMBERT AZIMUTHAL EQUAL AREA Scale Distortion Along Meridians (h) Scale Distortion Along Parallels (k) Longitude (°) Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E Latitude (°) 2 W 1 W 0 1 E 2 E 2 N 1.0000 0.9999 0.9998 0.9999 1.0000 2 N 1.0000 1.0001 1.0002 1.0001 1.0000 1 N 1.0001 1.0000 1.0000 1.0000 1.0001 1 N 0.9999 1.0000 1.0000 1.0000 0.9999 0 1.0002 1.0000 1.0000 1.0000 1.0002 0 0.9998 1.0000 1.0000 1.0000 0.9998 1 S 1.0001 1.0000 1.0000 1.0000 1.0001 1 S 0.9999 1.0000 1.0000 1.0000 0.9999 2 S 1.0000 0.9999 0.9998 0.9999 1.0000 2 S 1.0000 1.0001 1.0002 1.0001 1.0000 Longitude (°) Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E Latitude (°) 2 W 1 W 0 1 E 2 E 47 N 0.9999 0.9998 0.9998 0.9998 0.9999 47 N 1.0001 1.0002 1.0002 1.0002 1.0001 46 N 1.0000 1.0000 0.9999 1.0000 1.0000 46 N 1.0000 1.0000 1.0001 1.0000 1.0000 45 N 1.0001 1.0000 1.0000 1.0000 1.0001 45 N 0.9999 1.0000 1.0000 1.0000 0.9999 44 N 1.0001 1.0000 1.0000 1.0000 1.0001 44 N 0.9999 1.0000 1.0000 1.0000 0.9999 43 N 1.0000 0.9999 0.9999 0.9999 1.0000 43 N 1.0000 1.0001 1.0001 1.0001 1.0000 Longitude (°) Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E Latitude (°) 2 W 1 W 0 1 E 2 E 82 N 0.9998 0.9998 0.9998 0.9998 0.9998 82 N 1.0002 1.0002 1.0002 1.0002 1.0002 81 N 1.0000 1.0000 1.0000 1.0000 1.0000 81 N 1.0000 1.0000 1.0000 1.0000 1.0000 80 N 1.0000 1.0000 1.0000 1.0000 1.0000 80 N 1.0000 1.0000 1.0000 1.0000 1.0000 79 N 1.0000 1.0000 1.0000 1.0000 1.0000 79 N 1.0000 1.0000 1.0000 1.0000 1.0000 78 N 0.9999 0.9999 0.9999 0.9999 0.9999 78 N 1.0001 1.0001 1.0001 1.0001 1.0001 LAMBERT AZIMUTHAL EQUAL AREA Angular Deformation (w) Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E 2 N 2' 5.65'' 1' 18.54'' 1' 2.83'' 1' 18.54'' 2' 5.65'' 1 N 1' 18.54'' 31.41'' 15.71'' 31.41'' 1' 18.54'' 0 1' 2.84'' 15.71'' 0 15.71'' 1' 2.84'' 1 S 1' 18.54'' 31.41'' 15.71'' 31.41'' 1' 18.54'' 2 S 2' 5.65'' 1' 18.54'' 1' 2.83'' 1' 18.54'' 2' 5.65'' Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E 47 N 1' 49.95'' 1' 26.74'' 1' 18.95'' 1' 26.74'' 1' 49.95'' 46 N 55.90'' 32.01'' 23.76'' 32.01'' 55.90'' 45 N 33.55'' 9.73'' 0 9.73'' 33.55'' 44 N 41.41'' 16.71'' 7.65'' 16.71'' 41.41'' 43 N 1' 20.23'' 55.20'' 46.72'' 55.20'' 1' 20.23'' Longitude (°) Latitude (°) 2 W 1 W 0 1 E 2 E 82 N 1' 9.91'' 1' 8.76'' 1' 8.37'' 1' 8.76'' 1' 49.95'' 81 N 20.22'' 18.91'' 18.48'' 18.91'' 55.90'' 80 N 2.14'' 0.68'' 0 0.68'' 33.55'' 79 N 15.08'' 13.48'' 12.94'' 13.48'' 41.41'' 78 N 59.60'' 57.87'' 57.30'' 57.87'' 1' 20.23'' LAMBERT AZIMUTHAL EQUAL AREA TISSOT’S INDICATRICES If |a – b| < 0.001 ellipse is shown in green colour otherwise it is red Tangent Point: (φ, λ)= (0°, 0°) LAMBERT AZIMUTHAL EQUAL AREA TISSOT’S INDICATRICES If |a – b| < 0.001 ellipse is shown in green colour otherwise it is red Tangent Point: (φ, λ)= (45° N, 0°) LAMBERT AZIMUTHAL EQUAL AREA TISSOT’S INDICATRICES If |a – b| < 0.001 ellipse is shown in green colour otherwise it is red Tangent Point: (φ, λ)= (80° N, 0°) LAMBERT AZIMUTHAL EQUAL AREA TISSOT’S INDICATRICES Semi major - axis (a) Longitude (°) Latitude (°) 5 W 4 W 3 W 2 W 1 W 0 1 E 2 E 3 E 4 E 5 E 5N 1.0019 1.0016 1.0013 1.0011 1.0010 1.0010 1.0010 1.0011 1.0013 1.0016 1.0019 4 N 1.0016 1.0012 1.0010 1.0008 1.0006 1.0006 1.0006 1.0008 1.0010 1.0012 1.0016 3 N 1.0013 1.0010 1.0007 1.0005 1.0004 1.0003 1.0004 1.0005 1.0007 1.0010 1.0013 2 N 1.0011 1.0008 1.0005 1.0003 1.0002 1.0002 1.0002 1.0003 1.0005 1.0008 1.0011 1 N 1.0010 1.0006 1.0004 1.0002 1.0001 1.0000 1.0001 1.0002 1.0004 1.0006 1.0010 0 1.0010 1.0006 1.0003 1.0002 1.0000 1.0000 1.0000 1.0002 1.0003 1.0006 1.0010 1 S 1.0010 1.0006 1.0004 1.0002 1.0001 1.0000 1.0001 1.0002 1.0004 1.0006 1.0010 2 S 1.0011 1.0008 1.0005 1.0003 1.0002 1.0002 1.0002 1.0003 1.0005 1.0008 1.0011 3 S 1.0013 1.0010 1.0007 1.0005 1.0004 1.0003 1.0004 1.0005 1.0007 1.0010 1.0013 4S 1.0016 1.0012 1.0010 1.0008 1.0006 1.0006 1.0006 1.0008 1.0010 1.0012 1.0016 5 S 1.0019 1.0016 1.0013 1.0011 1.0010 1.0010 1.0010 1.0011 1.0013 1.0016 1.0019 Longitude (°) Latitude (°) 5 W 4 W 3 W 2 W 1 W 0 1 E 2 E 3 E 4 E 5 E 50 N 1.0015 1.0013 1.0012 1.0011 1.0011 1.0011 1.0011 1.0011 1.0012 1.0013 1.0015 49 N 1.0011 1.0010 1.0008 1.0008 1.0007 1.0007 1.0007 1.0008 1.0008 1.0010 1.0011 48 N 1.0009 1.0007 1.0006 1.0005 1.0004 1.0004 1.0004 1.0005 1.0006 1.0007 1.0009 47 N 1.0007 1.0005 1.0004 1.0003 1.0002 1.0002 1.0002 1.0003 1.0004 1.0005 1.0007 46 N 1.0005 1.0004 1.0002 1.0001 1.0001 1.0001 1.0001 1.0001 1.0002 1.0004 1.0005 45 N 1.0005 1.0003 1.0002 1.0001 1.0000 1.0000 1.0000 1.0001 1.0002 1.0003 1.0005 44 N 1.0005 1.0003 1.0002 1.0001 1.0000 1.0000 1.0000 1.0001 1.0002 1.0003 1.0005 43 N 1.0006 1.0004 1.0003 1.0002 1.0001 1.0001 1.0001 1.0002 1.0003 1.0004 1.0006 42 N 1.0008 1.0006 1.0005 1.0004 1.0003 1.0003 1.0003 1.0004 1.0005 1.0006 1.0008 41 N 1.0010 1.0009 1.0007 1.0006 1.0006 1.0005 1.0006 1.0006 1.0007 1.0009 1.0010 40 N 1.0014 1.0012 1.0010 1.0009 1.0009 1.0009 1.0009 1.0009 1.0010 1.0012 1.0014 Longitude (°) Latitude (°) 5 W 4 W 3 W 2 W 1 W 0 1 E 2 E 3 E 4 E 5 E 85N 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 1.0010 84 N 1.0007 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0007 83 N 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 1.0004 82 N 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 1.0002 81 N 1.0001 1.0001 1.0001 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 1.0001 80 N 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 79 N 1.0001 1.0001 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 78 N 1.0002 1.0002 1.0002 1.0001 1.0001 1.0001 1.0001 1.0001 1.0002 1.0002 1.0002 77 N 1.0004 1.0003 1.0003 1.0003 1.0003 1.0003 1.0003 1.0003 1.0003 1.0003 1.0004 76 N 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 1.0006 75 N 1.0010 1.0009 1.0009 1.0009 1.0009 1.0009 1.0009 1.0009 1.0009 1.0009 1.0010 LAMBERT AZIMUTHAL EQUAL AREA TISSOT’S INDICATRICES Semi minor - axis (b) Longitude (°) Latitude (°) 5 W 4 W 3 W 2 W 1 W 0 1 E 2 E 3 E 4 E 5 E 5N 0.9981 0.9984 0.9987 0.9989 0.9990 0.9990 0.9990 0.9989 0.9987 0.9984 0.9981 4 N 0.9984 0.9988 0.9990 0.9992 0.9994 0.9994 0.9994 0.9992 0.9990 0.9988 0.9984 3 N 0.9987 0.9990 0.9993 0.9995 0.9996 0.9997 0.9996 0.9995 0.9993 0.9990 0.9987 2 N 0.9989 0.9992 0.9995 0.9997 0.9998 0.9998 0.9998 0.9997 0.9995 0.9992 0.9989 1 N 0.9990 0.9994 0.9996 0.9998 0.9999 1.0000 0.9999 0.9998 0.9996 0.9994 0.9990 0 0.9990 0.9994 0.9997 0.9998 1.0000 1.0000 1.0000 0.9998 0.9997 0.9994 0.9990 1 S 0.9990 0.9994 0.9996 0.9998 0.9999 1.0000 0.9999 0.9998 0.9996 0.9994 0.9990 2 S 0.9989 0.9992 0.9995 0.9997 0.9998 0.9998 0.9998 0.9997 0.9995 0.9992 0.9989 3 S 0.9987 0.9990 0.9993 0.9995 0.9996 0.9997 0.9996 0.9995 0.9993 0.9990 0.9987 4S 0.9984 0.9988 0.9990 0.9992 0.9994 0.9994 0.9994 0.9992 0.9990 0.9988 0.9984 5 S 0.9981 0.9984 0.9987 0.9989 0.9990 0.9990 0.9990 0.9989 0.9987 0.9984 0.9981 Longitude (°) Latitude (°) 5 W 4 W 3 W 2 W 1 W 0 1 E 2 E 3 E 4 E 5 E 50 N 0.9985 0.9987 0.9988 0.9989 0.9989 0.9990 0.9989 0.9989 0.9988 0.9987 0.9985 49 N 0.9989 0.9990 0.9992 0.9992 0.9993 0.9993 0.9993 0.9992 0.9992 0.9990 0.9989 48 N 0.9991 0.9993 0.9994 0.9995 0.9996 0.9996 0.9996 0.9995
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