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Copyright, © Qiming Zhou GEOG1150. Cartography Earth-map Relations

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2 The earth Cartographic use of the sphere, ellipsoid and geoid Geographical coordinates Properties of the graticule Geodetic position determination For details on the contents of this lecture please read "Geodesy for the Layman", available on the website: for the Layman

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Earth-map Relations3 The Earth The earth is a very smooth geometrical figure. Imagine the earth reduced to a “sea level” ball 10in (25.4cm) in diameter: Mt. Everest would be a 0.007in (0.176mm) bump, and. Mariana trench a in (0.218mm) scratch in the ball. It would be smoother than any bowling ball yet made!

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Earth-map Relations4 Spherical Earth People know that the earth is spherical more than 2000 years ago. Pythagoras (6 century B.C.): Humans must live on a body of the “perfect shape”. Aristotle (4 century B.C.): Sailing ships disappear from view hull first, mast last. Eratosthenes (Greek, 250 B.C.): First calculation of the spherical earth’s size. Authalic sphere: 6,371km radius, 40,030.2km circumference.

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Earth-map Relations5 Aristotle's Observation Aristotle noted that sailing ships always disappear from view hull first, mast last, rather than becoming ever smaller dots on the horizon of a flat earth.

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Earth-map Relations6 Eratosthenes Measurement Summer solstice ~ 925km 7°12' = 1/50 circumference Thus: Circumference = 925 x 50 = 46250km (only 15% too large) The geometrical relationships that Eratosthenes used to calculate the circumference of the earth. From Robinson, et al., 1995

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Earth-map Relations7 Ellipsoidal Earth Newton (1670) proposed that the earth would be flattened because of rotation. The polar flattening would be 1/300th of the equatorial radius. The actual flattening is about 21.5km. The amount of the polar flattening (WGS [world geodetic system] 84) =

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Earth-map Relations8 Ellipsoidal Earth (Cont.) Equatorial Axis Polar Axis North Pole South Pole Equator a b WGS 84 ellipsoid: a = 6,378,137m b = 6,356,752.3m equatorial diameter = 12,756.3km polar diameter = 12,713.5km equatorial circumference = 40,075.1km surface area = 510,064,500km 2

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Earth-map Relations9 Geoidal Earth Geoid (“earth like”): an sea level equipotential surface. Gravity is everywhere equal to its strength at mean sea level. The surface is irregular, not smooth (-104 ~ 75m). The direction of gravity is not everywhere towards the centre of the earth.

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Earth-map Relations10 Geoidal Earth (Cont.) Geoid surface (EGM-96 Geoid). (Source:

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Spherical, Ellipsoidal and Geoidal Earth Earth-map Relations11 Source:

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Earth-map Relations12 Cartographic Use of the Sphere, Ellipsoid and Geoid Authalic sphere: the reference surface for small-scale maps Differences between sphere and ellipsoid is negligible Ellipsoid sphere: the reference surface large-scale maps Geoid: the reference surface for ground surveyed horizontal and vertical positions

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Earth-map Relations13 Geographical Coordinates Geographical coordinate system employs latitude and longitude Traced back to Hipparchus of Rhodes (2 century B.C.) Latitude Also called parallels, north-south Longitude Also called meridians, east-west

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Earth-map Relations14 Latitude Authalic latitude: based on the spherical earth. The angle formed by a pair of lines extending from the equator to the centre of the earth. Geodetic latitude: based on the ellipsoid earth. The angle formed by a line from the equator toward the centre of the earth, and a second line perpendicular to the ellipsoid surface at one’s location.

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Earth-map Relations15 Authalic Latitude and Longitude Authalic latitude and longitude. From Robinson, et al., 1995

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Earth-map Relations16 Geodetic Latitude P E N W S Equator Radius Polar Radius LatitudeKilometres 0

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Earth-map Relations17 Longitude Longitude is associated with an infinite set of meridians, arranged perpendicularly to the parallels. No meridian has a natural basis for being the starting line. Prime meridian: meridian of the royal observatory at Greenwich. Universally agreed in 1884 at the international meridian conference in Washington D.C.

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Earth-map Relations18 Longitude (Cont.) The angle formed by a line going from the intersection of the prime meridian and the equator to the centre of the earth, and then back to the intersection of the equator and the “local” meridian passing through he position.

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Earth-map Relations19 Length of a Degree of Longitude LatitudeKilometres 0 0.00 Where: d =ground distance D =ground distance at equator =latitude

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Earth-map Relations20 Properties of the Graticule The imaginary network of parallels and meridians on the earth is called graticule, as is their projection onto a flat map. The properties of the graticule deal with distance, direction and area. Assume the earth to be spherical.

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Earth-map Relations21 Distance The equator is the only complete great circle in the graticule. All meridians are one half a great circle in length. All parallels other than the equator are called small circles.

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Earth-map Relations22 The Great Circle The great circle is the intersection between the earth surface and a plane that passes the centre of the earth. An arc of the great circle joining two points is the shortest course between them on the spherical earth.

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Earth-map Relations23 Great Circle Distance Calculation Great circle arc distance = D R Where D = angle of the great circle arc (in radians) a and b = latitudes at A and B =the absolute value of the difference in longitude between A and B R = the radius of the globe (6,371 km)

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Earth-map Relations24 Direction Directions on the earth are arbitrary. North-south: along any meridian. East-west: along any parallel. The two directions are everywhere perpendicular except at poles. True azimuth: clockwise angle the arc of the great circle makes with the meridian at the starting point. Constant azimuth (rhumb line or loxodrome): a line that intersects each meridian at the same angle.

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Earth-map Relations25 True Azimuth A great circle arc on the earth's graticule. Note that the great circle arc intersects each meridian at a different angle. From Robinson, et al., 1995

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Earth-map Relations26 Constant Azimuth A constant heading of 30° will trace out a loxodromic curve. From Robinson, et al., 1995

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Earth-map Relations27 Computing the True Azimuth Where Z = the true azimuth a and b =latitudes at A and B =the absolute value of the difference in longitude between A and B Note:

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Earth-map Relations28 The Great Circle Route Two maps showing the same great circle arcs (solid line) and rhumbs (dashed lines). Map A is a gnomonic map projection in which the great circle arc appears as a straight line, while the rhumbs appear as longer "loops". In Map B, a Mercator map projection, the representation ahs been reversed so that the rhumbs appear as straight lines, with the great circle "deformed" into a longer curve on the map. From Robinson, et al., 1995

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Earth-map Relations29 Area The surface area of quadrilaterals is the areas bounded by pairs of parallels and meridians on the sphere. East-west: equally spaced. North-south: decrease from equator to pole.

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Earth-map Relations30 Computing the Surface Area of a Quadrilateral Lower LatitudeArea (km 2 ) 0 1,224, 1,188, 1,117, 1,011, 875, 711, 525, 322, 108,584 Right: Surface area of 10 x 10° quadrilaterals Where a and b = latitudes of the upper and lower bounding parallels =difference in longitude between the bounding meridians (in radians)

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Earth-map Relations31 Geodetic Position Determination Geodetic latitude and longitude determination Latitude: observing Polaris and the sun Longitude: time difference Horizontal control networks Survey monument Order of accuracy Vertical control Bench mark

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Earth-map Relations32 Geodetic Latitude Determination Latitude determination through observation of Polaris (A) and the sun (B). From Robinson, et al., 1995

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Earth-map Relations33 Horizontal Control Networks Horizontal control network near Meades Ranch, Kansas. From Robinson, et al., 1995

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Earth-map Relations34 Vertical Control The relationship between ellipsoid height, geoid- ellipsoid height difference, and elevation. From Robinson, et al., 1995

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