Earth-map Relations

1. Earth-map Relations

2. Earth-map Relations • 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: http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/toc.htm. Earth-map Relations

3. 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 0.0085in (0.218mm) scratch in the ball. • It would be smoother than any bowling ball yet made! Earth-map Relations

4. 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. Earth-map Relations

5. 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. Earth-map Relations

6. 7°12' = 1/50 circumference Thus: Circumference = 925 x 50 = 46250km (only 15% too large) ~ 925km Summer solstice Eratosthenes Measurement The geometrical relationships that Eratosthenes used to calculate the circumference of the earth. From Robinson, et al., 1995 Earth-map Relations

7. 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) = 298.257. Earth-map Relations

8. North Pole b Polar Axis a Equator Equatorial Axis South Pole Ellipsoidal Earth (Cont.) WGS 84 ellipsoid: a = 6,378,137mb = 6,356,752.3mequatorial diameter = 12,756.3kmpolar diameter = 12,713.5kmequatorial circumference = 40,075.1kmsurface area = 510,064,500km2 Earth-map Relations

9. 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. Earth-map Relations

10. Geoidal Earth (Cont.) Geoid surface (EGM-96 Geoid). (Source: http://www.geocities.ws/geodsci/geoidmaps.htm) Earth-map Relations

11. Spherical, Ellipsoidal and Geoidal Earth Source: http://instruct.uwo.ca/earth-sci/505/utms.htm Earth-map Relations

12. 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 Earth-map Relations

13. 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 Earth-map Relations

14. 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. Earth-map Relations

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

16. Latitude Kilometres 0 110.57 10 110.61 20 110.70 30 110.85 40 111.04 50 111.23 60 111.41 70 111.56 80 111.66 90 111.69 N P Polar Radius  W E Equator Radius S Geodetic Latitude Earth-map Relations

17. 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. Earth-map Relations

18. 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. Earth-map Relations

19. Latitude Kilometres 0 111.32 10 109.64 20 104.65 30 96.49 40 85.39 50 71.70 60 55.80 70 38.19 80 19.39 90 0.00 Length of a Degree of Longitude Where: d = ground distance D = ground distance at equator  = latitude Earth-map Relations

20. 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. Earth-map Relations

21. 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. Earth-map Relations

22. 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. Earth-map Relations

23. 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) Earth-map Relations

24. 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. Earth-map Relations

25. 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 Earth-map Relations

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

27. 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: Earth-map Relations

28. 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 Earth-map Relations

29. 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. Earth-map Relations

30. LowerLatitude Area (km2) 0 1,224,480 10 1,188,528 20 1,117,359 30 1,011,480 40 875,138 50 711,510 60 525,312 70 322,195 80 108,584 Computing the Surface Area of a Quadrilateral Where a and b = latitudes of the upper and lower bounding parallels  = difference in longitude between the bounding meridians (in radians) Right: Surface area of 10 x 10° quadrilaterals Earth-map Relations

31. 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 Earth-map Relations

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

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

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