1. Earth-map Relations

2. Earth-map Relations The earth
Qiming Zhou
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:
Earth-map Relations
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 in (0.218mm) scratch in the ball.
It would be smoother than any bowling ball yet made!
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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.
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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.
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6. 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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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) =
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8. Ellipsoidal Earth (Cont.)
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 = 510,064,500km2
Equatorial Axis
Polar Axis
North Pole
South Pole
Equator a b
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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.
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10. Geoidal Earth (Cont.) Geoid surface (EGM-96 Geoid).
(Source:
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11. Spherical, Ellipsoidal and Geoidal Earth
Source:
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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
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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
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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.
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15. Authalic Latitude and Longitude
From Robinson, et al., 1995
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16. Geodetic Latitude Latitude Kilometres
0           P E N W S
Equator Radius
Polar Radius
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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.
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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.
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19. Length of a Degree of Longitude
Latitude Kilometres
0          0.00
Where:
d = ground distance
D = ground distance at equator
 = latitude
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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.
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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.
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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.
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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)
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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.
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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
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26. Constant Azimuth
A constant heading of 30° will trace out a loxodromic curve.
From Robinson, et al., 1995
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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:
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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
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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.
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30. Computing the Surface Area of a Quadrilateral
Lower Latitude Area (km2)
0 1,224,  1,188,  1,117,  1,011,  875,  711,  525,  322,  108,584
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
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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
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32. Geodetic Latitude Determination
Latitude determination through observation of Polaris (A) and the sun (B).
From Robinson, et al., 1995
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33. Horizontal Control Networks
Horizontal control network near Meades Ranch, Kansas.
From Robinson, et al., 1995
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34. Vertical Control
The relationship between ellipsoid height, geoid-ellipsoid height difference, and elevation.
From Robinson, et al., 1995
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