Copyright, 1998-2013 © Qiming Zhou

GEOG1150. Cartography

Earth-map RelationsEarth-map Relations

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

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

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

Ellipsoidal Earth (Cont.)

6378137

3.63567526378137

a

baf

257.2981

f

Equatorial Axis

Pola

r A

xis

North Pole

South Pole

Equator a

b

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.

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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

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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

Geodetic Latitude

P

E

N

W

S

Equator Radius

Pola

r R

adiu

s

Latitude Kilometres0 110.57

10 110.6120 110.7030 110.8540 111.0450 111.2360 111.4170 111.5680 111.6690 111.69

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

Length of a Degree of Longitude

cosDdLatitude Kilometres

0 111.3210 109.6420 104.6530 96.4940 85.3950 71.7060 55.8070 38.1980 19.3990 0.00

Where:

d = ground distance

D =ground distance at equator

= latitude

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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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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.

cos2 RC

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

coscoscossinsincos babaD

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

cotsincsctancoscot abaZ Where

Z = the true azimuth

a and b = latitudes at A and B

= the absolute value of the difference in longitude between A and BNote:

sin

1csc

tan

1cot

Earth-map Relations 28

The Great Circle RouteTwo 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

Computing the Surface Area of a Quadrilateral

LowerLatitude Area (km2)

0 1,224,48010 1,188,52820 1,117,35930 1,011,48040 875,13850 711,51060 525,31270 322,19580 108,584

baRS sinsin2

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)

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