Download CHAPTER 2 COORDINATE SYSTEMS

CHAPTER 2
COORDINATE SYSTEMS
CE 316
2012
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2.0 Co-Ordinate Systems
2.1 RECTANGULAR (Cartesian) SYSTEMS
 Used to define positions and orientation
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4
D3
3
2
1
A
B
C
D
E
Descartes Cartesian co-ordinates are denoted as (X, Y),
or in this case (Numbers, Letters).
Rene Descartes – 17th C 12
French scientist and philosopher
2.1.1 Mine Coordinates – UTM,
Arbitrary, etc…
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2.2 Geographic Coordinates
 Used to define position and orientation, change in location or relative
distance between points on a sphere
 Based on latitude and longitude
 Equator: formed by the intersection of a plain bisection of the earth
at right angles to the axis of rotation
 Meridians: formed by the intersection of vertical planes passing
through the center of the earth and the sphere
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2.2 Geographic Coordinates
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2.2.1 Latitude
Measured in degrees north or south of the equator
0 to 90 North
0 to 90 South
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2.2.2 Longitude
Measured in degrees west or east of the Prime Meridian
The Meridian, which passes through Greenwich, England, is
known as the “Prime Meridian” for most of the world
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1899 Map
of Scotland
2.3 Introduction to Co-ordinate Systems
Simplification of the earth’s surfaces
Used for most forms of early navigation
Used for most land surveying applications
USA – Canada boundary
Provincial Boundaries
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2.4 History of the Radius of the Earth
Greek scholar Eratosthenese 220 B.C.
Keeper of the scrolls for Egypt - Alexandria
3 basic requirements for determining radius of the earth:
276-194 BC
1) Precisely measured N – S line
2) One of the baselines on the
Tropic of Cancer (summer solstice)
3) Length of noon shadow a
minimum at the other end of the
baseline
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2.4 History of the Radius of the Earth
(250,000 stadia)
(6,403,657 m)
Poseidonius 135 – 50 B.C.
Cearth = 24, 000 mi
R ~ 6,367,272 m
(R=6,147,316 m)
Greek philosopher Ptolemy Cearth = 18,000 mi
(R=4,610,430 m)
Columbus most likely used Ptolemy’s value
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2.5 Astronomical Coordinates
Used for navigating and early mapping
Positions determined on a celestial sphere
Celestial Equator – Plane made by Earth’s Equator on the celestial sphere
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2.5 Astronomic Coordinates
(Equatorial Coordinate System)
Sun’s center is directly over the earth’s equator
Ecliptic – Path made by the Earth travelling around the sun
Perihelion (Jan 4) and Aphelion (July 5) – Points on Ecliptic when the Earth is
farthest away from the Sun
Celestial Equator: The Vernal equinox and Autumnal equinox are points of
intersection between the ecliptic and the celestial equator. This plane also passes
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through the center of the sun
2.5.1 Dependant Astronomic Coordinates
Position of stars based on this method continually changing
Azimuth and Altitude
To Polaris
Azimuth = angle from the north
Zenith
Altitude h = angular position above the horizon
Saskatoon
Polaris
h
HORIZON
h≈f
h
N.P.
90-h
S.P.
Nadir
EARTH
Equator
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2.5.2 Independent Astronomic Coordinates
All stars are basically fixed in position on the celestial sphere
Upper Transit
Right ascension (R.A.) and declination (d)
Observer
Star
y
Hour
Angle
R.A.
Pole
Lower Transit
R.A. – measured in hours, minutes, and seconds westward along the
celestial equator from the vernal equinox to the intersection of the
meridian passing through the star. 1 hour = 15o
[A measure of time]
Declination – measured north (+) or south (-) from the celestial
equator along the meridian to the star
REF: http://www.physics.csbsju.edu/astro/SC1/SC1.01.html
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2.5.2 Independent Astronomic Coordinates
Values of R.A. and declination are provided in solar ephemeris or
Star Almanacs
Position of stars based on this method essentially fixed
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2.5.2 Independent Astronomic Coordinates
Independent position of the stars
http://www.astro.wisc.edu/~dolan/constellations/java/Cetus.html
Note: Sidereal Hour Angle = Distance measure eastward in degrees from vernal equinox.
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2.5.2 Independent Astronomic Coordinates
Meade 8” LX200 Telescope
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2.6 Relationship between Astronomic and
geographic coordinate systems
Stars are used to provide and determine latitude, longitude, and time
2.6.1 Latitude
Polaris
ZENITH
Saskatoon
h
HORIZON
S
T
E
N.P.
O
EARTH
S.P.
Equator
NADIR
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2.6.1 Latitude
Polaris
N.C.P.
d
North Celestial
Pole
U.C.
90 o- d
90 o- d
S.C.P.
L.C. Polaris
Measured altitude
of star “h”,
at L.C.
90-d
o
= h +( 90 - d)
Night horizon looking north
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2.6.2 Azimuth
Azimuth of reference line =
measured angle from Polaris
Polaris
W.E.
E.E.
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2.6.3 Time/Longitude
Solar Day – The interval between two
successive lower transits of the sun’s
center over the same meridian
360o Rotation in 24 hr.
15o per hour
Upper
Transit
Sun
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2.6.3 Time/Longitude
In 1759, an obsessed English clockmaker created a watch so
excruciatingly precise that he received a prize worth one million
dollars (U.S.). The prize
was awarded
because with this watch, the
Measuring
Time
clockmaker, John Harrison, had solved the problem of finding
Mogul Caliph in Jaipur, India
longitude at sea. But what does time have to do with knowing where
you are?
John Harrison (1693 – 1776)
H1
Sir Cloudesley Shovell’s fleet wrecked on the
Isles, 1707. Artist unknown.
http://www.rog.nmm.ac.uk/museum/harrison/harrison.html
Harrison's linked balance mechanism negates the
effects of motion of the clock.
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Animation © National Maritime Museum
2.6.3 Time/Longitude
Standard Time
• Sir Sandford Fleming , “The Father of Standard Time.”
• Proposed Standard Time – Prime Meridian Conference 1884.
• 1870’s 144 time zones in N.A.
• 12 mi E and 12 mi. W, 1 minute difference in time.
• Railways (i.e. Baltimore time)
• Irish travel guide – Slago – 5:35 pm, misprint, 5:35 a.m.
• Fleming was born on July 7, 1827, in Scotland, and emigrated to Quebec at the age of
17.
• In 1858 as the chief engineer of the Northwest Railway, he first proposed a railway to
the pacific.
• He was in charge of the initial survey for the first Canadian railway to span the
continent.
• Fleming also designed the first Canadian postage stamp. Issued in 1851, it cost 3
cents and depicted the beaver.
• Fleming took an active part in the intellectual and scientific life of Canada, throughout
his long career and received many honours.
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• He died in Halifax in 1915.
2.6.3 Time/Longitude
Time Zones
http://tycho.usno.navy.mil/tzones.htmlGMT
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2.6.3 Time/Longitude
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2.6.3 Time/Longitude
Greenwich Mean Time
Time at Greenwich, England – Sometimes
called Universal time (useful for
astronomers, astronauts and people dealing
with satellite data)
Daylight Savings Time
An Hour shift which is seasonally inserted
in some Time Zones.
Daylight savings time begins when clocks
are set from 2:00 a.m. to 3:00 a.m. on the
first Sunday in April and ends when clocks
are set from 3:00 a.m. to 2:00 a.m. on last
Sunday in October.
2007-U.S./Canada Second Sunday in
March – first Sunday in November
Europe – 2nd last Sunday in March – last Sunday in October @ 1:00 a.m. GMT
Note: Only Saudi Arabia uses local times because of religious considerations
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2.6.3 Time/Longitude
International Date Line
Imaginary line that separates
two calendar dates
Matter of convenience, no force
in international law
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2.6.3 Time/Longitude
Sidereal Time
The interval between two successive upper transits of the vernal
equinox over the same meridian
365.2422 Mean Solar Days per year
366.2422 Sidereal Days per year
One Sidereal Day = O.9973 Solar Days
= 23 hr. 56 m. 4.09 sec.
365.24 solar days
Upper Transit
Star
y
Hour
Angle
R.A.
Pole
Lower Transit
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2.7 Convergency Between Meridians
Correction required when trying to place a rectangular coordinate
system on a sphere
Assumptions: The earth is a sphere (R = 20,890,000 ft., = 6,367,272 m.)
CE = 40,006,749.88
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2.7.1 Angular Convergence
Definitions
q = angular convergence of meridians
f = latitude
d = distance between meridians
Dl = angular distance between meridians
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E
2.7.1 Angular Convergence
q = fn : ( D l and f )
P
q = CD / DE ( radians )
North
sin f = DO ' / DE
D
l = CD / DO ' ( radians )
q = ( CD / DE ) =
D l = d / R cos f
q = D l sin f
B’ B
DO ' Dl
DO ' / sin f
[ subst .
= D l sin f
into
.......... .......
O’
O
1 . 1)
1)
= ( d / R )(sin f / cos f ) = ( d / R )(tan f )
.. 1 . 2 )
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q ( rads ) = 1 . 571 x10 d tan f , for d in meters.
o
[ Note : 2p radians in 360 o , or 1rad = 57 . 2958 ]
q "=
360 deg.
2p rads
x
( 3600 " / deg .) d ( miles ) 5280 ft / mile
q " = 52 . 13 d tan f
(1/6,367,272) = 1.571E-7
tanf
20 ,890 , 000 ft
.......... .......... .......... .......... ..........
[Note : for q in " and d in miles ]
1 .3)
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2.7.1 Angular Convergence
Definitions
L = length of meridian between parallel of
latitudes
q = angular convergence between meridians at
latitude
s = distance along parallel between meridians
c = linear convergence along parallel
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2.7.1 Angular Convergence
Derivation for Linear Convergence
North
q=c
c
( Subst. into Eq. 1.2)
L
c s tan f
=
L
R
( sL tanf )
=
c
............. 1.4)
R
If d and L are expressed
in miles and R is the radius
q
North
Meridian
L
Mean Distance “s”
Parallel
q
of the earth in feet , then :
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=
c
s L tanf ................................... 1.5)
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2.8 Measurement of Distance on the
Earth’s Surface
Nautical Miles
N.P.
90 o
Earth
CE = 40,006,749.88 m
Equator
Origin of the meter
1 Nautical Mile = 1’ arc
S.P.
equal to one ten-millionth
of the distance from the
equator to the pole
measured on a meridian:
1 Nautical Mile = CE/(360x60)
1 Nautical Mile ≈ 6076.65 ft.= 1,852.16m [for CE = 40,006,749.88m]
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2.8 Measurement of Distance on the
Earth’s Surface
2.8 Measurement of Distance on the
Earth’s Surface
Spherical Trigonometry
B
Sin Law and Cosine Law
B
a
sin a sin b
sin c
=
=
.......... .................... 1.6)
sin A sin B sin C
O
c
c
a
b
cos a = cos b cos c + sin b sin c Cos A .... 1.7)
C
P’
C
A
P
b
M
A
Example
The latitude and longitude of Boston and Cape Town are
know to be:
Boston: f = 42o 21’ North and l = 71o 04’ West
Cape Town f= 33o 56’ South and l =18o 29’ East
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