Longitude and Spherical Triangles

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Longitude and Spherical Triangles
Materials presented here have been excerpted from
numerous sources, and are intended strictly for
in-class lecture/discussions.
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A few quick notes

• Nautical mile
• One minute of Earths circumference at the
  Equator
  
• Earths circumference at the equator is 360
  degrees
  
• Which is 36060 21,600 minutes
• So, Earths circumference at the equator is
  21,600 nautical miles
  
• One knot one nm per hour
• One nm 1.15 land mile

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More notes
Q. What do a row of Bacardi bottles and a
loxodrome have in common?A. Both are rum (rhumb)
lines."

• Every meridian is perpendicular to the equator
• Hence, the ease with which you could construct a
  spherical triangle with two right angles
  
• So, if we travel along a fixed direction that is
  other than due east-west or north-south but at an
  angle
  
• The journey results in a spherical spiral
• Also called a loxodrome

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Measuring Angles on a Sphere

• Lines on a sphere are great circles
• intersection of sphere with a plane through the
  spheres center
  
• Can define the angle between two lines as the
  angle made by the two planes that create them
  
• the smaller of the two possible choices
• Important to figure out what a given map
  projection does to angles

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Spherical Triangles

• Formed when arcs of three great circles meet in
  pairs
  
• Any two sides together greater than the third
  side
  
• Sum of interior angles can be (strictly) between
  180 and 540
  
• Very small triangle will be almost flat, so have
  just over 180
  
• Very large triangle can have almost 540 degrees
• Turns out that the area is directly proportional
  to the angle excess (how much more than 180
  degrees its angles add up to)

Triangle PAB v. triangle PCD
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Lunes

• A wedge (shaped like an orange slice) made by two
  intersecting great circles
  
• Area of a lune is directly proportional to angle
  d
  
• (d/360)(4pR2)

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Area of a Spherical Triangle

• See book for nice picture
• For each angle d, we can consider the lune that
  contains that angle
  
• The lune is the triangle another surface
• The area of triangle another surface equals d
  4pR2/360
  
• So the (area of the three other surfaces 3
  times the area of the triangle) (sum of the
  angles) 4pR2/360
  
• The area of the 3 other surfaces and the triangle
  is a hemisphere 2pR2 180(4pR2/360)
  
• So area of triangle is
• (1/2)(sum of angles 180)4pR2/360
• Manifestation of how much curvature is captured
  within the triangle
  

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Angles determine a lot about triangles on a sphere

• Two spherical triangles with the same angle sum
  have the same area
  
• totally different from plane situation where all
  triangles have 180
  
• If two spherical triangles have the same angles,
  then theyre not just similar...
  
• they have the same side lengths
• theyre congruent!

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There are no ideal maps

• An ideal map from the sphere to the plane would
  preserve both geodesics and angles
  
• What would it do to a spherical triangle?
• Would take great circles (geodesics on sphere) to
  straight lines (geodesics on plane)
  
• So it would take a spherical triangle to a plane
  triangle, preserving all the angles
  
• But plane triangle has 180 and spherical has
  180!
  
• The sphere is curved, and any triangle captures
  some of that
  
• thus, cannot be flattened out totally

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The Navigation Problem

• The ancient question Where am I?
• Earth coordinates latitude and longitude
• Latitude can be determined by Sun angle
• What about longitude?

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Latitude

• Comparatively easy
• Can use Eratosthenes method
• measure how far off from directly overhead the
  sun is, when it is at its highest point in the
  sky (local solar noon)
  
• Similar techniques using other astronomical
  bodies
  
• Latitude angle from horizon to North Star

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It is all in your stars!
Stars and other constellations helped sailors to
figure out their position. The red arrow is point
ing to the North Star, which is also known as
Polaris.
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This is a quadrant. A sailor would see the
North Star along one edge, and where the string
fell would tell approximately the ships latitude.
A sailor could also use this astrolabe.
Lined it up so the sun shone through one hole
onto another, and the pointer would determine the
latitude.
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On Land One could observe Natural clocks

• Motion of the moon against the background of the
  stars
  
• Motions of the moons of Jupiter
• But these were hard to observe from a ship,
  although they could be observed from land

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Techniques for measuring longitude

• Find some astronomical events that recur with
  known regularity
  
• Tables compiled by Galileo, Cassini of motion of
  moons of Jupiter (lo, Europa, Ganymede, and
  Callisto)
  
• The moons would have eclipses at regular
  intervals
  
• Tabulate exactly what time these eclipses
  occurred on given days
  
• e.g., you have a table that says something like
  Io will have an eclipse at 700 PM on Jan. 22 in
  Paris (Its actually rather messier than that)
  
• You have a clock set to local time
• On Jan. 22, you look through the telescope and
  see the eclipse at 300 PM local time
  
• So you are 4 hours 60 west of Paris

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Calculating the longitude

• Use stars or the Sun
• But in addition to making observations the need
  to know the time for some location of known
  longitude
  
• local time alone is not enough
• The development of the chronometer
• To find longitude to within 0.5 degree requires a
  clock that loses or gains no more than 3
  seconds/day

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Longitude

• Much more challenging
• Requires a way to determine how far you are from
  a fixed meridian
  
• Essentially the same question as what time is it
  in Paris when its noon here?
  
• Earth rotates at constant velocity, once around
  every 24 hours
  
• 1 hour 360 /24 15 longitude difference
• Thus, need to be able to tell time at your
  location
  
• e.g., pendulum clock, measuring local noon

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The problem of finding longitude at sea

• To the middle of the 18th century, no mechanical
  clock would keep accurate time in a sea-tossed
  ship
  

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Longitude Problem

• No easy way to determine longitude
• On July 8, 1714 the Longitude Act established in
  England to solve the longitude problem

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Odd Solutions

• Anchor a series of ships across the ocean that
  would shoot off flares and guns at set times
  
• Telepathic connection between animals on ship and
  those ashore

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Calculating longitude

• Requirements
• Clock showing base meridian time
• Record base meridian time when local noon (use
  sextant)
  
• Calculate time difference (3 hrs)
• Earth rotates 360 degrees in 24 hours
• 15 degrees in one hour
• Three hour difference is equal to 3x15 degree
  difference in longitude (45 degrees)

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The Chronometer

• Moons of Jupiter were too hard to observe on a
  ship
  
• Jupiters moons still used in the 1800s
• Chronometers fragile for land expeditions
• If we could just set a clock to Paris local time,
  and carry it with us, then when we figure out
  local noon, we can see what time it is in Paris
  
• Hard part is the implementation
• Pendulum clocks are sensitive to being jostled
• Materials expand and contract due to temperature,
  humidity, etc.

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Harrisons chronometer

• John Harrison (1693-1776) invented clocks that
  would keep good time at sea

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Culmination of the Sun

• Set your chronometer to some known time, say
  Eastern Standard Time, before you set sail

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Local noon vs. time zones

• Local noon is different at every longitude on the
  earth
  
• Standardize time zones so its the same time in a
  longitude region
  
• Set by political agreement
• e.g., Newfoundland is -330 from standard
• All of China is in one time zone, even though it
  has about 60 of longitude (bigger than US)

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Greenwich Meridian and International Date Line

• Greenwich Meridian (longitude through the Royal
  Observatory in Greenwich, England) chosen as the
  prime meridian (0)
  
• You can be up to 180 East (ahead)
• Or up to 180 West (behind)
• On the other side of the world, the International
  Date Line is where the discontinuity is

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Three important time-related dates

• 1761
• John Harrison builds a marine chronometer with
  error less than 1/5th of a second per day.
  
• Makes measurement of longitude possible while at
  sea.
  
• 1884
• The demands for readable railroad schedules
  requires adoption of Standard Time and time
  zones.
  
• 1905
• Albert Einstein shows that time is affected by
  motion

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GPS Segments

• Space Segment the constellation of satellites
• Control Segment control the satellites
• User Segment users with receivers

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GPS Orbits
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GPS Position

• By knowing how far one is from three satellites
  one can ideally find their 3D coordinates
  
• To correct for clock errors one needs to receive
  four satellites
  
• Illustration