Can Maths Tell Us Where We Are? Chris Budd
50
Can Maths Tell Us Where We Are? Chris Budd
Transcript of Can Maths Tell Us Where We Are? Chris Budd
Slide 1Chris Budd
Ballad of Gresham College v. 26
The College will the whole world measure Which most impossible conclude, And Navigation make a pleasure By fynding out the Longitude. Every Tarpaulin shall then with ease, Sayle any ship to the Antipodes.
Anon (circa 1660)
The need to advance navigation has stimulated mathematics
Geometry is crucial!
Knowing where you are and finding where to go safely is a hugely important part of human civilisation
Early navigation: dead reckoning
Now: Electronic means such as GPS
Ptolemy
To know where you are you first need a map
Mappa Mundi
First Trig Point, Cold Ashby 1936
Modern surveying using GPS
Navigation by dead reckoning
Cartesian coordinates (x,y) Map reference eg. Bath Centre ST 7509 6492
Polar coordinates: Compass bearings and distance estimates
CJB: 660 Double paces = 1 km
1802-1871 The Great Trigonometrical Survey of India
Triangles were large and the curvature of the Earth became important
Study using spherical trigonometry
fr.chachatelier.pierre.LaTeXiT
fr.chachatelier.pierre.LaTeXiT
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Map projections are needed to represent the curved surface of the Earth on a flat surface
Problem: Hairy Ball Theorem of geometry says that this will always lead to some distortion at some point
All projections are a compromise between accuracy, convenience and politics!
Mercator Projection: 1569
Easy to construct
Rhumb lines are straight
Great circle lines are curved and look longer
Gall Peters projection: 1973
Winkel-Tripel Bartholemew Projection, 1921
Used by the Times. Good compromise, but not very useful for navigation
Forecasting the weather: Icosahedral grid [Met Office +CJB]
Celestial Navigation
Finding location on the Earth
Need to find Latitude and Longitude
Columbus used a combination of dead reckoning and crude celestial navigation
Calculation of Latitude
This is relatively easy provided that we can measure the height Z of the Sun above the Horizon at Noon
Declination d: Angle Sun makes relative to the Equator
Finding Longitude
Obtained the reputation of an impossible problem!
Tackled by Newton, several Astronomers Royal, and Gresham Professor Robert Hooke
British Government offered a prize of up to £20,000 (equivalent to £2.89 million in 2018) under the 1714 Longitude Act
Key to Longitude is finding time accurately
Earth rotates 360 degrees in 24 Hours
1 hour = 15 degrees of Longitude
4s = 1 minute of Longitude = 1 Nautical Mile nm
Should time be determined by celestial or mechanical means?
Celestial method 1: Moons of Jupiter
Moons of Jupiter give an accurate celestial clock
Could be used to determine Longitude accurately on land
Louis XIV seeking to make his country the world leader in science commissioned astronomers to use measurements of Io's eclipses to improve the map of France
It was the most accurate map ever produced up to that time, and it revealed distances were actually shorter than had been believed.
France was smaller than thought and Louis complained that he was losing more of his realm to astronomers than to his enemies
Celestial method 2: Lunar Distances
1750s: map-maker Tobias Mayer devised the lunar distance method for finding longitude at sea.
Sailors measured the angle between the moon and a star to establish the time in Greenwich, and then compared it with the local time on board ship.
This required very precise observations
English Astronomer Royal Nevil Maskelyne was a keen advocate of the lunar method
Big problem: Calculating the exact location of the moon in advance
Solution: Determine and solve the differential equations describing the motion of the moon relative to the stars.
Solved by the great mathematician Leonhard Euler who was awarded £800 by the Longitude Board
£3000 to Mayer’s widow
Mechanical method: John Harrison
H1: 1736 H4: 1761
• Have an assumed position AP on the Earths surface
• Measure the height Ho of a celestial object at a precise time from your location
• Determine the geographical position GP of the object from Epheremides table
(point on the Earths surface directly below the object)
• Use spherical trigonometry to find the height Hc = 90-a and azimuth Z = C of the celestial body relative to the AP
GP AP
b: 90 – Latitude of AP (known)
c: 90 – Latitude of GP (known)
Correct AP along azimuth line by comparing Ho and Hc
Fix position by taking several sights
Navigation in WW2
German approach: 1940
Initially very effective
But could only direct a small number of aircraft to one location and was easy to jam
[R V Jones]
Master and two Slave stations transmitted simultaneous radio pulses.
Aircraft navigator measured the time differences of the arrival of the pulses
TA: Master and Slave A
TB: Master and Slave B
DA = c TA = difference in distance of aircraft from Master to Slave A
Curves of constant difference in distance are hyperbolae
Or a rotation of this
Aircraft lies at the intersection of the two hyperbolae
Accuracy was about 1 mile at a distance of 300 miles
Used from 1941 onwards. Effective at the start, then less so due to jamming
Supplemented with the Oboe system for more precise navigation and bomb aiming
Hyperbolic systems were first introduced in WW1 to detect guns by sound ranging
Still in use today in the Loran system
21st Century: Global Positioning System GPS
Very precise navigational system also based on time difference measurements.
Uses (typically 5) satellites to find a position
Each satellite transmits two signals at 1227 MHz and 1575MHz
Signal gives precise time and location of satellite at transmission
At X = (x,y,z) on Earth’s surface the time of reception of signals from 5 visible satellites is recorded
Times for the transmission are: T1,T2,T3,T4,T5
Satellite positions are: X1,X2,X3,X4,X5
True time: T = TR + TO TO the receiver time offset.
Distance Di from receiver to satellite Si is
Di = c (T – Ti)
Four unknowns: Position (x,y,z) and time offset TO
Hence given Di for four satellites we can locate the receiver
Fifth satellite allows for the assessment and correction of any positioning errors.
The GPS positioning accuracy is dependent on a number of interacting factors.
Some of these are similar to those in Celestial Navigation and include errors in calculating the position of the satellite and in the timing of the clocks on both the satellite and the receiver.
As all errors get multiplied by the (very large speed of light), a high precision is required.
Whereas Harrison’s chronometer had to be accurate to seconds per day, the clock in a GPS satellite has to be accurate to micro seconds per day
Errors due to Einstein’s Special and General Theories of Relativity
Satellite is moving fast: Clocks slow down
Satellite is in a reduced gravitational field: Clocks speed up
Combined effect = 38 micro seconds per day (11km)
Corrected in advance
1. Propagation paths are not always straight
2. The speed of light isn’t always constant due to electrons in the Ionosphere
Corrected by sending two signals
[Cathryn Mitchell, Bath]
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Ballad of Gresham College v. 26
The College will the whole world measure Which most impossible conclude, And Navigation make a pleasure By fynding out the Longitude. Every Tarpaulin shall then with ease, Sayle any ship to the Antipodes.
Anon (circa 1660)
The need to advance navigation has stimulated mathematics
Geometry is crucial!
Knowing where you are and finding where to go safely is a hugely important part of human civilisation
Early navigation: dead reckoning
Now: Electronic means such as GPS
Ptolemy
To know where you are you first need a map
Mappa Mundi
First Trig Point, Cold Ashby 1936
Modern surveying using GPS
Navigation by dead reckoning
Cartesian coordinates (x,y) Map reference eg. Bath Centre ST 7509 6492
Polar coordinates: Compass bearings and distance estimates
CJB: 660 Double paces = 1 km
1802-1871 The Great Trigonometrical Survey of India
Triangles were large and the curvature of the Earth became important
Study using spherical trigonometry
fr.chachatelier.pierre.LaTeXiT
fr.chachatelier.pierre.LaTeXiT
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Map projections are needed to represent the curved surface of the Earth on a flat surface
Problem: Hairy Ball Theorem of geometry says that this will always lead to some distortion at some point
All projections are a compromise between accuracy, convenience and politics!
Mercator Projection: 1569
Easy to construct
Rhumb lines are straight
Great circle lines are curved and look longer
Gall Peters projection: 1973
Winkel-Tripel Bartholemew Projection, 1921
Used by the Times. Good compromise, but not very useful for navigation
Forecasting the weather: Icosahedral grid [Met Office +CJB]
Celestial Navigation
Finding location on the Earth
Need to find Latitude and Longitude
Columbus used a combination of dead reckoning and crude celestial navigation
Calculation of Latitude
This is relatively easy provided that we can measure the height Z of the Sun above the Horizon at Noon
Declination d: Angle Sun makes relative to the Equator
Finding Longitude
Obtained the reputation of an impossible problem!
Tackled by Newton, several Astronomers Royal, and Gresham Professor Robert Hooke
British Government offered a prize of up to £20,000 (equivalent to £2.89 million in 2018) under the 1714 Longitude Act
Key to Longitude is finding time accurately
Earth rotates 360 degrees in 24 Hours
1 hour = 15 degrees of Longitude
4s = 1 minute of Longitude = 1 Nautical Mile nm
Should time be determined by celestial or mechanical means?
Celestial method 1: Moons of Jupiter
Moons of Jupiter give an accurate celestial clock
Could be used to determine Longitude accurately on land
Louis XIV seeking to make his country the world leader in science commissioned astronomers to use measurements of Io's eclipses to improve the map of France
It was the most accurate map ever produced up to that time, and it revealed distances were actually shorter than had been believed.
France was smaller than thought and Louis complained that he was losing more of his realm to astronomers than to his enemies
Celestial method 2: Lunar Distances
1750s: map-maker Tobias Mayer devised the lunar distance method for finding longitude at sea.
Sailors measured the angle between the moon and a star to establish the time in Greenwich, and then compared it with the local time on board ship.
This required very precise observations
English Astronomer Royal Nevil Maskelyne was a keen advocate of the lunar method
Big problem: Calculating the exact location of the moon in advance
Solution: Determine and solve the differential equations describing the motion of the moon relative to the stars.
Solved by the great mathematician Leonhard Euler who was awarded £800 by the Longitude Board
£3000 to Mayer’s widow
Mechanical method: John Harrison
H1: 1736 H4: 1761
• Have an assumed position AP on the Earths surface
• Measure the height Ho of a celestial object at a precise time from your location
• Determine the geographical position GP of the object from Epheremides table
(point on the Earths surface directly below the object)
• Use spherical trigonometry to find the height Hc = 90-a and azimuth Z = C of the celestial body relative to the AP
GP AP
b: 90 – Latitude of AP (known)
c: 90 – Latitude of GP (known)
Correct AP along azimuth line by comparing Ho and Hc
Fix position by taking several sights
Navigation in WW2
German approach: 1940
Initially very effective
But could only direct a small number of aircraft to one location and was easy to jam
[R V Jones]
Master and two Slave stations transmitted simultaneous radio pulses.
Aircraft navigator measured the time differences of the arrival of the pulses
TA: Master and Slave A
TB: Master and Slave B
DA = c TA = difference in distance of aircraft from Master to Slave A
Curves of constant difference in distance are hyperbolae
Or a rotation of this
Aircraft lies at the intersection of the two hyperbolae
Accuracy was about 1 mile at a distance of 300 miles
Used from 1941 onwards. Effective at the start, then less so due to jamming
Supplemented with the Oboe system for more precise navigation and bomb aiming
Hyperbolic systems were first introduced in WW1 to detect guns by sound ranging
Still in use today in the Loran system
21st Century: Global Positioning System GPS
Very precise navigational system also based on time difference measurements.
Uses (typically 5) satellites to find a position
Each satellite transmits two signals at 1227 MHz and 1575MHz
Signal gives precise time and location of satellite at transmission
At X = (x,y,z) on Earth’s surface the time of reception of signals from 5 visible satellites is recorded
Times for the transmission are: T1,T2,T3,T4,T5
Satellite positions are: X1,X2,X3,X4,X5
True time: T = TR + TO TO the receiver time offset.
Distance Di from receiver to satellite Si is
Di = c (T – Ti)
Four unknowns: Position (x,y,z) and time offset TO
Hence given Di for four satellites we can locate the receiver
Fifth satellite allows for the assessment and correction of any positioning errors.
The GPS positioning accuracy is dependent on a number of interacting factors.
Some of these are similar to those in Celestial Navigation and include errors in calculating the position of the satellite and in the timing of the clocks on both the satellite and the receiver.
As all errors get multiplied by the (very large speed of light), a high precision is required.
Whereas Harrison’s chronometer had to be accurate to seconds per day, the clock in a GPS satellite has to be accurate to micro seconds per day
Errors due to Einstein’s Special and General Theories of Relativity
Satellite is moving fast: Clocks slow down
Satellite is in a reduced gravitational field: Clocks speed up
Combined effect = 38 micro seconds per day (11km)
Corrected in advance
1. Propagation paths are not always straight
2. The speed of light isn’t always constant due to electrons in the Ionosphere
Corrected by sending two signals
[Cathryn Mitchell, Bath]
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