Lecture Notes On Spherical Trigo

Spherical Trigonometry
distances : angles : declination or latitude : sunset position : sundial
Definition : Spherical Trigonometry deals with triangles drawn on a sphere. History : The subject originated in the Islamic Caliphates of the Middle East, North Africa and Spain during the 8th to 14th centuries. It arose to solve an apparently simple problem: Which direction is Mecca? Application : Navigation Stellar map making Geographic map making Positions of sunrise and sunset Improvements to the sundial.
In the figure , a triangle, ABC, is drawn on a sphere
Each line of the triangle is a Great Circle.( These are circles drawn on a sphere with the same radius as the sphere. Great circles cover the shortest distance between two points.)
The capital letters (A, B, C) denote the angles between the great circle arcs of the triangle as measured on the surface of the sphere. The small letters (a, b, c) represent the lengths of the great circle arcs measured as angles from the centre of the sphere.
A spherical triangle, differs from a plane triangle in that the sum of the angles is more than 180 degrees.
Formulas : The Cosine Rule There is a Cosine Rule for spherical triangles:
Cos(a) = Cos(b) Cos(c) + Sin(b) Sin(c) Cos(A) Cos(b) = Cos(a) Cos(c) + Sin(a) Sin(c) Cos(B) Cos(c) = Cos(b) Cos(a) + Sin(b) Sin(a) Cos(C) Note : The Cosine Rule allows the length of one of the arcs of a spherical triangle to be evaluated if the other two arcs and the angle opposite the arc are known. The Sine Rule The Sine Rule for spherical triangles is: Sin(a) / Sin(A) = Sin(b) / Sin(B) = Sin(c) / Sin(C) Note : The Sine Rule can be used to find an angle if two sides and an angle are known OR to find a side if two angles and a side are known.
Distance Between Two Points

Definitions: The Cosine and Sine Rules can be used to solve the basic problems of navigation on the surface of the Earth.
In the figure above, the points B and C are two points on the surface of the Earth. We can define the following: Point A is the North Pole. The great circle joining points B and C is the shortest distance between them. The great circle (in blue) joining B' and C' is the Equator (Latitude 0). The great circle (red) joining ABB' is a line of Longitude. It is the Longitude of B. The great circle joining ACC' is another line of Longitude. It is the Longitude of C. The length of the great circle arc B'B is the Latitude of point B. The arc B'A is 90 (Equator to Pole).
The length of the great circle arc C'C is the Latitude of point C. The arc C'A is also 90. Latitude () is measured in degrees () measured from the Equator Northwards (marked N) or Southwards (S). The Latitude of the North Pole is 90N and the Latitude of the South Pole is 90S. Southern Latitudes are considered negative (by convention). The Equator is a natural line on the Earth as it represents the great circle bisecting the Earth's axis of rotation.
Longitude (L) is measured in degrees East (E) or West (W) of the Line of Longitude passing through Greenwich Observatory, in a suburb of London (UK). This is called the Prime Meridian of the Greenwich Meridian. It is not a natural line and has been chosen by convention. Using the Cosine Rule The Cosine Rule can be used to determine the distance between points B and C (the arc, a) as follows: Cos(a) = Cos(b) Cos(c) + Sin(b) Sin(c) Cos(A) where A is an angle measured in degrees. It is the difference in Longitude between points B and C. The great circle arc, b, is 90 minus the Latitude of C. This is called the Polar Distance.
The great circle arc, c, is 90 minus the Latitude of B. Spherical Triangle
A spherical triangle is specified as usual by its corner angles and its sides, but the sides are not given by their length, but by their arc angle.
A spherical triangle is defined when three planes pass through the surface of a sphere and through the sphere's center of volume
A spherical triangle has three surface angles and three central angles. In the figure, 'A', 'B', and 'C' label the surface angles while 'a', 'b', and 'c' label the central angles. The surface angles correspond to the angle at which two planes intersect each other. Note that the arcs are labeled by the central angles. In particular, notice that the arc opposite a surface angle is labeled with the
surface angle's LOWER case letter. For example, the arc opposite the surface angle 'A' is labeled by the lower case letter 'a'. This is just notation and need not be followed, but it helps in remembering the relations to follow.
Example Problem: 1.Find the distance between London (UK) and Baghdad (Iraq).
Solution : If we use the diagram below then B is London while C is Baghdad. From the table London has a Latitude of 51.30N and a Longitude of 0.10W. Baghdad has a Latitude of 33.20N and a Longitude of 44.26E.
We seek a, the angular distance along a great circle between London and Baghdad. This is the shortest distance between these two cities along the surface of the Earth. It is shown in red in the diagram. A (the difference in Longitude between London and Baghdad) is 44.26E + 0.10W = 44.36. The two cities lie on different sides of the Greenwich Meridian (just) so we add the Longitudes. b is the Polar Distance of Baghdad given by 90 minus the Latitude of Baghdad (90 - 33.20 = 56.80). c is is the Polar Distance of London: 90 minus the Latitude of London (90 - 51.30 = 38.70). By the Cosine Rule: Cos(a) = Cos(b) Cos(c) + Sin(b) Sin(c) Cos(A) Cos(a) = Cos(56.80) Cos(38.70) + Sin(56.80) Sin(38.70) Cos(44.36) which gives:
Cos(a) = 0.5476 0.7804 + 0.8368 0.6252 0.7150 Cos(a) = 0.4273 + 0.3741 = 0.8014 Therefore, the great circle arc joining London and Baghdad (a) is 36.74. The Earth's circumference is 40,074 km which is equivalent to 360. The value 36.74 is 36.74 / 360 of the Earth's circumference. Therefore the distance between London and Baghdad is given by 40,074 km 36.74 / 360 = 4089 km Cosine Rule With Latitudes For any angle, X, in degrees, Cos(90 - X) = Sin(X) and Sin(90 - X) = Cos(X) so we can re-write the Cosine Rule to use (the more readilly available) Latitudes instead of Polar Distances.
There are two forms of this rule depending on if the values of the two Longitudes. If the Longitudes are both on the same side of the Greenwich Meridian, (i.e both E or both W), the formula is given by: Cos(DistBC) = Sin(lC)Sin(lB) + Cos(lC)Cos(lB)Cos(LC - LB) If the Longitudes are on different sides of the Greenwich Meridian (i.e. One is E and the other is W), the formula is given by: Cos(DistBC) = Sin(lC)Sin(lB) + Cos(lC)Cos(lB)Cos(LC + LB) In either form: DistBC is the angular distance between B and C along an arc of a great circle. This is the shortest distance between the two points. lB is the Latitude of B (positive for N, negative for S). lC is the Latitude of C (positive for N, negative for S). LB is the Longitude of B. LC is the Longitude of C.
Example: Find the distance bewteen Chicago (USA) and Mexico City (Mexico). If we set B to Mexico City and C to Chicago in the diagram below and we use the Cosine Rule With Latitudes: Cos(DistBC) = Sin(lC)Sin(lB) + Cos(lC)Cos(lB)Cos(LC [+-] LB) DistBC is the required angular distance between Mexico City and Chicago. This is a on the diagram above. lB is the Latitude of Mexico City (Angle BB' = 19.25N). lC is the Latitude of Chicago (CC' = 41.50N). LB is the Longitude of Mexico City (99.10W). The great circle ABB' is this Longitude. LC is the Longitude of Chicago (87.45W). Great circle ACC'. The two Longitudes are both West of the Greenwich Meridian, so we use the following Cosine Rule With Latitudes: Cos(DistBC) = Sin(lC)Sin(lB) + Cos(lC)Cos(lB)Cos(LC - LB)
Putting in the values and taking the difference between the two Longitudes we have: Cos(DistBC) = Sin(41.50)Sin(19.25) + Cos(41.50)Cos(19.25)Cos(11.65) This gives: Cos(DistBC) = 0.6626 0.3297 + 0.7490 0.9441 0.9794 = 0.9110 Therefore: DistBC = 24.36 The distance between Chicago and Mexico City is given by: 40,074 km 24.36 / 360 = 2712 km

Lecture Notes On Spherical Trigo

Uploaded by

Copyright:

Available Formats

Lecture Notes On Spherical Trigo

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Lecture Notes On Spherical Trigo

Uploaded by

Copyright:

Available Formats

Spherical Trigonometry

distances : angles : declination or latitude : sunset position : sundial

In the figure , a triangle, ABC, is drawn on a sphere

Distance Between Two Points

The great circle arc, c, is 90 minus the Latitude of B. Spherical Triangle

You might also like