2. Spherical Trigonometry and Navigational Calculations
Badar Abbas
MS(CE)-57
College of EME

3. Outline Background Introduction History Navigational Terminology
Spherical Trigonometry
Navigational Calculations
Conclusion

4. Background

5. Introduction Navigation
Latin roots: navis (“ship”) and agere (“to move or direct”)
Coordinate System for Quantitative Calculations (Latitude and Longitude)
Spherical Trigonometry
Applications (Navigation, Mapping, INS, GPS and Astronomy)

6. History “Sphaerica” by Menelaus of Alexandria
Islamic Period (8th to 14th Century )
Abu al-Wafa al-Buzjani in 10th century (Angle addition identities and Law of Sines).
“The Book of Unknown Arcs of a Sphere” by Al-Jayyani (1060 AD).
Nasir al-Din al Tusi and al-Battani in 13th Century.
John Napier (Logarithms)

7. Navigational Terminology
Earth (Flattened Sphere or Spheroid)
6336 km at the equator and 6399 km at the poles.
Flattening ( (a-b)/a)
GPS Calculations (WGS-84) uses:-
Flattening = 1/
a = km
6370 km radius gives an error of up to about 0.5%.

8. Navigational Terminology
Two Angles Required
Degrees in geographic usage, radians in calculations
Latitude: The angle at the center of the Earth between the plane of the equator and a line through the center passing through the surface at the point.
North Pole: (+90° or 90° N).
South Pole: (- 90° or 90° S).
Parallels: Lines of constant latitude.

9. Navigational Terminology
Longitude: The angle at the center of the planet between two planes passing through the center and perpendicular to the plane of the Equator. One plane passes through the surface point in question, and the other plane is the prime meridian (0º longitude).
Range: -180º(180º W) to + 180º(180º E).
Meridians: Lines of constant longitude.
All meridians converge at poles.

11. Navigational Terminology
Azimuth/Bearing/True Course: The angle a line makes with a meridian, taken clockwise from north.
North=0°, East=90°, South=180°, West=270°
Rhumb Line: The curve that crosses each meridian at the same angle.
More distance, but is easier to navigate.
Complicated calculations.

13. Spherical Trigonometry
Great and Small Circles: A section of a sphere by a plane passing through the center is great circle. Other circles are called small circles.
All meridians are great circles
All parallels, with the exception of the equator, are small circles
Geodesic: The smaller arc of the great circle through two given points.
The shortest distance.
The “lines” in spherical trigonometry

14. Spherical Trigonometry
Spherical Triangle
Vertices
Sides (a, b and c)
Angles less than π
Each side correspond to a geodesic.
1 nm = 1 min of lat
Angles (A, B and C)
Each less than π.
If one point is North Pole, other angles give azimuth.

15. Spherical Trigonometry
Let the sphere be of unit radius.
Z-axis = OA
X-axis = OB projected into the plane perpendicular to Z-axis
From dot product rule:
This gives

16. Spherical Trigonometry
The Laws of Cosines
The Law of Sines
The Law of Tangets

17. Spherical Trigonometry
Girard’s Theorem
The sum of the angles is between π and 3π radians (180º and 540º).
The spherical excess (E) is:-
E = A + B + C – π
Then the area (A) with radius R is:-
In the Fig all angles are π/2, so E is also π/2. The area (A) is then πR2/2, which is 1/8 of the area of the Sphere (4πR2).

18. Spherical Trigonometry
Spherical geometry is a simplest model of elliptic geometry.
Elliptic geometry is one of the two forms of non-Euclidean geometry.
It is inconsistent with the famous “parallel postulate” of Euclid.
In elliptic geometry two distinct lines are never parallel and triangle sum is always greater than 180.
In the other form (hyperbolic) two distinct lines are always parallel and triangle sum is always less than 180.

19. Navigational Calculations
Distance and Bearing:
Follows directly from Law of Cosines:-
Bearing can be calculated by:-
Then using two-argument inverse tan function:-

20. Navigational Calculations
Dead Reckoning
Dead reckoning (DR) is the process of estimating one's current position based upon a previously determined position.
In studies of animal navigation, dead reckoning is more commonly known as path integration.
The algorithm to compute the position of the destination if the distance and azimuth from previous position is known is given in the paper.
Some links for software implementations can be found in the paper. The distance, reckon and dreckon function in MATLAB are also helpful.

21. Conclusion
Spherical trigonometry is a prerequisite for good understanding of navigation, astronomy, GPS, INS and GIS.
For the most accurate navigation and map projection calculation, ellipsoidal forms of the equations are used.
It is much more pertinent to integrate course of spherical trigonometry in the engineering curriculum.

22. Thank You