Transcript Presentation - Badar Abbas` Blog

Spherical Trigonometry and
Navigational Calculations
Badar Abbas
MS(CE)-57
College of EME
Outline
Background
Introduction
History
Navigational Terminology
Spherical Trigonometry
Navigational Calculations
Conclusion
Background
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)
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)
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 = 1/298.257222101
 a = 6378.137 km
6370 km radius gives an error of up to about
0.5%.
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.
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.
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.
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
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.
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:
cos a  OB  OC
cos a  (sin c,0, cos c)  (sin b cos A, sin b sin A, cos b)
This gives
cos a  cos b cos c  sin b sin c cos A
Spherical Trigonometry
The Laws of Cosines
cos a  cos b cos c  sin b sin c cos A
cos A   cos B cos C  sin B sin C cos a
The Law of Sines
sin A sin B sin C


sin a sin b sin c
The Law of Tangets
tan[( B  C ) / 2] tan[( b  c) / 2]

tan[( B  C ) / 2] tan[( b  c) / 2]
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:-
A  R2 E
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).
Spherical Trigonometry
Spherical geometry is a simplest model of elliptic
geometry.
Elliptic geometry is one of the two forms of nonEuclidean 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.
Navigational Calculations
Distance and Bearing:
Follows directly from Law of Cosines:cos a  sin( lat1) sin( lat 2)  cos(lat1) cos(lat 2) cos(lon 2  lon1)
Bearing can be calculated by:sin B  cos(lat 2) sin( lon 2  lon1)
cos B  cos(lat1) sin( lat 2)  sin( lat1) cos(lat 2) cos(lon 2  lon1)
Then using two-argument inverse tan function:-
B  tan 2 1 (sin B, cos B)
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.
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.
Thank You