Spherical Trigonometry and Navigational Calculations

Badar Abbas

MS(CE)-57

College of EME

Outline

BackgroundIntroductionHistoryNavigational TerminologySpherical TrigonometryNavigational CalculationsConclusion

Background

Introduction

NavigationLatin roots: navis (“ship”) and agere (“to

move or direct”)Coordinate System for Quantitative

Calculations (Latitude and Longitude)Spherical TrigonometryApplications (Navigation, Mapping, INS,

GPS and Astronomy)

History“Sphaerica” by Menelaus of AlexandriaIslamic 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 TrigonometryGreat 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 circlesGeodesic: 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 πEach 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 TrigonometryLet the sphere be of unit

radius.Z-axis = OAX-axis = OB projected into the

plane perpendicular to Z-axisFrom dot product rule:

)cos,sinsin,cos(sin)cos,0,(sincos

cos

bAbAbcca

OCOBa

Acbcba cossinsincoscoscos This gives

Spherical Trigonometry

The Laws of CosinesAcbcba cossinsincoscoscos

aCBCBA cossinsincoscoscos The Law of Sines

c

C

b

B

a

A

sin

sin

sin

sin

sin

sin

The Law of Tangets

]2/)tan[(

]2/)tan[(

]2/)tan[(

]2/)tan[(

cb

cb

CB

CB

Spherical TrigonometryGirard’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:-

ERA 2 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 TrigonometrySpherical 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.

Navigational Calculations

Distance and Bearing:Follows directly from Law of Cosines:-

)12cos()2cos()1cos()2sin()1sin(cos lonlonlatlatlatlata

Bearing can be calculated by:-

)12cos()2cos()1sin()2sin()1cos(cos

)12sin()2cos(sin

lonlonlatlatlatlatB

lonlonlatB

Then using two-argument inverse tan function:-

)cos,(sin2tan 1 BBB

Navigational CalculationsDead 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.

ConclusionSpherical 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