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ELEMENTRY SPHERICAL GEOMETRY

Great circle:-

Great circle on a sphere is any circle which divides the sphere in to two equal

hemispheres. All other circles are small circles.

Properties:-

1. Shortest distance between two points is an arc of a great circle.

2. A great circle is uniquely determined by any two points not 180o apart.

Spherical triangles are constructed from arcs of great circles.

For any spherical triangle, the sum Σ of the rotation angles formed by intersecting arcs is

always greater than 180o. The quantity (Σ – 180

o) is known as spherical excess and is directly

proportional to the area of a spherical triangle. Consider the triangle formed by two meridians

and equator. Because the two angles at the equator are both right angles, the rotation angle at the

pole φ between two meridians is equal to spherical excess. The area of the spherical triangle is

proportional to φ in this case.

This theorem is true for any spherical triangle. Sphere is uniformly curved. Sphere has

total symmetry. Rules of plane geometry hold for any infinitesimally small region on the surface

of a sphere.

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1 steradian = 1sr = solid angle subtended by or enclosed by

unit area of sphere at its centre

4π sterradian is total solid angle of a sphere.

Fundamental figure is the spherical triangle :

For any spherical triangle,

Law of sines : sinθ/sinΘ = sinλ/sinΛ= sinφ/sinΦ

Law of cosines for sides : cos θ = cosλ cosφ +sinλ sinφ cosΘ

Law of cosines for angles: cos Θ = - cosΛ cosΦ+ sinΛ sinΦ cos θ

Right Spherical Triangle:

One of the rotation angles is 90o. It contains five components.

Example:- Isoceles Right spherical triangle

ψ= sin-1

[ (2/3)1/2

] = 54.7356o

(Can be used for testing the relations)

Now, consider the right spherical triangle with the components and their complements

(90o - [ ]) arranged in a circle.

Rotational angle

Arc length of side

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Note that the complements are used for the three components farthest from the right angle.

Napier’s Rule:

The following relationships hold for right spherical triangle between the five parameters

in the circle.

The sine of any component in the circle is equal to the product of either

(a) the tangent of adjacent component or

(b) the cosines of the opposite component

sinλ = tanφ cotΦ = cos(90-θ) cos(90-Λ) = sinθ sinΛ

sinφ = tanλ cotΛ = sinθ sinΦ

cosθ = cotΦ cotΛ = cosφ cosλ

cosΛ = tanφ cotθ = cosλ sinΦ

cos Φ = tanλ cotθ = cosφ sinΛ

Quadrants for the solutions (the quadrant in which the angle lies) are determined as follows:

(a) An oblique angle and the side opposite are in the same quadrant

(b) The hypotenuse ( side ‘θ’ in the diagram) is less than 90 degree if and only if φ and λ are

in the same quadrant and is more than 90 degree if and only if φ and λ are in different

quadrants.

Napier’s Rule for quadrantal Spherical Triangle:

A quadrantal spherical triangle is one having one side of 90 degree. If the five

components and their complements are arranged in a circle the quadrantal spherical triangle is as

shown below.

90o -θ

φ

90o - Λ

λ

90o - Φ

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The same rule applies leading to the results,

sinΛ= tanΦ cotφ = sinΘ sinλ

sinΦ = tanΛcotλ = sinΘ sinφ

cosΘ = - cotφ cotλ = - cosΦ cosΛ

cosλ = - tanΦ cotΘ= cosΛ sinφ

cos φ = - tanΛ cotΘ = cosΦ sinλ

Rules for defining quadrants are modified as follows:

(a) an oblique angle ( other than Θ, the angle opposite to the 90 degree side) and its opposite

side are in the same quadrant.

(b) Angle Θ( the angle opposite to the 90 degree side) is more than 90 degree if and only if λ

and φ are in the same quadrant, and less than 90 degree if and only if λ and φ are in the

different quadrants.

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Application of spherical geometry in V-Slit sensor:- (p219 WERTZ)

Applying Napier’s rule to right spherical triangle SBC,

tanβ = tanθo/sin(ωΔt)

sin(ωΔt) = tanθo tanβ

Using Napier’s rule,

sin φ = tan λ cot ᴧ = sin θ sin Φ

sin(ωΔt) = tan(90-β) cot(90-θo) = cotβ tan θo

tanβ = tan θo/ sin(ωΔt)

Example : t = 0.1sec, N=60rpm , θo = 30o ,

tanβ = tan θo/ sin(ωΔt)

ω = (60/60) x 2π rad/sec = 6.28318 rad/sec

ωΔt = 6.28318 x 0.1 = 6.28318 rad = 36o

tanβ = tan 30/ sin(36) = 0.98225 β=44.49o

θ

φ

λ

Φ

ᴧ

λ = 90-β

ᴧ = 90- θo

φ = ω ∆t

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Spacecraft – centered celestial sphere

Recall that the spacecraft attitude is its orientation relative to the sun, the Earth, or the

stars regardless of the distances to these various objects. To think in terms of direction only, it is

convenient to form a mental construct of a sphere of unit radius centered on the spacecraft, called

the spacecraft-centered celestial sphere.

Each spherical coordinate system has two poles diametrically opposite each other on the

celestial sphere and an equator, or great circle, half way between the poles. The great circles

through the poles and perpendicular to the equator are called meridians and the small circles a

fixed distance above or below the equator are called parallels.

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General Requirements:

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