1 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 180 o 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 180 o . 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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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
3
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