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The surface area of the sphere is

The area of a lune bounded by two great circles whose inclination is © radians is

The area of a spherical triangle whose three rotation angles are ®2, and @3 is fi(=e, + 02+e3-ir (A-ll)

The area of a spherical polygon of n sides, where 0 is the sum of its rotation angles in radians, is ap=-®-(n-2}ir (A-12)

The area of a small circle of angular radius p is

The overlap area between two small circles of angular radii p and c, separated by a center-to-center distance, a, is

[ smpsma J

r cosp-cosccosa 1

r cosa-cosecosp l

Recall that area is measured on the curved surface.

AJ Right and Quadrantal Spherical Triangles

Example of an Exact Right Spherical Triangle. For testing formulas, the isosceles right spherical triangle shown in Fig. A-2 is convenient. The sides and angles shown are exact values.

Fig. A-2. Example of an Exact Right Spherical Triangle. * = arcsin^O «54.7356°«54°44'08".

Napier's Rules for Right Spherical Triangles. A right spherical triangle has five variable parts, as shown in Fig. A-3. If these components and their complements (complement of $=90° — 3>) are arranged in a circle, as illustrated in Fig. A-3, then die following relationships hold between the five components in the circle:

The sine of any component equals the product of either

1. The tangents of the adjacent components, or

2. The cosines of the opposite components

For example, sin X=tan<J>tan(90° -<&) = cos(90° - A)cos(90° - 9 )

Quadrants for the solutions are determined as follows:

1. An oblique angle and the side opposite are in the same quadrant

2. The hypotenuse (side 8) is less than 90 deg if and only if $ and X are in the same quadrant and more than 90 deg if and only if <f> and X are in different quadrants.

Note: Any two components in addition to the right angle completely determine the triangle, except that if the known components are an angle and its opposite side, then two distinct solutions are possible.

Fig. A-3. Diagram for Napier's Rules for Right Spherical Triangles. Note that the complements are used for the three components farthest from the right angle.

The following formulas can be derived from Napier's Rules for right spherical triangles:

Napier's Rules are discussed in Section 2.3. Proof of these rules can be found in most spherical geometry texts, such as those of Brink [1942]; Palmer, el a!., [1950]; or Smail [1952],

Napier's Rules for Quadrantal Spherical Triangles. A quadrantal spherical triangle is one having one side of 90 deg. If the five variable components of a quadrantal triangle are arranged in a circle, as shown in Fig. A-4, then Napier's Rules as quoted above apply to the relationships between the parts. (Note that the bottom component is minus the complement of 0.) The rules for defining quadrants are modified as follows:

1. An oblique angle (other than 0, the angle opposite the 90-deg side) and its opposite side are in the same quadrant.

2. Angle 0 (the angle opposite the 90-deg side) is more than 90 deg if and only if X and <j> are in the same quadrant and less than 90 deg if and only if X and 4> are in different quadrants.

90-deg side is minus the complement of 6.
The following formulas can be derived from Napier's Rules for quadrantal spherical triangles:

sin A = tan i> cot <l>=sin 0 sinX

(A-20)

sin = tan A cot X = sin 0 sin <¡>

( A-21 )

cos0= — cot£cotX= — cos $ cos A

(A-22)

cosX = — tan cot 0 = cos A sin £

(A-23)

cos$= -tan A cot© = cos 4» sin X

(A-24)

A3 Oblique Spherical Triangles

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