Spherical Triangle

A plane passing through the center of a sphere O cuts the surface in a circle, which is called a great circle. For any two points A and B on the sphere, if the line AB does not pass the center O, there is one and only one great circle which passes both points. The angle AOB, chosen as the one smaller than 180° or n in radians, is defined as the length of the arc AB. Given three points A, B, and C on the sphere, three great circles can be defined. The three arcs AB, BC, and CA, each less than 180° or n in radians, form a spherical triangle; see Fig. B.1.

Following standard notation, we denote the sides BC, CA, and AB by c, b, and a, respectively. The length of side a is defined as the angle E! OC, the length of side b is defined as the angle COA, and the length of side c is defined as the angle AOB. The vertex angles of the triangle are defined in a similar manner: The vertex angle A is defined as the angle between a straight line AD tangential to AB and another straight line AE tangential to AC, and so on.

Figure B.1 The spherical trian­gle. A great circle is defined by the intersection of a plane that cuts the sphere in two equal halves with the sphere. Given three points A, B, and C on the sphere, three great circles can be defined. The three arcs AB, BC, and CA, each less than 180° or n in rad, form a spherical triangle.

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