The relations between sides and angles in plane trigonometry do not hold in spherical trigonometry. (In any infinitesimal region of the sphere, plane geometry is. applicable.) However, three fundamental relations do hold among the components of any spherical triangle. Using the notation of Fig. 2-8, these are: The law of sines, sinfl = sinX = sin<j> sin© sinA = sin® The law of cosines for sides, cos 9 = cos X cos $+sin X sin 4» cos ©
The law of cosines for angles, cos 0 = — cos A cos G> + sin A sin G> cos 9 (2-3)
The rules of spherical trigonometry are considerably simplified in the special cases of right and quadrantal triangles. A right spherical triangle is one in which one of the rotation angles is 90 deg. A quadrantal spherical triangle is one in which one of the sides is 90 deg. In both cases, the relations between the five remaining sides and angles are given by a set of rules developed by John Napier, a 16th-century Scottish mathematician. These are presented in Appendix A. Napier's Rules and the laws of sines and cosines are particularly useful in attitude analysis and frequently provide simpler, exact analytic expressions than are available from plane geometry approximations. Napier's Rules may be derived from the law of cosines or may be used to derive it.
As an illustration of the application of spherical geometry, consider the problem illustrated in Fig. 2-9. Here, five solid-angle Sun sensors (also called two-axis sensors; see Section 6.1), A-E, are arranged with the optical axis of one toward each pole of a suitably defined coordinate system and the three others equally spaced around the equator. The problem is to determine the maximum angle that a point on the sphere can be from the axis of the closest Sun sensor. By symmetry, one such farthest point must lie along the meridian half way between C and D. Further, it must lie along the meridian at a point, P, such that y, = y2=y3. (If, for example, y, were greater than y2, the point could be moved away from A and the distance from the nearest sensor would be increased.) Because y, = y2=y, the triangle APD is isosceles and, therefore, r, = r2=r. Because the sensors are symmetrically placed about A, T, = 360°/6=60°. Because ^ is 90 deg, we now know three components of spherical triangle APD and the quantity of interest, y, can be found by several methods. With no further analysis, we may go directly to Table A-l in Appendix A for a triangle with two angles and the included side known to obtain:
where S is defined by tana=tanr,cos^ (2-4)
In this case, the equation cannot be evaluated directly, since cos^ = 0 and tan^ = + oo.
Alternatively, we may apply the law of cosines for sides to APD to obtain cosTr^cos-ftCOS^ + sinyjsin^cosr, (2-5)
Because yi=*y2 and ^=90°, this reduces to cosy = sinycosr coty = cosr (2-6)
We may also create a right spherical triangle by constructing the perpendicular bisector of arc AD which will pass through P, since APD is an isosceles triangle
and must be symmetric. Then, from Napier's rules for right spherical triangles (Appendix A), we obtain for the hypotenuse, y2
Finally, we may also solve the problem by recognizing immediately that APD is a quadrantal triangle and using Napier's rules for quadrantal triangles (Appendix A) to obtain directly sin r,=tan r2tan(90° - y2)
Because r=60°, y=arccot(0.5)=63.42o. (This suggests why the field of view of solid angle Sun sensors is often approximately a circle of 64-deg radius.) This analysis remains valid for a Sun sensor at each pole and any number of sensors (greater than 1) uniformly distributed about the equator. Thus, with a total of six sensors, four are distributed along the equator, r=45°, and y=arccot^ =54.73°.
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