Spherical Geometry

James R. Wertz

A.I Basic Equations

A.2 Right and Quadras ta) Spherical Triangles A.3 Obliqbe Spherical Triangles A.4 Differential Spherical Trigonometry A.S Haversines

Finding convenient reference material, in- spherical geometry is difficult. This appendix provides a compilation of the most useful equations for spacecraft work. A brief discussion of the basic concepts of spherical geometry is given in Section 23. The references at the end of this appendix contain further discussion and proofs of most of the results presented here.

A.l Basic Equations

Algebraic Formulas. Let Pt be a point on the unit sphere with coordinates (a,,Sj). The arc-length distance, 9(PVP^, between Pt and P2 is given by:

= si n 8, sin S2 + cos 8, cos \$2cos( a t — a2) 0< 0< 180° (A-l)

The rotation angle, h(PltP2,P3), from Pt to P2 about a third point, P3, is cumbersome to calculate and is most easily obtained from spherical triangles (Sections A.2 and A.3) if any of the triangle components are already known. To calculate directly from coordinates, obtain as intermediaries the arc-length distances 0(P,, Pj), between the three pairs of points. Then cos 9(P., P2) - cos 01 P., P3)cos9(P2, P3) --sin0(P„ P3)sin0(P2, P3)

with the quadrant determined by inspection.

The equation for a small circle of angular radius p and centered at (a^fig) in terms of the coordinates, (a,8), of the points on the small circle is, from Eq. (A-l), cosps=sinfisind0+cosficosS(/cos(a- a0) (A-3)

The arc length, /?, along the aic of a small circle of angular radius p between two points on the circle separated by the rotation angle, \$ (\$ measured at the center of the circle) is j8=\$sin p (A-4)

The chord length, y, along the great circle chord of an arc of a small circle of angular radius p is given'by cos y = I - (I - cos <&)sin2p 0 < y < 180" (A-5)

where \$ is as defined above.

The equation for a great circle with pole at (a^S^) is, from Eq. (A-3) with p=90°, tan 6 = - cot50cos(a - a0) (A-6a)

The inclination, /, and azimuth of the ascending node (point crossing the equator from south to north when moving along the great circle toward increasing azimuth), 4>q, of the great circle are i=90° —50

Therefore, the equation for the great circle in terms of inclination and ascending node is tan 6 = tan i sin(a - <f>0) (A-6c)

The equation of a great circle through two arbitrary points is given below. Along a great circle, the arc length, the chord length, and the rotation angle, 4>, are all equal, as shown by Eqs. (A-4) and (A-5) with p=90°.

Finally, the direction of the cross product between two unit vectors associated with points P, and P2 on the unit sphere is the pole of the great circle passing through the two points. Find the intermediary, /?„ from

of the great circle through /», and P2. The coordinates, (ac,8c), of the cross product P, x P2 are given by

Combining Eqs. (A-7b) and (A-6a) gives the equation for a great circle through points P, and P2:

Area Formulas. All areas are measured on the curved surface of the unit sphere. For a sphere of radius R, multiply each area formula by R1. All arc lengths are in radians and all angular areas are in steradians (sr), where

1 sr=solid angle enclosing an area equal to the square of the radius v2

0 0