SPHERICAL HYPERBOLA FORMULAS AND PLANE EQUIVALENTS 



Spherical Plane 



tan ^a (sin ^c - sin^a) a^(c^ — a^) 



(1) tan 'r 



sin c cos a — sin a c cos a — a 



/o\ -2 sin a cos c 



(2) sin X = — ; — ■_ — — sin y + sin a 



2 2 



in c - sin a c — a 



a — c 



c cos D — 



,„, „ cos 2c ± cos 2a 



(3) tan R = -^-- ^ R = 



sin 2c cos p ± sin 2a 



fA\ . 21 a /^\ sin (c -a) sin (R + c + a) 2 , n ,css (c - a) (R + c + a) 



(4) tan (;8/2) = . . ^ , . ,„ r- tan' ((8/2) =-7 — r- 



sin (c + a) sin (R - c + a) (c + a) (R - c + a) 



In (1) and (2) the origin of coordinates is the midpoint of Q Qj , see Figure 5. Equations (3) 

 and (4) are two polar forms with origin at a focus Qi , see Figures (5) and (6), Appendix 1 has 

 computations based on the plane equivalent of (3). 



DISTANCE AND AZIMUTH FORMULAS 



Normal section azimuths (Geodetic latitude, <p) 



[sin 02 - (Ni/N,) sin ^Je^ cos (p^ sec 02 + (sin 01 cos AA - tan 02 cos 0;) 



^AB 



«BA 



sin AA 

 jsin 0, - (N2/N1) sin 02^ e' cos 02 sec 0^ + (sin 02 cos AA - tan 0i cos 02) 



sin AA 

 Normal Section Azimuths (parametric latitude 6) 



sin ^1 cos AA- cos 6^ tan 6^ + e^ (sin 62 - sin 0,) cos 6^ sec 



'AB 



d-e'cos'e'J''' sin AA 



sin Q^cos AA - cos d^ tan 0, + e' (sin 6^ - sin 62) cos 0, sec 6. 



cot ao A =-^ — , L 



^^ d-e' cos'02)'^'sin AA 



Great Elliptic Section Azimuths (Geodetic latitude 0) 



Ni (tan 01 cos AA — tan 02) cos 0, 

 cota^g = d-e')— ^ 



sin A A 

 N2 (tan 01 — tan 02 cos AA) cos 0j 



cotaBA = d-e') ^ 



a sin AA 



Great Elliptic Section Azimuths (parametric latitude d) 



(tan e, cos AA - tan 6^) (cos d,) {I - i cos^ 6,)'/^ 



^AB 



sin AA 

 (tan di - tan 6^ cos AA) (cos (^ ) (1 - e' cos' d^Y^ 



^BA - 



sin AA 



