GREAT ELLIPTIC ARC LENGTH IN TERMS OF PARAMETRIC LATITUDE 6 



Equation (41) gives the arc length, but the modulus k, dj and dj, and vertex cf)„ must be 



expressed in terms of parametric latitude, 6, if the geographic latitudes 0i, <;62 oi the given 



points P,, Pj have been first converted to parametric latitudes 0^, 0^. 



tan a 

 The relationships tan <h = , N sin (b = , sin 6, applied to 



k = (e \/ 1 - e^/a) No sin 0^, 



di = arc cos (N, sin (^i/Np sin cf}^, A^ = arc cos (N2 sin c^j/No sin (f}^), and the last 

 of equations (21) give 



e,, = k = e sin ^^ , dj = arc cos (sin 6^/sm 0^), d^ = arc cos (sin ^j/sin dg), 



tan 00 = (tan^^i + tarf^j- 2 tan 0, tan 0j cos AXY^^ /sin AX , 



whence 



sin 6lo = tan 6L/(1 + tan'l9o) '^' , (42) 



1/2 



tan^^j + tan ^0j - 2 tan 0, tan 0^ cos A A 

 sin Ob =/ 



I tan^^j + tan ^6^-2 tan 6^ tan 0j cos A A + sin^ A A 

 Equations (41) and (42) give then the arc length along the great elliptic arc when geographic 

 latitudes have been converted to parametric latitudes. 

 THE CHORD DISTANCE 



The chord distance between the points Q, (xi, 0, Zj), Q2 (xj, yj, Zj) as shown in Figures (13) 

 and (14) is given by the usual distance formula where the coordinates may be expressed in 

 terms of either cf) or 9, that is from (1) 



Xj = N, cos </>!, yj = 0, Zj = N, (1 - e^) sin (fi^ (in terms of cf>) 



Xj = N2 cos cj)^ cos A A, y2 = N2 cos <^2 sin A A, Zj = N2 (1 ~ e^) sin <p^ , (43) 



cos ^1, y = 0, z,= a \/ 1 — e^ sin 0^ (in terms of 



X2 = a cos 02 cos A A, y2 = a cos 02 sin AA, Z2 = a\/1 — e^ sin 9^ . 

 Applying the distance formula to each set of formulas in (43) for coordinates one obtains 



C = [(Ni cosc^i-Nj cos <^2Cos AA)' +N2'cos'(/)2sin'AA + (l-e')'(N, sin0,-N,sin02)']'^^ 

 and in terms of 



C = a[(cos6l2 cosAA-cos 6I1)' + cos'02 sin 'AA + (l-e')(sin ^j- sin 0,^]"^' (45) 



In (45), expand the quantities in the brackets combining terms to obtain 



C = a [2 - 2 (sin 9, sin 02+cos 0,cos 9^008 AA) - e' (sin 9^ - sin (9J']'^' . (46) 



Now cos (dj + dj) = sin 9^ sin 2+ cos 0, cos 02^03 AA and with sin 0,= sin 0^ cos d,, 

 sin 02 = sin 0^ cos d2, k' = e^sin'^o from (42), equation (46) can be written 



C = a[2il - cos (d. + d2)i - k' (cos d. - cos d2)']'^'. (47) 



53 



