A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
From: Sean C
Date: 2017 Dec 4, 22:25 -0800
My spreadsheet uses the law of cosines formulae (the same ones used for sight reduction) to find the total distance, initial course, waypoints and intermediate courses. I can't remember where I originally read about it, but the tabular method is explained in Section D: Other Aplications of Pub. 229 and also in the 2002 ed. of Bowditch, Chapter 24: The Sailings. This can, of course, be adapted for use with the formulae quite easily. Following are instructions on how to do just that...
To find G.C. distance and initial course:
Substitute the declination with the latitude of the destination and the L.H.A. with the difference in longitude. Thus, we have:
asin(sin(φ₂) ∙ sin(φ₁) + cos(φ₂) ∙ cos(δλ) ∙ cos(φ₁)) = D ; 60 ∙ (90 - D) = G.C. distance
acos((sin(φ₂) ∙ cos(φ₁) - cos(φ₂) ∙ cos(δλ) ∙ sin(φ₁)) / cos(D)) = C ; If δλ < 180°, course = 360° - C; otherwise course = C
To find waypoints along the G.C. route:
To find the latitude and difference in longitude of the waypoint: substitute the declination with 90° - distance to the waypoint (e.g. use 5° for legs of 300 NM) and substitute the L.H.A. with the initial course. thus:
asin(sin(85°) ∙ sin(φ₁) + cos(85°) ∙ cos(I.C.A.) ∙ cos(φ₁)) = latitude of the waypoint 300 NM from the departure point
acos((sin(85°) ∙ cos(φ₁) - cos(85°) ∙ cos(I.C.A.) ∙ sin(φ₁)) / cos(φ wpt.)) = difference in longitude of the waypoint from the departure point
To find the next waypoint (600 NM from the departure point), one would simply substitute 85° with 80°. The I.C.A. remains unchanged for all of the waypoints. As you realized, this method yields waypoints which are 300 NM apart along the G.C.course, but not necessarily 5° of longitude apart. This has the advantage of keeping each leg of the route the same distance no matter what angle at which the course cuts the meridians. The "distance remaining" column in my spreadsheet was a quick and dirty solution - subtracting 300 NM from the total distance for each waypoint. I didn't really check to see if that was absolutely accurate. To find the intermediate courses, one simply needs to run through the first set of equations again with the waypoints as the departure and destination.
As for why the waypoints stop with more than 300 NM left ... well, that was either due to a limitation of Excel or my ineptitude at getting it to do what I wanted or a little of both. I can't quite remember. I'm going with the former, though, because that sounds better to me. :D
Seriously, though, I'll have to go back and look at it and see if I can't figure out how to fix that.