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Vertex, The Master-key?

Vertex the key to all secrets
Information about the coordinates of vertex is the master key to all the secrets of a great circle. Any calculation in respect of a great circle can be accomplished, if the position of vertex is known. Be it the intermediate position; intermediate course; or the position where a parallel of latitude or meridian is cut; everything and anything can be found. Vertex, though described in a simple way as most poleward point of a great circle, contains many secrets within. Let me reveal a few to my readers.

Great Circle Sailing
Higher probability of bad weather and lower temperatures on a great circle route than what would be experienced in relevant Rhumb Line sailing, are some of the factors which come to mind when a Master decides about options to cross an ocean, particularly in winters. Most mariners, generally resort to rhumb-line sailing only. Fear of rough weather; bad visibility; probability of confronting ice; and the mention of limiting latitude in crew agreements are some of the deterrents in following a great circle route. Those who have been doing GC sailings or have an experience of such sailing, often talk big about having the knowledge and experience of GC sailing, as if GC sailing is a very special extra qualification.

Finding the way-points
Indeed, the excitement of altering course every noon or at an intermediate way-point, during an ocean passage and then eventually reaching the destination, is unique. To do a great circle sailing, positions of intermediate waypoints must be found. These waypoints may be at fixed d’long intervals. Finding the latitude at some intermediate meridian can be done in three ways. The first method uses, gnomonic sheet to find the intermediate way-points. Waypoints may be planned say every 5 degrees of d’long. A straight line drawn on the gnomonic sheet between initial and final positions is actually a great circle route. Latitudes every 5 degrees off are noted down. The coordinates of waypoints are transferred on Mercator chart. Courses between these way-points are traversed by plane or rhumb-line sailing.

In the other two methods involving calculations, first, initial course is found using initial and final position. In the next step:

1. Using co-lat of initial position A, initial course, and angle V being 900 , triangle APV  is solved to find position of vertex using Napier’s method. Using the co-lat of vertex and d’long with the pre-decided meridians, the respective co-lats on these meridians are found, thereby giving intermediate waypoints.  Navigation between adjacent waypoints is done by rhumb-line sailing.

2. Using initial course, initial co-lat & d’long to any predetermined meridian, the co-lat of way-point can be calculated by four parts formula. This way co-lat for d’longs of 50, 100, 150, etc are found. Thus, waypoints every 50 d’long off are found. Navigation between adjacent way-points is done by rhumb-line sailing.

Four Part Formula:
A spherical triangle has 6 parts. Four part formula can be used to find the missing outer part (only), if the 4 parts used are in continuity. Thus, in above fig angle A, side AP, angle B & side PC are in continuity. Angle A & side PC are outer parts. (Outer angle and outer side respectively), whereas side AP & angle P are inner parts. Let angle A be 400, side AP be 520.Then  for any d’long or say a d’long of 100, PC can be calculated by following formula.

Thus, Cot PC x Sine PA = Cot 400 x Sine 100 + Cos 520 x Cos 100

Some Interesting points

A few points about composite circle sailing:

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