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

  • Vertex represents the maximum pole-ward position of a great circle.
    Vertex really does not mean that one is in ice latitudes. The equator has no vertex. On a great circle that crosses equator on a course of 0850 will not go very high in terms of latitude. In fact the vertex latitude will be 50N. Following a course of 0900 is doing a great circle course, whose vertex is the pole itself.
  • Latitude of vertex is actually equal to the smaller angle between the ship’s head and the equator at equator crossing.
    This can be visualized from the following picture of a cricket ball.
  • Every great circle has 2 vertices one in the northern hemisphere and the second in southern hemisphere.
  • If the course at one vertex is 090°, the course at the second vertex is also 90°
    Course on the GC is continuously changing and at vertex, North component changes to South and vice versa. But this is not true about East or West component. The East component never changes to West (and vice versa) in a great circle route. Throughout the passage, either the vessel has easterly component or westerly component.
  • For the vertex to be in between initial and final position a course of 090° or 270° must be there between initial and final course.
    This is because the course at vertex is either 0900 or 2700. In northern hemisphere if the course has northerly component, the vertex is ahead. If the course however, has a southerly component, the vertex is passed behind. Well, there has to be a vertex in northern hemisphere,, either ahead or behind. A similar rule would apply in southern hemisphere.
  • D’long between the equator crossing and vertex is 90°
    In a complete 3600 traverse of a great circle, there are two equator crossings and two vertices. In fact they come alternately, every 900 off. This means, after 900 d’long of an equator crossing there will be vertex and after 900 d’long of vertex, there will be the equator crossing.

Some Interesting points

  • If we know the longitude of one of the vertex the longitude of the vertex in other hemisphere is found by adding 180°, then subtracting from 360° and changing the name.
  • If initial and final positions are given, the vertex latitude can be calculated in three steps.
  • Knowledge of course and longitude at equator crossing is enough to know the latitude and longitude of vertex without doing any spherical triangle calculation.
  • If the d’long between the initial and final position is more than 90° and both the positions are in the same hemisphere the vertex lies between the initial and final positions.
  • If V1 & V2 are two vertices of a great circle, there can be infinite number of Great circles which can be drawn passing through the two vertices.
  • If the initial and final positions are in the same hemisphere and have the same latitude, the vertex is equidistant from the two positions and is at the mid-long of the two positions.

A few points about composite circle sailing:

  • In composite circle sailing, the first and third leg of the passage are actually parts of two different great circles. This is the reason that on the limiting latitude there are two different vertices.
  • However, if the initial and final latitudes were same, the two legs of great circles would have same size.
  • The entire composite circle calculations, are only done by Napier’s method. Use of cosine formula is not needed.
  • Initial course of composite circle route is always equatorwards of the initial course if it were single GC.
  • In a situation when limiting latitude is same as the latitude of destination and the d’long to the vertex V of composite circle route is more than d’long to destination, then composite route is same as single GC route.

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