Formulas For Solving Great Circle Sailing Problems
There are many formulas which can be used to solve the types of great circle sailing problems found on USCG license exams. It is very difficult to memorize all of them. Fortunately, you don't have to do this. But, I would suggest the formulas for determining great circle distance and initial course angle be memorized.
These formulas are:
For Distance (D) Cos D = Sin L1 X Sin L2 + Cos L1 X Cos L2 X Cos Dlo
For Initial Course Angle (C) Cos C = Distance must be found first in order to be used in the second equation.
The other formulas needed to solve USCG exam problems are found in Bowditch, Volume II, Chapter X, The Sailings. Remember, this book is available in the exam room so you can reference it for the formulas when you take your exam. Adapting the Bowditch formulas to a hand-held calculator requires basic level understanding of the relationships between the different functions of trigonometry (sine, cosine, tangent, cotangent, secant, and cosecant). This is because calculators only have keys for the Sine, Cosine, and Tangents of angles.
Secant (angle X) = 1 divided by cosine (angle X)
Cosecant (angle X) = 1 divided by sine (angle X)
Cotangent (angle X) = 1 divided by tangent (angle X)
Sine (angle X) = cos (90° - angle X)
Cos (angle X) = sine (90° - angle X)
Procedures For Solving Great Circle Sailing Problems
A hand-held calculator with trig functions and at least three memory spaces is a must for solving great circle sailing problems. There are long hand mathematical methods which can be used, such as with logarithms, but they are time consuming and you might make a simple math mistake. There are several different types of great circle sailing problems that you could be asked on your examination, but most are the type which ask for great circle distance (D) and initial course (Cn). The other types are:
1. Finding the latitude of the upper vertex (Lv).
2. Finding the longitude of the upper vertex (Lov).
3. Finding the distance of the upper vertex (Dv).
4. Determining the latitude and or longitude of point X (Lx and Lox) along the great circle track.
Note: Each USCG question should be read carefully because many have given more information than is needed to solve the problem. You need to have a format to go by. A format that you can use to tabulate values and record results. The use of a neat format for these problems will help you from making mistakes.
Great Circle Sailing Charts and Voyage Planning
Below is some information on gnomonic charts in planning voyages which incorporate great circle tracks for all or a portion of the voyage, and the procedures for answering typical questions found on USCG license exams which relate to gnomonic charts.
The Gnomonic Projection
The gnomonic chart projection is a geometrical projection in which the surface features of a certain portion of the earth are projected outward from the center of the earth to a tangent plane. There are three types of gnomonic charts depending on the location of the point of tangency. In equatorial gnomomic charts, the point where the plane is tangent is located at some point on the equator. A polar gnomonic chart is tangent at one of the poles. Oblique gnomonic charts have their point of tangency at some latitude and longitude between the equator and the poles. The land features of gnomonic charts become more distorted as the distance from the point of tangency increases. The most often used gnomonic charts are oblique charts with a point of tangency located in the center of an ocean basin. Gnomonic tracking charts WOXZC 5270 and WOXZC 5274, have their points of tangency in the North Pacific and North Atlantic Ocean basins.
The chief advantage of gnomonic charts is that a straight line between two points represents the great circle between the points. If you proceed by great circle, all you need to do is connect the points of departure and arrival with a straight line. The latitudes and longitudes of points at intervals along this line may be measured on the gnomonic chart and then transferred to a Mercator chart. When these points are connected with straight lines (rhumb lines), you have the "legs" of the great circle track. Once transferred to the Mercator chart, the course on each leg can be found and the sum of the distances along each leg gives the total great circle distance. The main disadvantages of the gnomomic projection are that courses, direction and distance can not be readily measured. For this reason gnomonic projections are not used for day to day navigation and plotting, and should be used as an aid to voyage planning.
Use of Gnomonic Charts for Track Planning
The use of gnomonic charts for voyage planning provides for quick checking of planned great circle tracks for intervening land, iceberg hazards, and other factors. Example, if you wished to determine the advantage of proceeding by great circle from Cape Flattery, at the entrance to Puget Sound, to the entrance to Tokyo Bay, Japan, you would draw a straight line between the two places on chart WOXZC 5270. Inspection of the chart would reveal that this track would require your vessel to pass through the Aleutian Chain twice, and part of the voyage would be in the Bering Sea. Knowledge of the average poor weather, violent storms, uncharted rocks, and fog in these northern waters makes the saving in distance of a great circle over a rhumb line less attractive.
The great circle chart can be a big help to plan composite sailing tracks. Composite sailing consists of proceeding from the point of departure until the great circle reaches a limiting latitude. You then proceed by rhumb line (parallel sailing) along that latitude to the point where a great circle from the point of arrival is tangent to the limiting latitude line. In a later blog I will show how you can solve Coast Guard Great Circle Sailing exam questions using Pub. 229.
