# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Great Circle Computation**

**From:**William Trayfors

**Date:**2000 May 23, 7:16 AM

Frank: The following, taken from one of my old navigation courses, may be helpful: GREAT CIRCLE SAILING INTRODUCTION A great circle course is the shortest distance between any two points on earth. It is of interest to the navigator because he can often cut the total distance sailed from point to point considerably by following a great circle course. Great circle courses always traverse higher latitudes than do rhumb line courses, i.e., further north in the Northern Hemisphere and further south in the Southern Hemisphere. Where obstructions or navigational hazards lie in the path of the great circle course, a combination of great circle sailing and rhumb line or traverse sailing can often be used to advantage. This is called composite sailing. It is possible to solve great circle course and distance problems in several ways. By using a gnomic projection chart, great circle routes can be plotted as straight lines. A series of points on this chart can then be transferred to a regular mercator chart -- say at intervals of 5 degrees of longitude -- and then the rhumb lines between each point can be taken directly off the mercator chart and used for navigation. Using rhumb line sailing between each 5 degree point on the great circle track introduces only a small error, but requires less frequent course changes than would be necessary to follow a great circle track precisely. Another practical way to solve great circle course and distance problems is by the use of mathematical formulae. This method is fine if you're handy with a scientific calculator or have a computer handy, particularly when a gnomic projection chart is not available. In the following examples, two simple equations and a table of natural trigonometric functions (such as Table 31 in Bowditch) are used. It is only required that the longitude and latitude of the points of departure and destination be accurately known. These can be taken from charts, Sailing Directions, or from Appendix S in Bowditch. The equations shown were taken from the A.R.R.L. Antenna Book, pp 284-285. They can also be used to compute radio bearing and correction factors. SOLUTION OF GREAT CIRCLE DISTANCE PROBLEMS Equation: cosD = sinL1 sinL2 + cosL1 cosL2 cosDLo where... D = great circle distance in degrees of arc L1 = latitude of departure L2 = latitude of destination DLo = difference in longitude between departure and destination Example: It is desired to compute the great circle distance between the entrance to Chesapeake Bay (Cape Henry Light) and Bermuda (North Rock Light). Solution: From Bowditch, Appendix S, obtain the maritime positions of the departure and destination points. Cape Henry Light = 36 deg 56 min North 76 deg 00 min West North Rock Light = 32 deg 28 min North 64 deg 46 min West L1 = 36 56 N sinL1 = .60089 cosL1 = .79934 L2 = 32 28 N sinL2 = .53681 cosL2 = .84370 DLo = 11 deg 14 min cosDLo = .98084 Substituting in formula: cosD = .60089 x .53681 + .79934 x .84370 x .98084 cosD = .32256 + .66148 or cos D = .98404 By inspection of trig. tables, .98404 corresponds with the cosine of 10 deg 15.0 min. Therefore, D = 10 deg 15 min or 10 x 60 + 15 = 615 nautical miles. Note: Equation is used as shown above only when L1 and L2 are of the same name, i.e., when both lie in the northern or southern hemisphere. When they are of opposite names, the value of sinL2 is minus. For example, if in the above illustration L2 were 32 deg 28 min South instead of North, the substitution would be made as follows... cosD = .60089 x (-.53681) + .77934 x .84370 x .98084 cosD = -.32256 + .66148 or cosD = .33892 D = 70 deg 11.5 min or 70 x 60 + 11.5 = 4211.5 nautical miles When extracting the value for D, be careful of the ambiguity, particularly when the value is fairly high. In this case, .33892 could be either the cosine of 70 deg 12 min or of 109 deg 49 min. GREAT CIRCLE SAILING Finding Initial Course Equation: sinC = cosL2 cscD sinDLo where... C = initial great circle course (true) L2 = latitude of destination D = great circle distance between departure and destination DLo = difference in longitude between departure & destination Example: A navigator wishes to determine the initial true course along the great circle track from Cape Henry Light (36 deg North, 56 min West) to North Rock Light, Bermuda (32 deg 28 min North, 64 deg 46 min West). Solution: (Assumes D is known or has been previously computed.) L2 = 32 28 N cosL2 = .84370 D = 10 deg 15 min (615 miles) cscD = 5.61976 DLo = 11 deg 14 min sinDLo = .19481 Substituting in equation... sinC = .84370 x 5.61976 x .19481 sinC = .92367 C = 112.32 deg True Caution: .92367 is SIN of both 112.32 degrees and of 67.28 degrees; care must be exercised to be sure the correct one is selected. Hope this helps, Bill At 04:04 PM 5/23/00 +0200, you wrote: >Frank Dinkelaar wrote: > >> In regards to the attachment (navigation formulas) send by Ed Kitchin >> the formula in #1 of the great circle course >> it seems to me it looks like this >> cos D = abs(sinL1 - sinL2) - abs(cosL1 x cosL2 x cos Dlo) >> it would save me many hours of experimenting if somebody >> could confirm or send me the formula in a clearer form. >> thanks > __________________________________ Bill TrayforsThe Washington Decision Support Group, Inc. 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