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I was looking at the distance calculations from various list members 
responding to Silicon Sea Leg 70 and was puzzled by the differences in the 
same calculations. I looked into the equation used for calculating distance 
and determined that it is very sensitive to round off errors in the course.

Given a departure and destination point, the equations used to determine 
course and distance for Mercator sailing is:

C = arctan (Dlo/DMP), where C is course, Dlo is difference in Longitude in 
minutes, and DMP is difference in meridional parts.

D = Dlat cos C, where Dlat is difference in latitude in minutes.

Typically in writing the course C, it is rounded to 1 decimal place. If this 
rounded value of C is used in the calculation D, the error can be significant 
if C is near east or west. The rounding is not an issue if the calculation is 
being done by a computer program where the full precision of C is maintained.

The differential of D is dD = D tan C dC. For typical rounding of C to 1 
decimal place, the error dC is ~+/- 0.05. The resulting error in D, dD, is 
highly dependent on C. If C is near east or west, tan C is very large. Thus 
explains the differences in the distances from the various list members for 
SS 70.

As an alternate approach for those of us who like to use pencil, paper, and 
tables to do the calculations, we can first calculate the departure before we 
calculate D. But we need a calculation of departure not explicitly dependent 
on course. From Roy Williams, "Geometry of Navigation", we have p = Dlo 
(Dlat/DMP) where p is the departure. Then distance is then:

D = sqrt(Dlat^2 + p^2)

The differential error in this format is dD = (Dlat d(Dlat) + p dp)/D. The 
error is significantly insensitive to errors in Dlat and p.

Sam Chan

   

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