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Posting about Earhart and Wartime navigation in WW II required me to do
some rhumb line calculations. I had programed my Casio PB-1000
calculator in about 1989 to do many navigational calculations including
planning great circle and rhumb line courses. I input the starting and
ending points and the program computes great circle course and distance
followed by rhumb line course and distance and the coordinates of the
vertex. Then I put in the longitude of the first intermediate point and
the interval of subsequent points and the program returns the latitude
of the GC at the selected longitude, the rhumb line course and distance
between the prior point and the new point and the cumulative rhumb line
distances. I then repeat this for the rest of the intermediate points.
At the end it also displays the extra distance of this approximation of
the  GC over a perfect GC course. Bellow is the subroutine that computes
great circle and rhumb line courses and distances. Line 28 computes
rhumb line course (RCRS); line 50 rhumb line distance; 60 and 70 the GC
distance; and 80 and 90 the GC course.

I got the formula for the rhumb line course from AFM 51-40, attached.

It is interesting that an Air Force manual would contain a significant
typo. The formula given to compute the rhumb line course doesn't work
because it is missing a pair of parentheses. I remember that it took me
a while to find this problem. The 180 in the bottom of the formula
multiplies both terms, not just the first one. (Ln is natural log, log
base e in AFM 51-40) The problem persists in the current navigation
manual, AFPAM 11-216.

gl




5 ON ERROR GOTO900
10 CLS:PRINT"COMPUTING"
11 IF LT1>=90 THENLT1 =89.9999
12 IF LT1<=-90 THENLT1=-89.9999
13 IF LT2>=90 THENLT2 =89.9999
14 IF LT2<=-90 THENLT2=-89.9999
15 IF LT1=0 THENLT1=.0001
16 IF LT2=0 THENLT2=.0001
20 IF LL1<0 THEN LL1=360+LL1
21 IF LL2<0 THEN LL2=360+LL2
23 LHA=LL2-LL1
24 IF LHA<0 THEN LHA=LHA+360
25 IF LHA=180 THEN LHA=179.99982
26 T=LHA
27 IFT>180 THEN T=360-T
28
RCRS=ABS(ATN((PI*(T))/(((LOG(TAN(45+LT2/2)))-(LOG(TAN(45+LT1/2))))*180)))
30 IF LT1>LT2 THEN RCRS=180-RCRS
50 RDIST=ABS(((1/(ABS(COSRCRS)))*60)*(LT2-LT1))
60 ZD=ACS(SINLT1*SINLT2+COSLT1*COSLT2*COS(LHA))
70 GDIST=60*ZD
80 AZZ=ACS((COSLT1*SINLT2-SINLT1*COSLT2*COS(LHA))/COS(90-ZD))
90 GCRS=AZZ:IF LHA<180 THENGCRS=360-AZZ:RCRS=360-RCRS
800 RETURN
900 LT2=LT2+.005
920 RESUME 28

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