# NavList:

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

**Re: Barrie Hudson Challenge**

**From:**George Huxtable

**Date:**2002 Nov 9, 22:55 +0000

I agree with Peter Fogg that there's something amiss with Dan Allen's program, in that it finds itself making a sudden dog-leg right at the end ov the voyage. I also agree with Peter's result, which he has achieved by some process of successive approximation. However, it's not necessary to do it that way. It can be done in a single shot, remembering that a great-circle crosses the equator in two places exactly 90� away, in longitude, from the longitude of each vertex. The vertex is the point on a great circle which is nearest to the North (or South) pole. As usual, Bowditch provides the information about how to do it. Once the initial course C has been calculated (which Dan and Peter have done without problems) passing through the lat and long of Valparaiso, that defines the great circle which will arrive at San Francisco. There's no further need to consider San Francisco's coordinates. Mine is the 2-volume (1981) edition of Bowditch. In vol 1, the relevant diagrams are fig. 1016, a and b, on page 605. The formulae are on page 604, in which we can stir together equations 25 and 27, to give the cos of the difference in long from Valparaiso to the equator crossing as cos course / sqrt(1- (cos lat * sin course)^2). lat is the latitude of Valparaiso, and course is the initial great-circle course from Valparaiso to San Francisco. Once the long of the Equator crossing has been established (I make it W95.2687�, the same as Peter does ) then the result can be easily checked by working out the great-circle course from that point to Valparaiso and to San Francisco, which should differ by exactly 180� (they do). George Huxtable. ======================== Peter Fogg said- >a) The initial course is 321.9d and distance 5117.9 nm. >b) At the equator I get a longitude of W95d 16' and course 327.8d >This is very close to Dan's 95.26867, which is W95d 16m 7.2s, or W95d >16.12' >I have no idea why the course is different. To find the equator I had to >slice the segments into smaller and smaller lengths, they are all 327.8d >around those waters. Dan's courses seem to make sense until the last one >which may be just a quirk of his program. > >Dan Allen wrote: > >> On Friday, October 18, 2002, at 05:12 PM, bhudson wrote: >> >> > Hi, >> > Now that we can handle the Cos formula and GC sailing here is a poser! >> > A ship sails from a position off Valpraiso Lat 33�01'S, Long 72�10'W >> > to a >> > position off San Francisco Lat 37�50'N, Long 123�14'W >> > a) Find the initial course and GC distance. >> > b) Find the position of the ship as it crosses the equator and the >> > course it >> > crosses the equator. >> > Barrie Hudson >> >> I got the following results: >> >> DistNMI Latitude Longitude Course >> --------------------------------------- >> 0 -33.01667 72.16667 321.94335 Valpariso >> 1000 -19.42858 82.97198 326.76300 >> 2000 -5.30354 92.05548 328.72709 >> 2371 0.00000 95.26867 328.87595 Equator >> 2559 2.66837 96.88125 328.83839 Midpoint >> 3000 8.94774 100.72401 328.44879 >> 4000 22.98133 110.10570 325.84358 >> 5000 36.33635 121.63659 320.08411 >> 5118 37.83333 123.23333 26.75367 San Francisco >> --------------------------------------- >> >> which were generated by the following Awk program. >> >> # Usage: awk -f hudson.awk >> >> BEGIN { # all arguments and results are in decimal degrees >> CONVFMT = OFMT = "%10.5f" >> PI = 4*atan2(1,1) >> DTOR = PI/180 >> RTOD = 180/PI >> # Near Valpariso, Chile >> lat1 = -(33 + 1/60) >> lon1 = 72+10/60 >> # Near SF, CA >> lat2 = 37+50/60 >> lon2 = 123+14/60 >> # Let's go! >> d = GCDistance(lat1,lon1,lat2,lon2) >> c = GCCourse(lat1,lon1,lat2,lon2) >> print >> print "DistNMI Latitude Longitude Course" >> print "---------------------------------------" >> for (i = 0; i <= d; i += 1000) { >> s = GCPoint(lat1,lon1,c,i) >> printf("%6d %s%s\n",i,s,i == 0 ? " Valpariso" : "") >> if (i == 2000) { >> EquatorCrossing(lat1,lon1,lat2,lon2) >> printf("%6.0f %s Midpoint\n",d/2,GCPoint(lat1,lon1,c,d/2)) >> } >> } >> s = GCPoint(lat1,lon1,c,d) >> printf("%6.0f %s San Francisco\n",d,s) >> print "---------------------------------------" >> } >> >> function Abs(x) { return x < 0 ? -x : x } >> function Floor(x) { return x < 0 ? int(x) - 1 : int(x) } >> function Round(x) { return Floor(x+0.5) } # Everyday round > 0 >> function Mod(x,y) { return x - y * Floor(x/y) } >> function Sin(x) { return sin(x*DTOR) } >> function Cos(x) { return cos(x*DTOR) } >> function Tan(x) { return Sin(x)/Cos(x) } >> function ASin(x) { return atan2(x,sqrt(1 - x * x))*RTOD } >> function ACos(x) { return atan2(sqrt(1 - x * x),x)*RTOD } >> function ATan2(y,x){ return atan2(y,x)*RTOD } >> >> function GCDistance(lat1,lon1,lat2,lon2) { >> return 60*2*ASin(sqrt((Sin((lat1-lat2)/2))^2 + >> Cos(lat1)*Cos(lat2)*(Sin((lon1-lon2)/2))^2)) >> } >> >> function GCCourse(lat1,lon1,lat2,lon2) { >> return >> Mod(ATan2(Sin(lon1-lon2)*Cos(lat2),Cos(lat1)*Sin(lat2)- >> Sin(lat1)*Cos(lat2)*Cos(lon1-lon2)),360) >> } >> >> function GCPoint(lat1,lon1,c,distance, d,dlon) { # lat,lon need to be >> globals >> d = distance/60 >> lat = ASin(Sin(lat1)*Cos(d) + Cos(lat1)*Sin(d)*Cos(c)) >> dlon = ATan2(Sin(c)*Sin(d)*Cos(lat1),Cos(d)-Sin(lat1)*Sin(lat)) >> lon = Mod(lon1 - dlon + 180,360) - 180 >> return lat " " lon " " GCCourse(lat,lon,lat2,lon2) # lat2, lon2 need >> to be globals >> } >> >> function GCLatitude(lat1,lon1,lat2,lon2,lon3) { >> return >> ATan2(Sin(lat1)*Cos(lat2)*Sin(lon3-lon2)-Sin(lat2)*Cos(lat1)*Sin(lon3- >> lon1), >> Cos(lat1)*Cos(lat2)*Sin(lon1-lon2)) >> } >> >> function GCLongitude(lat1,lon1,lat2,lon2,lat3, >> dlon,lon,l12,lon3_1,lon3_2,A,B,C) { >> l12 = lon1 - lon2 >> A = Sin(lat1)*Cos(lat2)*Cos(lat3)*Sin(l12) >> B = Sin(lat1)*Cos(lat2)*Cos(lat3)*Cos(l12) - >> Cos(lat1)*Sin(lat2)*Cos(lat3) >> C = Cos(lat1)*Cos(lat2)*Sin(lat3)*Sin(l12) >> lon = ATan2(B,A) >> if (Abs(sqrt(A^2+B^2) - C) < 0.0000001) >> dlon = 0 >> else >> dlon = ACos(C/sqrt(A^2+B^2)) >> lon3_1 = Mod(lon1+dlon+lon+180,360) - 180 >> lon3_2 = Mod(lon1-dlon+lon+180,360) - 180 >> if (lon3_1 > lon1 && lon3_1 < lon2) >> return lon3_1 >> else >> return lon3_2 >> } >> >> function EquatorCrossing(lat1,lon1,lat2,lon2, d,lonV) { >> lonV = GCLongitude(lat1,lon1,lat2,lon2,0) >> d = GCDistance(lat1,lon1,0,lonV) >> printf("%6d ",d) >> print GCPoint(lat1,lon1,GCCourse(lat1,lon1,0,lonV),d), " Equator" >> } ------------------------------ george@huxtable.u-net.com George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. Tel. 01865 820222 or (int.) +44 1865 820222. ------------------------------