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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"
}

   

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