       Re: "Traveling salesman on a hemisphere" problem

• To: mathgroup at smc.vnet.net
• Subject: [mg121474] Re: "Traveling salesman on a hemisphere" problem
• From: Olaf <olaf.rogalsky at googlemail.com>
• Date: Fri, 16 Sep 2011 05:49:03 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j4see5\$sp4\$1@smc.vnet.net>

```Hi Peter,

with the arclen from http://en.wikipedia.org/wiki/Great-circle_distance,
the following code will give you the shortest route.

stars = {
{"M51", {21, 44.7}},
{"NGC2721", {4, -17}},
{"a funny comet", {57.3, 7}},
{"absolutely must see", {23, -176.3}}};

arclen[{_, {F1_, L1_}}, {_, {F2_, L2_}}] :=
Module[{f1 = F1 Degree, l1 = L1 Degree, f2 = F2 Degree, l2 = L2
Degree, dl = (L1 - L2) Degree},
ArcTan[Sin[f1] Sin[f2] + Cos[f1] Cos[f2] Cos[dl],
Sqrt[(Cos[f2] Sin[dl])^2 + (Cos[f1] Sin[f2] - Sin[f1]
Cos[f2] Cos[dl])^2]]]

tour = FindShortestTour[stars, DistanceFunction -> arclen]

sortedStars = starlist[[tour[]]]

And here some eye-candy:

p2c[{_, {f_, l_}}] := {Cos[f Degree] Cos[l Degree],
Sin[f Degree] Cos[l Degree], Sin[l Degree]}
greatCircleArc[{q_, p_}] :=
Module[{u = p2c[q], v = p2c[p], a}, a = VectorAngle[u, v];
Table[Evaluate[RotationTransform[t, {u, v}][u]], {t, 0, a,
a/Ceiling[10 a]}]]

Graphics3D[
{Sphere[{0, 0, 0}, 0.97],
{Black, Thick, Arrow[{{0, 0, -1.3}, {0, 0, 1.3}}]},
{Red, PointSize[Medium], Point[Map[p2c, sortedStars]]},
{Blue, Thick,
Map[Line, Map[greatCircleArc, Partition[sortedStars, 2, 1]]]}
}, SphericalRegion -> True]

Regards, Olaf

```

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