"Traveling salesman on a hemisphere" problem

• To: mathgroup at smc.vnet.net
• Subject: [mg121454] "Traveling salesman on a hemisphere" problem
• From: Peter Sisak <p-kun80 at hotmail.com>
• Date: Thu, 15 Sep 2011 04:42:28 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```I need to find an optimised roundtrip path for a list of objects given with polar coordinates on a unit sphere (actually, they are sky objects). I have a list of objects, such as:

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

In which list, the name of the celestial object is the string (may contain spaces, hashes, slashes, hyphens &c.), then its position is given with the pair {elevation(inclination/altitude),azimuth}. Elevation, since we are talking about sky objects, will be between 0 and 90 inclusive (0 being right on the horizon, 90 right above the observer's head; items with a negative elevation are under the horizon and they are not considered in the current problem); azimuth ranges from 180 to 180 inclusive, and gives the compass direction of the same object (0 being North, +-180 being South, +90 East, 90 West).
If one simply maps the coordinates onto an orthogonal coordinate system, it helps a little, but is not exact, for the reason that the linear distance between two objects is not constant (it grows as you proceed towards the horizon/decrease elevation) with this mapping. There doesn't exist such a mapping that would convert the angular distances into linear distances without distortion while being able to map a full hemisphere.
What I want to get as an output is the ordered list of the entire input object list, preferably with all data unchanged apart from their order. If a total length of roundrip path (in total angular length) would be calculated, that would be even more helpful. Text/spreadsheet file import for the list of objects, even better.