Logbook entry

J.Bauer / 04 May 3305
WK1: Returning Home - Galaxy Navigation & Long Range Route Planning

With the Distant Worlds 2 expedition complete, thousands of Commanders are now charting their own way home, or for the more daring, onward. The crew of the USNS Chicago however has made plans to head toward Colonia. The reason is two fold.  First, we want to return to civilized space to offload our exploration data.  Explorer's Anchoridge is the closest station from our present location.  The second is to explore Colonia's viability as a permanent home. Rather than fly to Explorer's Anchoridge then "west" toward Colonia; we may be able to save time by crossing the galaxy diagonally and heading there directly. Determining if this is a worthwhile effort requires a bit of math.

Galaxy Navigation
The Galaxy Map divides the Milky Way into four quadrants with Sol centered at the origin.  Cartesian coordinates are used to navigate the galaxy map with the X and Z axis parallel to the horizontal plane of the galaxy, and the Y axis perpendicular.  The axis are measured in light years.  In general, these coordinates can be ignored as the navigation computer can report your current distance to any selected destination. However, it is useful to know route lengths between points for long range route planning.

Using vector algebra we can determine the distance between two sets of coordinates.  For example Beagle Point is located at -1,111.5625 / -134.21875 / 65,269.75.  My two potential destinations are:
By subtracting the destination coordinates from the source coordinates we arrive at the vectors to each destination.  We then calculate the magnitude of that vector using M2 = X2 + Y2 + Z2.

For Beagle to Explorer's Anchorage:
CEA - CBP = VBP-EA
[29 / -20 / 25,900] - [-1,112 / -134 / 65,270] = [1,140 / 114 / -39,370]
MBP-EA2 = 11402 + 1142 + 393702
MBP-EA = 39,387 Ly

Explorer's Anchorage to Colonia:
CC - CEA = VEA-C
[-9,531 / -910 / 19,808] - [29 / -20 / 25,900] = [-9560 / -890 / -6092]
MEA-C2 = 95602 + 8902 + 60922
MEA-C =  11,371 Ly

For Beagle to Colonia:
CC - CBP = VBP-C
[-9,531 / -910 / 19,808] - [-1,112 / -134 / 65,270] = [-8419 / -776 / -45462]
MBP-C2 = 84192 + 7762 + 454622
MBP-C = 46,241 Ly

Traveling directly to Colonia will take 46,241Ly instead of 50,758Ly saving 4,517 Ly, or 8.9%. While that appears to be a small improvement given the total distance of the return journey, some of the waypoints during the Distant Worlds expedition were themselves only 5k Ly.   Assuming a modest travel rate of 5,000 Ly per week, we should arrive in Colonia in approximate 9.25 weeks. 

Long Range Route Planning
Calculating the best route and plotting it are two different things entirely.  The navigation computer appears to have a hard limit of 20k Ly, however it often fails to plot routes at much shorter distances.  Best practices is to plot routes at 5k-10k Ly at a time.  To maximize efficiency, we should aim to plot a course as straight as possible to our destination.  That means that each waypoint should lie along the calculated vector.

To calculate waypoint coordinates, we will scale our destination vector to 5k Ly and add our scaled vector to our current coordinates:
CScaleFactor = MWaypoint / MBP-C
CScaleFactor = 5000 / 46241
CScaleFactor  = 0.108

VWaypoint = VBP-C * CScaleFactor
VWaypoint = [-8,419 / -776 / -45,462] * 0.108
VWaypoint = [-909 / 84 / -4,910]

CWaypoint = CBP + VWaypoint
CWaypoint = [29 / -20 / 25,900] + [-909 / 84 / -4,910]
CWaypoint = [-880 / 64 / 20,990]

Since we are rounding, you will find the the solution above does not provide a perfectly straight line toward our destination, however increasing waypoint distance or performing calculations to six decimal places will provide a more accurate solution.

Using the method I've described above, I've calculated the next 9 waypoints to Colonia from our current location:

WaypointX-coordinateY-coordinateZ-coordinateEstimated Arrival Date
Start-1139-163578May 4, 3305
1 SKUEQIAE JG-W C18-0-2080-10358668May 11, 3305
2-3021-20553759May 18, 3305
3-3963-30748849May 25, 3305
4-4904-40943940June 1, 3305
5-5845-51139030June 8, 3305
6-6786-61334121June 15, 3305
7-7727-71529211June 22, 3305
8-8669-81724302June 29, 3305
Colonia-9530-91019808July 6, 3305

All waypoints are approximately 5k Ly in length except for the last one which is only 4577 Ly.  For WP1, we've set the navigation computer to target SKUEQIAE JG-W C18-0.  We will reach our destination in 157 jumps.  Our next check in will be in approximately one weeks time.

CMDR J.Bauer signing off.
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