Great Circle Sailings

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Because Everyone Loves Their Curves

Because the earth is not flat, but a sphere (roughly), the shortest distance between two points is not a straight line, but a curved one.
The Great Circle is the closest approximation we have to steering along the exact curve of the earth. Despite what you may think about GC sailings, they're not all that tough, but they are fairly tedious depending on the distance you travel and the number of waypoints along your track.
But, your captains will be much happier having you steer a GC than they would if you went straight across an ocean, seeing as you would have just cost them thousands of extra dollars in fuel.

A Few Things To Remember:
-All variables are described in degrees, and from there are converted
-When entering into the equations, if L1 and L2 are contrary name, L2 is entered as a negative number
-If after solving an equation, C is found to be negative, add 180 to find the true value
-C is based off the elevated pole, not the direction of travel as with the other sailings
-If the formulas look familiar, they should, they're basically the same as used in celestial, but with new variables. So if you're lost, see the Celestial Section for an alternate explanation

To Solve For A Given Variable:
-First establish which variable it is you are solving for. If you need to find the distance (D) from P1 to P2, use the equation that solves for D.
-Next establish what you know and what you need to know. Ex., for D, you need to know L1, L2, and DLo. 
-Find the remaining unknown variables. If you only know your starting and final positions, solve for DLo, then input into the equation.
-Lastly, solve for the given variable by solving the appropriate equation.

To Find Waypoints Along Trackline:
-The waypoints along the GC track can be solved using the equations for X, Lx, and DLovx, where X is the waypoint you are seeking.
-First determine either the number of waypoints or the spacing of waypoints that you are seeking. For example, if you need waypoints to be 5 of longitude apart, use the Lx equation that uses DLovx. If you need the waypoints to be 60nm apart, use the Lx equation that uses Lv.
-Solve for Lx, then solve for DLovx, and lastly λx.

Back To Sailings

D: GC dist (in ) of arc
Dv: Dist (in ) from P1 to V
Cn: Initial true course from P1 along GC
V: Vertex of GC Track
C: Initial course angle from P1 along GC, measured from elevated pole
Lv: Latitude of Vertex
DLov: Distance (in ) between λ1 and λv
λv: Longtitude of vertex

X: any point on GC track between P1 and P2

Lx: Latitude of point along GC track

Dvx: Distance (in ) from V to X along GC track

DLovx: Longitudinal distance (in ) from V to X

Great Circle Sailings are all about filling in the blank.
Start with what you know and find what you don't. Pick the equation that only has one unknown to use first.

GC Equations
(found on/near Bowditch page 332)
cos D = sinL1sinL2 + cosL1cosL2cosDLo
tan C = (sin DLo) / ((cosL1tanL2) - (sinL1cosDLo))
cos Lv = cosL1sinC
sin DLov = (cos C) / (sin Lv)
sin Dv = cos L1sinDLov
tan Lx = (cosDLovx)(tanLv)
sin Lx = (sinLv)(cosDvx)

sin DLovx = (sin Dvx) / (cos Lx)

Example 1

Example 2