# Is the Line-Of-Sight Bearing equal to the Great Circle Path Initial Heading?

If you are at a known location (you know your precise latitude and longitude for example) and have an unobstructed view of another known location you can:
A: Take a precise visual bearing to the other location (for example with a compass and correcting for compass variation) – This is the Line-of-Sight bearing.
B: Calculate the Initial Heading of a Great Circle Path between the two points using a known formula. In each case you arrive at a true bearing.
My Question: Are these two equal?

At My location magnetic variation (from current aviation map) is approximately 14 ½ degrees East. Converting from observed magnetic bearing to true (astronomical) bearing you add the easterly variation.

where φ1,λ1 is the start point, φ2,λ2 the end point (Δλ is the difference in longitude)

JavaScript:
var y = Math.sin(λ2-λ1) * Math.cos(φ2);
var x = Math.cos(φ1)*Math.sin(φ2) -
Math.sin(φ1)*Math.cos(φ2)*Math.cos(λ2-λ1);
var brng = Math.atan2(y, x).toDegrees();

My reasoning for why I believe they are equal:

A Great Circle path between two points lies in a plane which passes through each point and the center of the earth. A straight line between the two points (the notional line-of-sight) also lies in this plane.
Therefore the line-of-sight is directly above the great circle path, and
the line of sight bearing is equal to the great circle initial bearing.

Is this reasoning correct?

No, they are not. Take two points at $$80^\circ N$$, one at $$0^\circ$$ and one at $$179^\circ E$$. The bearing from the first to the second is due East, but the great circle is almost over the North pole so the initial heading is almost North.
• No, it is just an extreme example. Even if you have two points close together on the $80^{th}$ parallel the great circle will go North of East even though the bearing is due East. The line of latitude is not a great circle, so it cannot be the shortest route. – Ross Millikan Oct 14 '18 at 23:09