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.
From: https://www.movable-type.co.uk/scripts/latlong.html
where
φ1,λ1 is the start point, φ2,λ2 the end point (Δλ is the difference in longitude)
JavaScript:
(all angles
in radians)
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?
Best Answer
For a spherical representation of the Earth, yes, your reasoning is correct. A great circle is defined as being the intersection of the sphere and a plane that passes through the sphere's center point. A line of sight, for most practical purposes, can be considered a straight line, and of course, lies on the same plane as its corresponding great circle plane. The initial bearings are angles with respect to North and are measured on the sphere's surface from the projection of the line along the normal. The angles are equal for both line of sight and great circle, and for all altitudes as well.
In reality, the Earth being slightly squashed, it is actually closer to an ellipsoid of revolution. Bearings and distances on an ellipsoid are computed on a geodesic. A geodesic on the ellipsoid does not lie on a plane (except for the equator and meridians), and the line of sight would not necessarily have the same start and end bearings as the geodesic, but we are talking about very small differences here, especially for short distances, negligible in most practical cases.