A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antoine Couëtte
Date: 2020 May 13, 00:40 -0700
Hello to all,
Dave Walden's initial problem recently asked for determining a Departure Point on Earth given its Departure Track Angle and Distance towards one Arrival Point.
Various solutions, both direct and iterative have been published in the course of the ensuing discussions.
Some "Direct Solutions" first start solving for the Longitude Difference between Departure point and Arrival Point through solving from a reverse-sine. This mathematically yields 2 solutions which can be very close : angles " α " and " π-α " and difficult to immediately keep or reject. Well known problem and Peter also indicated various earlier references dealing with this "possibly uncomfortable existence" of 2 such mathematical solutions.
Nonetheless from one submitted case of a similar problem Peter Hakel had the bright idea to modify his "one-body fix" spreadsheet to separately accommodate each of these 2 angles " α " and " π-α ".
And lo and behold, for this specific example Peter discovered the existence of 2 different Departure Points fulfilling both Azimut and Distance Conditions to one same Arrival point .
As subsequently mentionned here, this is a good example of an Equal Azimuth Line (030° towards one unique arrival point) having 2 different intersections with a Circle of Equal Altitudes (55°) / Equal Distances (35° or 2100 NM) centered on one same and unique Arrival Point.
Here we are ... Equal Azimuth Lines
Have they been earlier addressed onto NavList ? What is their equation, if any similar to the Cosine Formula for a great circle ? How do they look like ? Can we easily draw them on a chart ?
Probably trivial problem to Sun Burnt Land Surveyors.
As I certainly do no wish to die as a stupid ignorant here, it would be certainly interesting to further investigate this subject together.
Positive Contributions more than welcome.
Best Navigational Maritime and Land-Based Regards.