NavList:

Lars Bergman, you wrote:
"It is not an exact 3:4:5 triangle."

Exactly --not exactly! ;) As I noted in my first message, we could make a true 3:4:5 triangle on the ground with laser-ranging or other modern ranging tools or with fairly good accuracy using many historical methods. This can be done, and then we can get as close as we want to a proper Euclidean plane triangle, limited only by our patience and budget. The part that I think is surprising for most navigation enthusiasts is that a simple triangle of latitude and longitude on a small scale like this doesn't quite work! This is distinct from the issue of the curvature of the globe that we talked about last month. 

Curvature of the globe comes into play as we go larger. A small triangle a few miles on a side is flat by almost any measure, but when you get to dozens and then hundreds of miles, curvature turns the supposedly flat triangle into a three-dimensional object. It's like a "tortilla chip", for us Americans, or maybe like a triangular chip of an eggshell -- it has real shape in three dimensions, not merely a flat outline, and if you peel it off the globe, you can't press it flat without breaking it. That, of course, was the point of the 3:4:5 discussion a couple of weeks ago. For surprisingly large dimensions, up to a few hundred miles (maybe surprising...), we can treat a spherical triangle as a flat triangle, and the results will be very good.

For small triangles, the match between plane triangles and spherical triangles is essentially perfect. If we physically measure the distances on the ground for distances up to five or ten miles, we would find no meaningful difference between a plane triangle and a spherical triangle. We can ditch the spherical triangle math [Note: code doesn't care... but for manual calculation and for common-sense "mental" analysis, the plane triangle is essentially perfect]. Instead of globe curvature, on a small scale, when drawing a triangle or analyzing any other problem using latitude and longitude coordinates, we run into the subtleties of the ellipsoidal definition of coordinates on the globe. 

Lars B., in your first post, you suggested "using Vincenty's formula". This is absolutely the "gold standard" for comparison, but it's also more than we need because it solves two distinct problems at once, both connected to the ellipsoidal spheroid -- in other terms, the "oblateness" of the Earth. Vincenty's formula, as well as a few related solutions, incorporates the ellipsoidal definition of latitude --which we want-- but it also (in fact, primarily) solves the problem of a "geodesic" on the ellipsoidal shape of the true globe. A "geodesic" is a geometric generalization of the concept of a great circle, but the ellipsoidal solution is far less useful on many levels. The practical applications of the ellipsoidal geodesics are few and far between (a couple of weeks ago, Robin Stuart mentioned an important exception in the history of radio navigation ...more in another post). I have not been able to come up with any practical benefit to knowing or even contemplating the long-distance ellipsoidal geodesics.

In your analysis yesterday, Lars, you referred to "an initial great circle course of 53°18.9' ". By that you mean the course given by Vincenty's formula, right? It's not a "great circle course" necessarily and certainly not in longer distance Vincenty solutions.

And here's where Vincenty's formula is sneaking in that second issue, which is absolutely important for local piloting calculations, as well as surveying. Latitude is not what it seems. It is not the angle at the Earth's center measured up from the equator to a point on the Earth's surface. Almost everyone who studied "some" navigation has run into the question of defining. Instead of the simple definition of ordinary "spherical coordinates", the latitude that we use in everything we do in navigation was carefully defined (or re-defined) a couple of centuries ago in a way that specifically respects astronomically observed latitudes: in its most basic form, the latitude of a point on the globe is determined by the celestial sphere, pulled down from the heavens on the local vertical --the local "normal", the line perpendicular to the local level, the local direction of gravity in such a way that the Declination of a star matches the latitude of a point below it on the globe. And because the Earth is an ellipsoidal spheroid (really, not the Earth but rather its net gravitational field), not quite a sphere, this forces the lines of latitude near the equator (especially) to be somewhat closer together than we expect. 

This has gotten long, so I will follow up with more detail in another post.

Lars, you concluded:
"Whether WGS84 is a reasonably good model for this specific part of the earth, or not, I have no idea."

That whole business of having different datums for different parts of the globe is defunct, extinct, as "dead as Julius Caesar" and "never to be resurrectionised" ;) [that was from Lecky's "Wrinkles" in 1883, talking about lunars, but it fits here, too]. The latitudes and longitudes we use in the 21st century all reference a simple global ellipsoid. It goes by the dreadfully clumsy techno name "WGS84". Maybe someday the GIS community, in search of funding, will put out a press release and announce a new, NASA-style backronym --the Terrestrial Harmonized Earth-reference datum (*)-- so it can finally be THE datum. And I know you, Lars, noted this quite correctly in a follow-up message. I'm just adding this paragraph for the continuing conversation. :)

Frank Reed

* Credit where due: I asked ChatGPT to invent this backronym. For any of you who have not sat down for a chat with the ChatGPT chatbot recently, I do now recommend it. On factual matters, it still has flaws, but try asking it something about a navigation history topic, like Lecky's "Wrinkles". You may be surprised. Don't forget to come up for air...