How to illustrate geodesic line reflecting bearing between two points + third projected point

Question

I'm using Spatialite + ST_Project to extrapolate a line with an implied bearing between two points to some distance beyond those two points. Here's an example sql snippet:

SELECT AsGeoJSON(ST_Project(
    GeomFromText('POINT (-73.87172699 40.76195099)'),
    5000000, 
    ST_Azimuth(
        ST_GeomFromText('POINT (-73.87172699 40.76195099)'), ST_GeomFromText('POINT (-73.887831 40.804936)')
    )
));

The third point produced from this reflects a geodesic, at least according to the PostGIS documentation for ST_Project. And you can see in the images below that the "bearing" as illustrated on a spherical projection looks to be in line with the original two points. But when I try to create a line string from joining these three points — either using ST_MakeLine or by manually assembling the new point with the existing two points in WKT text — the resulting line appears to use a great circle, and on a spherical projection it looks wrong, and doesn't align with the original two points' bearing:

		as GeoJSON
Two original points		{"type":"FeatureCollection","features":[{"type":"Feature","properties":{},"geometry":{"type":"Point","coordinates":[-73.87172698999998,40.76195099]}},{"type":"Feature","properties":{},"geometry":{"type":"Point","coordinates":[-73.887831,40.80493599999999]}}]}
Projected third point. Azimuth looks good!		{"type":"FeatureCollection","features":[{"type":"Feature","properties":{},"geometry":{"type":"Point","coordinates":[-73.87172698999998,40.76195099]}},{"type":"Feature","properties":{},"geometry":{"type":"Point","coordinates":[-73.887831,40.80493599999999]}},{"type":"Feature","properties":{},"geometry":{"type":"Point","coordinates":[155.052772,71.177753]}}]}
Linestring between all three points. Wha happened?		{"type":"FeatureCollection","features":[{"type":"Feature","properties":{},"geometry":{"type":"LineString","coordinates":[[-73.87172698999998,40.76195099],[-73.887831,40.80493599999999],[155.052772,71.177753]]}}]}

Is there a way to draw a line, with Spatialite or PostGIS, that would follow the projected geodesic used by ST_Project? I'm trying to illustrate what it would look like to walk in a visually straight line between the two original points to some arbitrary distance beyond those two points; I understand the great circle line here to be the shorter path, but I'm not sure I understand if they also reflect the path you'd take were you to walk along the earth from the azimuth/bearing between the two original points. Is there a different PostGIS/Spatialite function I should be using to illustrate this geodesic? Am I thinking about this illustration wrong, and somehow the great circle line here is somehow more "correct" to reflect the path you'd take were you to walk in a straight line from the original two points?

Answer 1

The problem is not in your data, it's the rendering. Depending of the tool you use, you will have potentially different results. For example, if you use a planar projection like with QGIS in 3857, you will have something like this (with straight arrow):

But the result is not what you want, this path is the shortest path along the projected map, not on the earth. You can instead use a map around the north pole, like the projection 102016 for example, you will have a more straight line and you can more easily see that it's the shortest path:

It's a general rule to use a projection centered on the center of your data if you want it to be easily readable and representative.

But if you want a 3D representation, I think that can be obtained with Google Earth. Save your geojson as a file, convert it in KML (you can find tools to do that quite easily), go to Google Earth, in project import your KML file and you can have this kind of result: