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A geodesic is the shortest path between two points on a curved surface; in this context applies to questions where the curved surface is the earth's surface.
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Modifying straight line segments to follow geodesic / great circle paths using QGIS?
You have to reproject every line into an azimutal equidistant projection based on one of its points, then densify the line.
Since all points of your grid are connected to three lines, you can densify …