A geodesic is the shortest path between two points, taking into account the local geometry.
For example, let’s say you wanted to get from one side of a field to the other. In theory the shortest path would be the straight line through the middle. But if the middle of the field is all swampy that might slow you down – so the actual shortest path might involve going around the edge of the field.
Geodesics are particularly useful for long-distance travel, as they tell you the shortest route to take (particularly for ships and aeroplanes), accounting for the curvature of the Earth. It is why transatlantic flights tend to go very far North – it may look like a longer route on a map, but it is the shortest distance on a sphere.
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