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Why are great circles considered straight paths on a sphere?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In the context of differential geometry, great circles are considered "straight" paths on a sphere because they are geodesics—curves that are intrinsically straight.1

While no curve on a sphere appears straight from an extrinsic (outside) perspective, a great circle is intrinsically straight because an object moving along one does not turn or curve with respect to the surface itself. In spherical geometry, these paths function as the equivalent of straight lines in Euclidean geometry.12

It is important to note that while great circles are intrinsically straight, they are not always the shortest distance between two points; an arc longer than a half-circle is still intrinsically straight but does not represent the shortest path. Generally, on a smooth and complete surface, every shortest curve is intrinsically straight, and every intrinsically straight curve is the shortest path between nearby points.12