Britannica Sources used in this answer

Why is a great-circle arc longer than a half circle not the shortest path between its endpoints?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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 differential geometry, a geodesic is defined as a curve that is intrinsically straight on a surface. On a sphere, these geodesics are arcs of great circles.12

While a great-circle arc is intrinsically straight, it is not necessarily the shortest path between two points. A great circle is divided into two arcs by any two points on it. If one arc is longer than a half circle, it represents a path that "wraps around" the sphere, whereas the other arc (the minor arc) provides a shorter route between the same two endpoints. Consequently, the shorter of the two arcs is the true shortest path, or great circle route, between those points.23

In summary, while both arcs are intrinsically straight, the property of being a "shortest path" is distinct from the property of being "intrinsically straight" when considering the global topology of the sphere.23