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What is the relationship between geodesics and shortest paths on smooth surfaces?
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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 differential geometry, a geodesic is defined as a curve on a surface that is "intrinsically straight," meaning it possesses no curvature relative to the surface itself.1
The relationship between geodesics and shortest paths is nuanced:
- Local Property: On a complete, smooth surface, every shortest path between two points is a geodesic. Conversely, every geodesic is the shortest path between its endpoints, provided the points are sufficiently close to one another.1
- Global Distinction: A geodesic is not always the shortest path globally. For example, on a sphere, a great circle arc longer than a semicircle is intrinsically straight (a geodesic) but does not represent the shortest distance between its endpoints.1
While a geodesic is often described as the shortest line between two points on a surface, this definition strictly applies to local distances. In summary, while all shortest paths are geodesics, not all geodesics are necessarily the shortest path between two distant points.21