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This topic has appeared in the trending rankings 1 time(s) in the past year. While it does not trend frequently, its appearance suggests a renewed or concentrated surge of public interest.
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This topic has appeared in the English Wikipedia rankings 1 time. It first appeared on 2026-06-06 and was most recently seen on 2026-06-06.
In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are locally length-minimizing, but not necessarily globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) globally length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.
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