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In differential geometry, conjugate points or focal points[1][2] 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 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.[3]
^Bishop, Richard L. and Crittenden, Richard J. Geometry of Manifolds. AMS Chelsea Publishing, 2001, pp.224-225.
^Hawking, Stephen; Ellis, George (1973). The large scale structure of space-time. Cambridge university press.
^Cheeger, Ebin. Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, 1975, pp. 17-18.
In differential geometry, conjugatepoints or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example...
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{f}}_{2}={\frac {e}{\sqrt {d^{2}-c^{2}}}}{-c \choose 1}} point to two conjugatepoints and the tools developed above are applicable. Example: For the ellipse...
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ombilic consists of a pair of complex conjugatepoints (which are double points). A real plane contains these two points if and only if it is perpendicular...
projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction:...
to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and...
example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter...
the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1 {\displaystyle y^{2}-x^{2}=1} to complement it...
Assume that γ ~ ( 0 ) {\displaystyle {\widetilde {\gamma }}(0)} has no conjugatepoints along γ ~ {\displaystyle {\widetilde {\gamma }}} , and let J , J ~...
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