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I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented parametricaly, how does one determine the $n$-sphere of curvature at any given point?

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    Space curves do have osculating spheres and osculating circles (the circle formed by the intersection of the osculating sphere and osculating plane).2011-09-13

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