Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|cite|improve this question
The 3-sphere of curvature is also called an osculating sphere. – Mike Spivey Sep 13 '11 at 4:00
But note that this is a sphere in contact with a curve, not with a two-dimensional surface. In general, a (hyper)surface doesn't have a sphere of curvature but a (hyper)ellipsoid of curvature, since it can have different curvature in different directions. By the way, if a (hyper)surface in $n$-dimensional space had a sphere of curvature, it would be an $(n-1)$-sphere, not an $n$-sphere. – joriki Sep 13 '11 at 8:07
Space curves do have osculating spheres and osculating circles (the circle formed by the intersection of the osculating sphere and osculating plane). – J. M. Sep 13 '11 at 8:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.