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Osculating Sphere


The center of any sphere which has a contact of (at least) first-order with a curve C at a point P lies in the normal plane to C at P. The center of any sphere which has a contact of (at least) second-order with C at point P, where the curvature kappa>0, lies on the polar axis of C corresponding to P. All these spheres intersect the osculating plane of C at P along a circle of curvature at P. The osculating sphere has center

 a=x+rhoN^^+(rho^.)/tauB^^

where N^^ is the unit normal vector, B^^ is the unit binormal vector, rho is the radius of curvature, and tau is the torsion, and radius

 R=sqrt(rho^2+((rho^.)/tau)^2),

and has contact of (at least) third order with C.


See also

Curvature, Osculating Plane, Radius of Curvature, Sphere, Torsion

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References

Kreyszig, E. Differential Geometry. New York: Dover, pp. 54-55, 1991.

Referenced on Wolfram|Alpha

Osculating Sphere

Cite this as:

Weisstein, Eric W. "Osculating Sphere." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OsculatingSphere.html

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