Osculating Sphere

osculating sphere [¦äs·kyə‚lād·iŋ ′sfir]

(mathematics)

For a curveCat a pointp, the limiting sphere obtained by taking the sphere that passes throughpand three other points onCand then letting these three points approachpindependently alongC.

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Osculating Sphere 

The osculating sphere of a space curve l at a point M is the sphere having contact of order n ≥ 3 with l at M. The osculating sphere can also be defined as the limit of a variable sphere passing through four points of l as the points tend to M.

The radius of the osculating sphere is

where ρ is the radius of curvature of l at M, ̓ is the torsion of l, and ds is the differential of arc of l.

REFERENCE

Rashevskii, P. K. Kurs differentsial’noi geomelrii, 4th ed. Moscow, 1956.