Osculating Sphere

osculating sphere

[¦äs·kyə‚lād·iŋ ′sfir] (mathematics) For a curve C at a point p, the limiting sphere obtained by taking the sphere that passes through p and three other points on C and then letting these three points approach p independently along C.

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.