osculating

Osculating is a term used in geometry and astronomy to describe a high-order contact between a curve or trajectory and a simple geometric object at a point, historically implying a “kiss” or close touch.

In differential geometry, the osculating circle of a plane curve at a given point is the circle that best approximates the curve there. It has second-order contact with the curve, meaning the circle agrees with the curve up to the second derivative at that point. The center of this circle is the center of curvature, and its reciprocal lies in the curvature κ = 1/R, where R is the radius of curvature. The osculating circle lies in the osculating plane, the plane spanned by the curve’s tangent and principal normal. This circle provides a local quadratic approximation to the curve.

For space curves, concepts extend similarly: the osculating plane at a point is defined by the tangent and principal normal, and the osculating circle, when it exists, lies in that plane and matches the curve’s behavior up to second order there. The Frenet frame (tangent, normal, binormal) describes how the curve twists and bends, with curvature and torsion quantifying departure from a straight line and a plane, respectively.

In astronomy, an osculating orbit is the instantaneous Keplerian orbit that precisely matches a body’s position and velocity at a given time. As the actual path is perturbed, the osculating orbital elements evolve to reflect the current state, with the term indicating the local, touching fit to the true trajectory.