osculation

Osculation is the concept of contact of higher order between geometric objects, typically curves, meaning they agree not only in position but also in direction and rate of change at a point. The term comes from Latin osculari, “to kiss,” reflecting the idea of close contact.

In plane geometry, the most common example is the osculating circle. At a given point on a smooth plane curve, the osculating circle has second-order contact with the curve: the two share the same position, the same tangent direction, and the same curvature. The center of this circle is the center of curvature, and its radius equals the reciprocal of the curvature at that point. The collection of all such centers forms the evolute of the curve, a locus that encodes how the curvature varies along the curve.

For space curves, the concept extends to the osculating plane. If a curve has nonzero curvature, its osculating plane is the plane spanned by the tangent and the principal normal vectors at a point, representing the best second-order approximation to the curve there. The binormal direction is perpendicular to this osculating plane, and the torsion measures how the curve departs from lying in that plane.

In astronomy and orbital mechanics, “osculation” describes an instantaneous orbit or set of orbital elements that is tangent to the actual trajectory at a given time. The osculating orbit matches the position and velocity at that moment, and the osculating elements are updated as the state evolves to maintain tangency with the real path.

See also: curvature, evolute, Frenet-Serret apparatus, contact order.