Osculációs

Osculational is a term originating from differential geometry and calculus, referring to a high degree of contact between two curves or surfaces. When two curves are osculating, they not only touch at a point but also share the same tangent line and the same curvature at that point. This means they locally "mimic" each other very closely.

The concept can be extended to surfaces, where osculating surfaces share a common tangent plane and curvature at a point of contact. This implies a more profound geometric relationship than simple tangency, which only requires a shared tangent line or plane.

A key application of osculation is in the study of curvature. For a plane curve, the osculating circle is the circle that best approximates the curve at a given point. This circle has the same tangent and curvature as the curve at that point, and its radius is the reciprocal of the curvature, known as the radius of curvature.

The degree of contact can be quantified. Two curves have contact of order n if their first n derivatives match at the point of osculation. Simple tangency is contact of order at least 1. Osculation, in the strictest sense, implies contact of order at least 2, meaning the second derivatives also match, signifying a shared curvature. This high degree of contact makes osculating curves or surfaces behave similarly in their immediate vicinity.