Tangensiell

Tangensiell, or tangential in English, is an adjective used to describe anything related to a tangent. In mathematics, it denotes a direction, line, or vector that touches a curve at a point without crossing the curve there, or a quantity measured along that touching direction. The term also applies more broadly to the tangent concept in higher dimensions.

Etymology: The word derives from Latin tangens, “touching,” from tangere, “to touch.” The root is shared with related terms such as tangent line, tangent space, and tangential components in physics and engineering.

In geometry and calculus: A tangent to a smooth curve at a given point is the unique line that has the same first-order behavior as the curve there. For a parametric curve r(t) = (x(t), y(t)), the tangent vector at t0 is r′(t0). Two curves are tangent at a point if they share a common tangent line there. For a circle with center at the origin and radius r, the tangent at (x0, y0) is perpendicular to the radius, with the tangent line satisfying x0 x + y0 y = r^2 and a tangent vector of (-y0, x0).

In higher dimensions: The tangent space or tangent plane at a point on a surface consists of all possible tangent vectors to curves on the surface through that point. Tangent concepts provide first-order approximations of geometric objects.

Physics and engineering: Tangential components describe motion, velocity, or force along the tangent direction to a path or to a rotating object. Tangential speed, tangential acceleration, and tangential force are common terms, often contrasted with radial components.

Applications: Tangency concepts are central in computer graphics, integration of curves, differential geometry, and mechanics, where distinguishing tangent directions aids in modeling motion, contact, and local linear approximations.