Bogenlängenkonvention

The Bogenlängenkonvention, or arc length convention, refers to the standard method of measuring the length of a curve in mathematics. In Euclidean geometry, a curve is typically defined by a parametric representation in terms of a parameter, often denoted by 't'. The arc length 's' is then calculated by integrating the magnitude of the derivative of the position vector with respect to the parameter over a specified interval. This integral represents the total distance traveled along the curve as the parameter changes.

This convention is crucial in various fields, including calculus, differential geometry, and physics. For instance, in physics, it's used to describe the path of a moving object, where the arc length at any given time represents the total distance covered. In differential geometry, the arc length serves as a natural parameter for curves, simplifying many differential equations and geometric properties.

The formal definition involves a curve C parameterized by r(t) = (x(t), y(t), z(t)) for t in [a, b]. The arc length from t=a to any parameter value t is given by the integral: s(t) = ∫[a to t] ||r'(u)|| du, where ||r'(u)|| is the magnitude of the velocity vector r'(u) = (x'(u), y'(u), z'(u)). This convention ensures a consistent and unambiguous measure of curve length.