Arclängige

Arclängige is a mathematical term used to describe quantities or parameterizations that are expressed with respect to arc length, the length along a curve between two points. In English-language mathematics, arc length is a standard geometric concept; in German texts the adjective arclängige appears in phrases such as arclängige Parametrisierung or arclängige Koordinaten to indicate a dependence on the curve’s length rather than on an arbitrary parameter.

For a smooth parametric curve r(t) = (x(t), y(t), z(t)) defined on an interval [a, b], the arc length s between t = a and t = t is s(t) = ∫_a^t ||r'(u)|| du, where ||r'(u)|| is the speed of the curve. A curve is said to be parameterized by arc length if its parameter s itself serves as the parameter, so that ||dr/ds|| = 1 for all s in [0, L], where L is the total arc length. In such an arclängigen parameterization, distances along the curve correspond directly to differences in the parameter.

Applications of arclängige descriptions include simplifying the computation of lengths, facilitating certain analyses in differential geometry, and aiding numerical methods in physics and computer graphics. In German-language literature, arclängige concepts are often discussed alongside Bogenlänge (arc length) and Bogenlängenparametrisierung (arc-length parametrization). See also arc length, unit tangent vector, and curvature.