totallengden

Totallengden, often written totallængden in Danish, is a term that denotes the total length of a geometric object such as a curve, a path, or a network. In mathematics it corresponds to the concept of arc length. For a smooth space curve r(t) = (x(t), y(t), z(t)) with t in [a, b], totallengden L is L = ∫_a^b ||r'(t)|| dt. For a plane curve described by y = f(x), x in [a, b], L = ∫_a^b sqrt(1 + (f'(x))^2) dx. For a polygonal path composed of straight segments with lengths li, L = sum li.

Properties of totallengden include being nonnegative and invariant under reparameterization; it is additive for concatenated curves. It reflects the actual physical or geometric length of the object, not merely its coordinate representation.

Computationally, exact formulas exist for many standard curves, while for general shapes the length is typically computed numerically or approximated by approximating the curve with a polyline and summing segment lengths. In discrete data sets, totallengden is often estimated by summing the distances between consecutive samples.

Applications span several fields. In geometry and analysis, totallengden is used to compare curves and study their properties. In computer graphics and GIS, it informs rendering, path planning, and spatial analysis. In robotics and motion analysis, arc length relates to travel distance and timing.

See also: arc length, length of a curve, circumference, polyline length, metric.