contourlengte

Contourlengte, or contour length, is a measure of the length of a curve or contour. It is the geometric quantity that describes how long a curve extends in space, regardless of how it is traced or parameterized.

In mathematics, for a parameterized curve r(t) = (x(t), y(t), z(t)) with t ranging from a to b, the contourlengte is defined as L = ∫_a^b ||r′(t)|| dt. In two dimensions, where r(t) = (x(t), y(t)), this becomes L = ∫_a^b sqrt((dx/dt)² + (dy/dt)²) dt. If a curve is given as y = f(x) on [a, b], then L = ∫_a^b sqrt(1 + (dy/dx)²) dx. In practice, curves are often sampled discretely, and the length is approximated by L ≈ Σ_i |P_i+1 − P_i| for successive points P_i.

Properties include that contour lengte is invariant under reparameterization, depends on the actual geometry of the curve, and is finite for rectifiable curves. For closed contours, the length is related to the perimeter of the enclosed region, with the isoperimetric principle stating that among all shapes of a given area, the circle minimizes length.

Applications span several fields. In digital image analysis, contourlengte serves as a boundary descriptor for objects. In geographic information systems, it corresponds to the length of contour lines or object perimeters. In polymer physics, the contour length is the fully stretched length of a polymer chain, L = Nℓ, where N is the number of segments and ℓ their length, distinguishing it from the end-to-end distance.