Lengthsuch

Lengthsuch is a term used in theoretical and computational geometry to describe a proposed measure of curve complexity that is tied to the length of polygonal approximations rather than to the classical arc length alone. The core idea is to quantify how long a curve effectively appears when it is approximated by a polyline within a specified tolerance, reflecting both the geometry of the curve and the limits of discretization.

Formally, for a rectifiable curve γ in a Euclidean space and a tolerance ε > 0, lengthsuch(γ, ε) can be defined as the infimum of the total length of polygonal paths P whose vertices lie on γ and such that every point of γ lies within a distance ε of P. In this sense, lengthsuch depends on both the curve and the chosen tolerance; as ε decreases, lengthsuch(γ, ε) is nondecreasing and approaches a value related to the traditional arc length when the curve is well-behaved. The concept is invariant under reparametrization of the curve and under similarity transforms, focusing on the geometric path rather than its particular representation.

In practice, lengthsuch offers a framework for analyzing discretization quality in applications such as computer graphics, path planning, and data compression, where curves are often sampled or approximated by polylines. It provides a way to compare curves with respect to how their complexity manifests under finite representations.

Lengthsuch remains a relatively informal or exploratory concept in published literature, often used to discuss the relationship between true geometric length and discretized approximations. See also arc length, curve simplification, and polygonal approximation.