Delintervals

Delintervals is a term used in theoretical discussions to describe intervals that carry an explicit uncertainty or delay component. In its common interpretation, a delinterval generalizes ordinary intervals on the real line by attaching a non-negative tolerance that accounts for measurement error, delay, or variation. The concept is frequently encountered in studies of interval arithmetic, robust geometry, and uncertain data management, though its exact formalism can vary between authors.

Definition and variants: Let I = [l, r] be a closed interval and d ≥ 0 a tolerance. A widely used form defines the delinterval associated with I and d as I_d = [l − d, r + d]. This expanded interval represents all values that could lie within the base interval after accounting for the specified uncertainty. Some authors describe delintervals in terms of the set of points within distance d of I, which on the real line yields the same expanded interval [l − d, r + d].

Operations: Standard interval operations extend to delintervals by applying them to the expanded endpoints. For two delintervals I_d and J_e, containment, union, and intersection follow the corresponding rules for real intervals, with endpoints adjusted by their tolerances. In particular, the intersection I_d ∩ J_e is nonempty when max(l − d, L − e) ≤ min(r + d, R + e).

Applications: Delintervals are used to model uncertain or delayed measurements in computational geometry, scheduling with imprecise data, and temporal databases. They provide a simple and conservative way to propagate uncertainty through interval-based computations.

History and terminology: The term delintervals appears in various papers and discussions as a lightweight way to encode tolerance in interval data. Because definitions can differ, practitioners should confirm the exact interpretation of d in a given source.

See also: interval, interval arithmetic, fuzzy intervals, interval graphs, uncertain data.