Curveswhether

Curveswhether is a term that appears in some mathematical and computational discussions to denote a boolean-valued property associated with a curve. It is not a standardized concept with a single accepted definition, and usages may vary between authors and contexts.

Definition and variants. Broadly, a curveswhether is a function that assigns a true/false value to a curve based on whether the curve satisfies a prescribed predicate P. If a curve is given by a smooth map gamma: [a,b] -> R^n, a curveswhether W(gamma) may be defined in different ways. In a universal (for all) variant, W(gamma) is true if P(gamma(t)) holds for every t in [a,b]. In an existential (there exists) variant, W(gamma) is true if P(gamma(t)) holds for some t in [a,b]. Other formulations combine pointwise checks with aggregation rules, such as counting the measure of t for which P holds or using threshold criteria.

Examples. P can express a geometric or analytic condition, such as gamma(t) lying inside a specified region, the curvature kappa(t) staying within a range, or gamma(t) avoiding a set of forbidden zones. In computer graphics and path planning, a curveswhether might encode feasibility of a trajectory with respect to obstacles or safety constraints.

Relation to other concepts. Curveswhether relates to predicate evaluation along curves, indicator functions, and boolean signals in dynamical systems. It is distinct from classical invariants of a curve (like curvature or torsion) and is typically used as a practical check within algorithms rather than a fundamental geometric quantity.

Notes. Because the term is informal and not widely standardized, definitions and applications should be clarified in context. See also: curves, curvature, predicate logic, boolean function.