Subpaving¶
Nonoverlapping boxes representing or approximating a region in interval analysis.
Core Idea¶
A subpaving represents a region by a finite family of nonoverlapping interval boxes in R^n. Rather than describing every point individually, interval analysis classifies boxes as definitely inside, definitely outside or unresolved relative to a target set. Unions of inner and outer box families can bracket the true set, with smaller boxes improving boundary precision at additional computational cost.
Jaulin's planar ring construction exhibits the method directly. Drevelle, Jaulin and Zerr then use subpavings to bound the space an uncertain robot sensor has definitely or possibly explored in a simulated underwater mission. The term does not imply exhaustive coverage of all R^n, exact equality with the target, or a fielded robot deployment. The previously inherited R+ notation is corrected to R^n, as the author's sources state.
Scope of Application¶
A box family can approximate a set without covering the whole domain or the exact target.
- Interval analysis. Represent solution sets with certified enclosures.
- Robot localization. Bound feasible poses under sensor uncertainty.
- Set inversion. Classify cells against nonlinear constraints.
- Geometric computation. Trade resolution against storage and runtime.
Clarity¶
A subpaving is a nonoverlapping family of interval boxes in R^n. Its union may form an inner or outer set approximation, not necessarily an exact partition of the target or whole domain. Jaulin's ring construction and a simulated underwater-robot exploration study show the mathematical and applied uses, respectively.
Manages Complexity¶
Box classification replaces a continuum of points with a finite, refineable representation. Inner boxes certify membership and outer boxes preserve possible solutions; unresolved boundary cells carry approximation error. Refinement sharpens the boundary but grows memory and computation. Shared closed-box faces also require care before asserting a strict partition.
Abstract Reasoning¶
Choose a framed target set, classify or bisect interval boxes, preserve nonoverlapping interiors, take their union as an inner or outer approximation, and state resolution and boundary conventions.
Knowledge Transfer¶
The finite-cell approximation pattern transfers to uncertainty mapping and numerical set computation, but literal subpavings require interval boxes and the stated nonoverlap convention. A polygon mesh or overlapping particle cloud may serve a similar purpose without being this structure.
Neighborhood in Abstraction Space¶
Subpaving sits in a sparse region of the domain-specific corpus (65th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Algebraic Varieties & Topological Invariants (27 abstractions)
Nearest neighbors
- Solid Modeling — 0.86
- Algebraic Surface — 0.85
- Convex body — 0.84
- Assouad–Nagata Dimension — 0.84
- Urysohn's lemma — 0.84
Computed from structural-signature embeddings · 2026-10-08