Relative convex hull¶
The smallest geodesically convex set containing given points while constrained to remain inside a surrounding polygon or simple closed region.
Core Idea¶
The relative convex hull of X within P is the intersection of all subsets of P that contain X and every P-geodesic between their points. Ordinary straight chords blocked by the boundary are replaced by shortest paths inside P, and the hull closes X under those geodesics. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of computational geometry. It is obstacle-aware convex enclosure under intrinsic shortest paths. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.
Scope of Application¶
Relative convex hull belongs to computational geometry and is useful where the analyst can specify a simple polygon or rectifiable closed curve P, point set X inside it, shortest paths constrained to P, relatively convex subsets, hull boundary and reflex obstacles, then evaluate ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity. The scope is broad within that domain but bounded by the need for ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Relative convex hull can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Relative convex hull. Relative convex hull compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a simple polygon or rectifiable closed curve P, point set X inside it, shortest paths constrained to P, relatively convex subsets, hull boundary and reflex obstacles. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of computational geometry because they reuse a simple polygon or rectifiable closed curve P, point set X inside it, shortest paths constrained to P, relatively convex subsets, hull boundary and reflex obstacles, Ordinary straight chords blocked by the boundary are replaced by shortest paths inside P, and the hull closes X under those geodesics., and type the carrier, state every parameter and convention in the definition, test that ambient region and geodesic metric are fixed and the hull remains inside P while satisfying relative convexity, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Relative convex hull Domain-specific
Parents (1) — more general patterns this builds on
-
Relative convex hull is a kind of Boundary Prime
The proposed strict upward parent is
prime:boundary.
Hierarchy path (1) — routes to 1 parentless root
- Relative convex hull → Boundary
Neighborhood in Abstraction Space¶
Relative convex hull sits in a crowded region of the domain-specific corpus (16th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Convex Geometry & Spatial Partition (35 abstractions)
Nearest neighbors
- Convex hull — 0.94
- Curve — 0.91
- Polyhedral space — 0.91
- Equichordal point problem — 0.91
- Geodesic convexity — 0.91
Computed from structural-signature embeddings · 2026-09-08