Spatial Data Structures & Meshing

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Abstractions about representing and partitioning geometric space computationally, covering hierarchical spatial structures (quadtrees, binary space partitioning, unstructured and corner-point grids), mesh and surface construction (mesh generation, subdivision surfaces, polygon partition), and geometric-relation problems like visibility and motion planning.

17 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Binary space partitioning — A recursive geometric partition that divides a space by hyperplanes into two half-spaces and stores the resulting hierarchy in a binary tree.
• Chain code — A contour representation that encodes a connected digital boundary as a start coordinate followed by symbols for successive grid-neighbor directions.
• Contact graph — Represent a family of geometric objects by one vertex per object and an edge exactly when two objects satisfy a declared boundary-contact relation without prohibited interior overlap or crossing, making the permitted notion of touching part of the graph class.
• Corner-point grid — A three-dimensional hexahedral grid whose cells are defined by ordered pillars and independently positioned corner points along those pillars.
• Crofton formula — An integral-geometric identity recovering a curve’s length from the invariant-measure average number of intersections with lines.
• Line–line intersection — The classification and computation of the common-point set of two lines under a declared geometry, yielding no point, one point, or coincident lines in ordinary Euclidean settings.
• Mesh generation — The construction of a discrete cell complex approximating a continuous geometric domain for numerical simulation, balancing geometric fidelity, element quality and computational size.
• Motion planning — The computational problem of finding a collision-free sequence of valid configurations connecting a system’s start and goal states.
• Opaque set — A set of planar curves or segments intersecting every line that crosses a specified convex body.
• Passthrough (architecture) — A wall opening that transfers food, objects, service or communication between adjacent rooms while retaining more enclosure and storage than an open plan.
• Polygon partition — A decomposition of a polygon into nonoverlapping primitive polygons whose union is the original polygon, often optimized by piece count, boundary length or another criterion.
• Quadtree — A hierarchical spatial data structure whose internal nodes recursively divide a two-dimensional region into four child regions.
• Shape analysis (digital geometry) — The computational representation, measurement and comparison of geometric form independently of irrelevant pose, sampling and sometimes scale.
• Subdivision surface — A smooth limit surface generated by repeatedly refining a coarse polygonal control mesh with a fixed local subdivision rule.
• Unstructured grid — A mesh that tessellates a domain with irregularly connected cells such as triangles or tetrahedra, allowing local refinement around complex geometry.
• Vertex enumeration problem — The computational problem of listing all vertices of a polyhedral or discrete-geometric object from an implicit constraint representation.
• Visibility (geometry) — A geometric relation in which two points see one another when the line segment joining them remains inside free space and avoids declared obstacles.