Kuboidset

Kuboidset is a formal construct in discrete geometry and voxel-based modeling that represents a three-dimensional region as a finite collection of kuboids. A kuboid is an axis-aligned rectangular prism in three-dimensional integer space, specified by two opposite corners (x1, y1, z1) and (x2, y2, z2) with integer coordinates and x1 ≤ x2, y1 ≤ y2, z1 ≤ z2. The edges align with the coordinate axes, and the interior contains all lattice points within those bounds.

A kuboidset stores a region as a finite list of kuboids. The kuboids may overlap or abut; operations such as union, intersection, and set difference are defined on the collection, producing another kuboidset. In practice, implementations often reduce redundancy by merging overlapping or touching kuboids into larger ones when possible, aiming to produce a compact canonical representation.

Key computations include evaluating the volume of a kuboid, given by (x2−x1+1)(y2−y1+1)(z2−z1+1), and the volume of a kuboidset, which accounts for overlaps to avoid double counting. Queries commonly include containment checks (whether a point lies in any member), overlap tests between sets, and adjacency checks. Projections along an axis yield 2D representations that simplify certain analyses.

Applications of kuboidsets appear in voxel-based rendering and modeling, 3D printing preparation, collision detection in physics engines, and geographic information systems that rely on axis-aligned spatial partitioning. They offer a precise, scalable alternative to unstructured meshes when the region of interest is naturally decomposed into axis-aligned blocks.

History and terminology note that kuboidset is sometimes called cuboidset or voxelset in related literature. The term kuboid emphasizes the axis-aligned rectangular nature of each element, while kuboidset highlights its role as a collection that defines a larger region.