Convex Geometry & Spatial Partition

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Abstractions about shapes, hulls, partitions, intersections, visibility, packing, meshing, and motion through geometric space. They include convexity and support structures, polyhedral constructions, digital shape analysis, spatial optimization, and incidence problems.

35 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.

• Acute and obtuse triangles — A Euclidean triangle classification based on whether all angles are below a right angle or exactly one angle exceeds it.
• Affine plank problem — The conjecture that planks covering a convex body have total relative width at least one.
• 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.
• 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.
• Convex hull — The smallest convex set containing a given set, equivalently all finite convex combinations of its points.
• Corner-point grid — A three-dimensional hexahedral grid whose cells are defined by ordered pillars and independently positioned corner points along those pillars.
• Danzer set — A set of points in Euclidean space that intersects every convex body of unit volume, studied through the unresolved question of whether bounded-density examples exist.
• Datum reference — A declared geometrically significant point, axis, plane or feature used as the origin or constraint for defining and measuring other features of an object.
• Equichordal point problem — The plane-geometry question whether a convex body can have two distinct interior points through each of which every chord has the same point-specific length, answered negatively.
• Fat object (geometry) — A geometric object whose extent is comparable in every direction under a declared fatness criterion, excluding arbitrarily thin or needle-like shapes and enabling stronger algorithmic bounds.
• Hilbert metric — A projectively invariant metric on the interior of a bounded convex domain, defined by a logarithmic cross ratio of the two boundary intersections on the line through a point pair.
• Kissing number — The maximum number of nonoverlapping congruent spheres that can simultaneously touch one congruent central sphere in a specified space or dimension.
• 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.
• One-seventh area triangle — The inner triangle formed by three one-third cevians of a triangle, whose area is exactly one seventh of the original.
• Opaque set — A set of planar curves or segments intersecting every line that crosses a specified convex body.
• Orthocentric system — A planar set of four points in which each point is the orthocenter of the triangle formed by the other three.
• Perfect rectangle — A rectangle tiled exactly by finitely many squares whose side lengths are all distinct.
• 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.
• Quadrisecant — A line meeting a spatial curve or related geometric set in four distinct points, a maximal generic multi-secant whose existence and ordering carry knot- and algebraic-geometric information.
• Relative convex hull — The smallest geodesically convex set containing given points while constrained to remain inside a surrounding polygon or simple closed region.
• Reuleaux polygon — A convex constant-width curve assembled from an odd number of equal-radius circular arcs, with each arc centered at an opposite vertex of its generating polygon.
• Shape analysis (digital geometry) — The computational representation, measurement and comparison of geometric form independently of irrelevant pose, sampling and sometimes scale.
• Simplex — The convex hull of n+1 affinely independent points in n-dimensional space, generalizing a point, segment, triangle and tetrahedron as the simplest full-dimensional polytope.
• Star polyhedron — A nonconvex polyhedron with systematic star-like self-intersection or alternating salient and reentrant geometry.
• Subdivision surface — A smooth limit surface generated by repeatedly refining a coarse polygonal control mesh with a fixed local subdivision rule.
• Supporting hyperplane — A hyperplane meeting a set while the entire set lies in one of the two closed half-spaces it bounds.
• Tangential quadrilateral — A convex quadrilateral whose four sides are tangent to one inscribed circle.
• Tetrahedron packing — The geometric optimization problem of arranging congruent regular tetrahedra without overlapping so as to maximize the fraction of three-dimensional space they occupy.
• Uniform polyhedron — A polyhedron with regular polygonal faces and a symmetry group transitive on vertices, including convex, star and self-intersecting examples.
• Unstructured grid — A mesh that tessellates a domain with irregularly connected cells such as triangles or tetrahedra, allowing local refinement around complex geometry.
• Variation diminishing property — A transformation is variation diminishing when it cannot increase sign changes, oscillations, or intersection counts, making its output no more geometrically or algebraically variable than its input control data.
• 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.