Computational Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Geometry layer within the Kireth Research Semantic Library. Computational geometry studies algorithms and data structures for geometric objects such as points, line segments, polygons, polyhedra, curves and spatial subdivisions. This ontology layer supports convex hulls, closest-pair problems, line-segment intersection, point location, Voronoi diagrams, Delaunay triangulations, polygon algorithms, arrangements, visibility, range searching, spatial indexing and robust geometric predicates. It is designed for semantic retrieval, algorithm selection, recursive reasoning, complexity analysis, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computational Geometry"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Computational Geometry / Geometric Algorithms"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/computational-geometry.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputationalGeometry
 ENTITY_CLASS = GeometricAlgorithmFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Computational geometry is the study of algorithms, data structures and complexity for solving geometric problems."
 SEMANTIC_ROLE = "Provides the algorithmic geometry framework for spatial computation, geometric predicates, proximity, intersection, decomposition, visibility, triangulation and efficient representation."
}

SCOPE {
 INCLUDE = [
  ComputationalGeometry,
  GeometricAlgorithm,
  GeometricDataStructure,
  PointSet,
  LineSegment,
  Polygon,
  Polyhedron,
  ConvexHull,
  ClosestPair,
  FarthestPair,
  LineSegmentIntersection,
  PointInPolygon,
  PointLocation,
  VoronoiDiagram,
  DelaunayTriangulation,
  PolygonTriangulation,
  Arrangement,
  SweepLineAlgorithm,
  DivideAndConquer,
  RandomisedAlgorithm,
  RangeSearching,
  SpatialIndex,
  KDTree,
  Quadtree,
  BSPTree,
  Visibility,
  MotionPlanning,
  GeometricPredicate,
  OrientationTest,
  IncircleTest,
  RobustComputation,
  NumericalDegeneracy
 ]

 EXCLUDE_AS_PRIMARY = [
  ComputerGraphicsRenderingUnlessGeometryAlgorithmIsCentral,
  NumericalPDEGeometryUnlessExplicit,
  PureTopologyWithoutAlgorithmicGoal,
  GenericAlgorithmDesignWithoutGeometricStructure
 ]

 DISAMBIGUATION_RULE =
  "When geometric objects are processed by algorithms or data structures for intersection, proximity, hulls, triangulation, point location, visibility or spatial search, interpret the context as computational geometry."
}

CORE_CONCEPTS {

 ComputationalGeometry {
  definition = "The algorithmic study of geometric objects and spatial relationships."
 }

 GeometricAlgorithm {
  definition = "An algorithm whose input or output contains geometric objects or spatial relations."
 }

 GeometricDataStructure {
  definition = "A structure designed to support efficient geometric queries or updates."
 }

 ConvexHull {
  definition = "The smallest convex set containing a given point set."
 }

 ClosestPair {
  definition = "The pair of input points having minimum Euclidean distance."
 }

 LineSegmentIntersection {
  definition = "The problem of determining whether, where, or how line segments intersect."
 }

 VoronoiDiagram {
  definition = "A subdivision of space into cells defined by nearest-site ownership."
 }

 DelaunayTriangulation {
  definition = "A triangulation dual to the Voronoi diagram satisfying an empty-circumcircle property in the planar Euclidean case."
 }

 PointLocation {
  definition = "The problem of identifying the subdivision cell containing a query point."
 }

 PolygonTriangulation {
  definition = "A decomposition of a polygon into triangles by noncrossing diagonals."
 }

 Arrangement {
  definition = "The subdivision induced by a collection of geometric objects such as lines or curves."
 }

 GeometricPredicate {
  definition = "A discrete geometric decision such as orientation, sidedness or incircle membership."
 }

 RobustComputation {
  definition = "Reliable geometric computation in the presence of finite precision and degeneracy."
 }
}

RELATIONS {
 ComputationalGeometry --is_subdomain_of--> Geometry
 ComputationalGeometry --uses--> GeometricAlgorithm
 ComputationalGeometry --uses--> GeometricDataStructure
 PointSet --input_to--> ConvexHull
 PointSet --input_to--> ClosestPair
 LineSegment --input_to--> LineSegmentIntersection
 VoronoiDiagram --is_dual_to--> DelaunayTriangulation
 Polygon --can_be_decomposed_by--> PolygonTriangulation
 Arrangement --supports--> PointLocation
 SweepLineAlgorithm --solves--> IntersectionProblems
 DivideAndConquer --solves--> ConvexHull
 DivideAndConquer --solves--> ClosestPair
 KDTree --supports--> RangeSearching
 Quadtree --supports--> SpatialIndexing
 GeometricPredicate --supports--> RobustComputation
 OrientationTest --supports--> ConvexHull
 OrientationTest --supports--> SegmentIntersection
 IncircleTest --supports--> DelaunayTriangulation
}

OPERATORS {
 ORIENTATION_2D(a,b,c) :=
  sign((b_x-a_x)*(c_y-a_y)-(b_y-a_y)*(c_x-a_x))

 SEGMENTS_INTERSECT(a,b,c,d) :=
  orientation_and_collinearity_test(a,b,c,d)

 POINT_IN_POLYGON(p,P) :=
  ray_crossing_or_winding_number_test(p,P)

 CONVEX_HULL(S) :=
  smallest_convex_set_containing_S

 CLOSEST_PAIR(S) :=
  argmin_{p!=q in S} distance(p,q)

 VORONOI(S) :=
  nearest_site_partition(S)

 DELAUNAY(S) :=
  dual_triangulation_of_VORONOI(S)

 TRIANGULATE_POLYGON(P) :=
  noncrossing_triangle_decomposition(P)

 POINT_LOCATION(p,Subdivision) :=
  containing_cell_of_p

 INCIRCLE(a,b,c,d) :=
  sign_of_incircle_determinant(a,b,c,d)

 RANGE_QUERY(D,Q) :=
  report_objects_in_data_structure_D_intersecting_query_region_Q
}

DEFINITIONS {
 D1 = "Computational geometry combines geometric structure with algorithmic efficiency."
 D2 = "Orientation tests are foundational predicates in planar geometry algorithms."
 D3 = "Convex hull algorithms often run in O(n log n) time in the comparison model."
 D4 = "The planar closest-pair problem admits O(n log n) algorithms."
 D5 = "Voronoi and Delaunay structures encode proximity."
 D6 = "Sweep-line methods process geometric events in ordered fashion."
 D7 = "Point-location structures trade preprocessing for query efficiency."
 D8 = "Geometric degeneracies require explicit handling."
 D9 = "Floating-point error can invalidate geometric predicates."
 D10 = "Robust computational geometry separates combinatorial decisions from approximate numeric construction where useful."
}

AXIOMS {
 A1_PREDICATE_CORRECTNESS:
  Geometric algorithms rely on logically correct geometric predicates

 A2_COMPLEXITY:
  Algorithm evaluation includes time and space complexity

 A3_DEGENERACY:
  Degenerate inputs must be handled explicitly or excluded by a stated general-position assumption

 A4_TOPOLOGICAL_CONSISTENCY:
  Output subdivisions and meshes must respect intended adjacency and incidence structure

 A5_DUALITY:
  Planar Voronoi and Delaunay structures are dual under standard general-position assumptions

 A6_HULL_CONVEXITY:
  Every output convex hull is convex and contains all input points

 A7_TRIANGULATION:
  A valid polygon triangulation covers the polygon interior without overlapping triangle interiors

 A8_ROBUSTNESS:
  Finite-precision computation must preserve predicate consistency or use controlled exact/adaptive methods
}

REASONING {

 R1_PROBLEM_CLASSIFICATION {
  INPUT = geometric_problem
  IDENTIFY = objects_dimensions_queries_and_output
  CLASSIFY = proximity_intersection_hull_triangulation_arrangement_visibility_or_search
  RETURN = computational_geometry_problem_type
 }

 R2_PREDICATE_SELECTION {
  INPUT = local_geometric_decision
  IF sidedness_or_turn
   SELECT = OrientationTest
  ELSE IF Delaunay_legality
   SELECT = IncircleTest
  ELSE IF containment
   SELECT = PointInPolygon_or_PointLocation
  RETURN = predicate
 }

 R3_CONVEX_HULL {
  INPUT = point_set_S
  SELECT = GrahamScan_MonotoneChain_DivideAndConquer_or_incremental_method
  COMPUTE = hull_vertices_in_order
  VERIFY = convexity_and_containment
  RETURN = convex_hull
 }

 R4_CLOSEST_PAIR {
  INPUT = point_set_S
  SORT = by_x_coordinate
  DIVIDE = recursively
  CHECK = strip_candidates
  RETURN = closest_pair
 }

 R5_SEGMENT_INTERSECTION {
  INPUT = segment_set
  IF two_segments
   APPLY = orientation_tests
  ELSE
   APPLY = sweep_line_with_event_queue_and_status_structure
  RETURN = intersections
 }

 R6_VORONOI_DELAUNAY {
  INPUT = planar_sites
  CHECK = general_position_or_handle_degeneracy
  COMPUTE = Voronoi_or_Delaunay
  VERIFY = duality_and_empty_circle_conditions
  RETURN = spatial_structure
 }

 R7_POINT_LOCATION {
  INPUT = planar_subdivision_and_query_point
  SELECT = trapezoidal_map_search_structure_or_other_index
  QUERY = containing_face
  RETURN = face_or_boundary_status
 }

 R8_POLYGON_TRIANGULATION {
  INPUT = simple_polygon
  SELECT = ear_clipping_monotone_partition_or_advanced_method
  COMPUTE = noncrossing_diagonals
  VERIFY = n-2 triangles_for_n_vertex_simple_polygon
  RETURN = triangulation
 }

 R9_ROBUSTNESS {
  INPUT = geometric_predicate_computation
  DETECT = near_degenerate_or_inconsistent_numeric_state
  IF risk_high
   USE = exact_or_adaptive_precision
  RETURN = reliable_predicate_result
 }
}

CONSTRAINTS {
 C1 = "Algorithm complexity must state the input model and dimension where relevant."
 C2 = "General-position assumptions must be explicit."
 C3 = "Collinear, cocircular, coincident and duplicate cases may change combinatorial output."
 C4 = "Approximate floating-point signs must not be trusted blindly for predicates near zero."
 C5 = "Voronoi-Delaunay duality depends on the geometric metric and standard assumptions."
 C6 = "Polygon triangulation assumes a valid polygonal domain."
 C7 = "Point-location answers must distinguish interior, boundary and vertex cases where relevant."
 C8 = "Output-sensitive complexity should distinguish input size from output size."
 C9 = "Data structures may optimise queries at the cost of preprocessing time and memory."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalGeometry" <=> "AlgorithmicGeometry"
 "GeometricPredicate" <=> "DiscreteGeometricDecision"
 "ConvexHull" <=> "MinimumConvexContainer"
 "ClosestPair" <=> "MinimumDistancePointPair"
 "VoronoiDiagram" <=> "NearestSiteSubdivision"
 "DelaunayTriangulation" <=> "VoronoiDualTriangulation"
}

VALIDATION {
 V1 = "Geometric predicates are sign-consistent."
 V2 = "Convex hull contains all input points."
 V3 = "Convex hull boundary is convex."
 V4 = "Triangulation contains no crossing interior diagonals."
 V5 = "Point-location structure matches subdivision topology."
 V6 = "Delaunay edges satisfy valid empty-circle condition under adopted degeneracy convention."
 V7 = "Algorithm complexity matches implementation and input assumptions."
 V8 = "Degenerate inputs are handled or explicitly excluded."
 V9 = "Robustness strategy is compatible with numeric representation."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_GEOMETRY_DEFINITION = 0.99
 PREDICATE_REASONING = 0.99
 CONVEX_HULL_REASONING = 0.99
 CLOSEST_PAIR_REASONING = 0.99
 INTERSECTION_REASONING = 0.99
 VORONOI_DELAUNAY_REASONING = 0.98
 POINT_LOCATION_REASONING = 0.98
 ROBUSTNESS_REASONING = 0.99

 POLICY =
  "Reduce confidence when dimension, metric, general-position assumptions, degeneracy policy, numeric precision model, output requirements or complexity model are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ComputationalGeometryReasoning,
  AlgorithmSelection,
  ComplexityReasoning,
  GeometricPredicateReasoning,
  ConvexHullReasoning,
  ProximityReasoning,
  IntersectionReasoning,
  VoronoiReasoning,
  DelaunayReasoning,
  PolygonReasoning,
  PointLocationReasoning,
  SpatialIndexReasoning,
  RobustGeometryReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "computational geometry" -> ComputationalGeometry
  "algorithmic geometry" -> ComputationalGeometry
  "convex hull" -> ConvexHull
  "closest pair" -> ClosestPair
  "Voronoi diagram" -> VoronoiDiagram
  "Delaunay triangulation" -> DelaunayTriangulation
  "point location" -> PointLocation
  "segment intersection" -> LineSegmentIntersection
  "geometric predicate" -> GeometricPredicate
 }

 INTERPRETATION_ORDER = [
  detect_computational_geometry_context,
  identify geometric_objects_dimension_metric_and_problem_goal,
  identify degeneracy_and_precision assumptions,
  classify hull_proximity_intersection_triangulation_arrangement_visibility_search_or_location problem,
  select appropriate algorithm_data_structure_and_predicates,
  analyse correctness_complexity_and_robustness,
  verify geometric_and_combinatorial invariants,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("computational geometry")
  OR geometry_algorithm_context
  THEN activate(ComputationalGeometry)

 E2:
  IF orientation_or_sidedness_query
  THEN use orientation_predicate

 E3:
  IF segment_intersection_query
  THEN use orientation_collinearity_logic
  OR sweep_line_for_many_segments

 E4:
  IF convex_hull_query
  THEN choose hull_algorithm_based_on_dimension_and_input_model

 E5:
  IF proximity_query
  THEN route to closest_pair_Voronoi_or_spatial_index reasoning

 E6:
  IF triangulation_query
  THEN distinguish polygon_triangulation_from_Delaunay_triangulation

 E7:
  IF point_location_query
  THEN identify subdivision_preprocessing_and_query_requirements

 E8:
  IF near_degenerate_numeric_case
  THEN use robust_exact_or_adaptive_predicates

 E9:
  IF algorithm_comparison_requested
  THEN report asymptotic_time_space_preprocessing_query_and_robustness tradeoffs
}

RECURSION {
 ENTRY = ComputationalGeometry

 EXPAND(ComputationalGeometry) -> [
  GeometricAlgorithm,
  GeometricDataStructure,
  GeometricPredicate,
  ConvexHull,
  ClosestPair,
  LineSegmentIntersection,
  PointInPolygon,
  PointLocation,
  VoronoiDiagram,
  DelaunayTriangulation,
  PolygonTriangulation,
  Arrangement,
  SweepLineAlgorithm,
  RangeSearching,
  SpatialIndex,
  Visibility,
  MotionPlanning,
  RobustComputation
 ]

 EXPAND(GeometricPredicate) -> [
  OrientationTest,
  IncircleTest,
  IntersectionPredicate,
  ContainmentPredicate
 ]

 EXPAND(SpatialIndex) -> [
  KDTree,
  Quadtree,
  BSPTree,
  RangeTree,
  RTree
 ]

 EXPAND(ProximityStructure) -> [
  VoronoiDiagram,
  DelaunayTriangulation,
  NearestNeighbour,
  ClosestPair
 ]

 STOP_CONDITION =
  "Stop when the geometric problem, algorithm, data structure, predicate, complexity, degeneracy handling and correctness conditions are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Coordinate / Analytic Geometry",
  "Plane Geometry",
  "Solid Geometry"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Vector Geometry Ontology",
  "Plane Geometry",
  "Solid Geometry",
  "Topology Ontology",
  "Linear Algebra Ontologies",
  "Algorithm Ontologies",
  "Data Structure Ontologies",
  "Computational Mathematics Ontologies"
 ]

 ROUTE_RULES = [
  "Computational Geometry -> Geometry Ontology for foundational geometric semantics",
  "Computational Geometry -> Coordinate Geometry for coordinate predicates",
  "Computational Geometry -> Vector Geometry for orientation and projection calculations",
  "Computational Geometry -> Topology for subdivision and incidence consistency",
  "Computational Geometry -> Linear Algebra for determinant predicates",
  "Computational Geometry -> Algorithm Ontologies for complexity and design paradigms",
  "Computational Geometry -> Data Structures for spatial indexing",
  "Computational Geometry -> Computational Mathematics for numerical robustness"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible algorithms, predicates, data structures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve geometry-plus-algorithm semantics.",
  "Preserve predicate correctness requirements.",
  "Preserve explicit degeneracy handling.",
  "Preserve algorithm complexity semantics.",
  "Preserve distinction between geometric objects and their computational representations.",
  "Preserve numerical robustness requirements.",
  "Preserve machine-readable computational-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Computational geometry studies algorithms and data structures for geometric objects and spatial problems."

 CONVEX_HULL_SUMMARY =
  "The convex hull is the smallest convex set containing the input points."

 CLOSEST_PAIR_SUMMARY =
  "The closest-pair problem finds the two input points with minimum distance."

 INTERSECTION_SUMMARY =
  "Segment-intersection algorithms rely on orientation and collinearity predicates."

 VORONOI_SUMMARY =
  "A Voronoi diagram partitions space by nearest-site ownership."

 DELAUNAY_SUMMARY =
  "The Delaunay triangulation is dual to the Voronoi diagram under standard planar assumptions."

 POINT_LOCATION_SUMMARY =
  "Point location identifies the subdivision cell containing a query point."

 ROBUSTNESS_SUMMARY =
  "Reliable computational geometry requires explicit treatment of degeneracy and finite-precision predicate error."
}

FLOW {
 INPUT
 -> detect_computational_geometry_context
 -> identify geometric_objects_dimension_metric_and_input_size
 -> identify problem_type_query_model_and_output_requirements
 -> identify degeneracy_general_position_and_numeric_precision assumptions
 -> select algorithm_data_structure_and_geometric_predicates
 -> analyse correctness_time_space_and_query complexity
 -> verify combinatorial_geometric_and_robustness constraints
 -> route to geometry coordinate vector topology linear_algebra algorithms data_structures_or_computational_mathematics ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Geometry layer provides the algorithmic branch of geometry, covering geometric predicates, convex hulls, closest-pair problems, segment intersections, Voronoi diagrams, Delaunay triangulations, polygon triangulation, arrangements, point location, spatial search, visibility and robust geometric computation. It combines geometric correctness with algorithmic complexity and explicit handling of degeneracy and numerical precision."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-COMPUTATIONAL-GEOMETRY-V1"
}

}

END_FLOW