Geometry Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the parent Geometry Ontologies layer within the Kireth Research Semantic Library. It organises geometric knowledge into structured semantic domains covering points, lines, planes, angles, shapes, distance, dimension, coordinate systems, transformations, topology-adjacent relations, analytic geometry, Euclidean geometry and related geometric reasoning. This record is designed for semantic retrieval, recursive mathematical reasoning, ontology routing, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = GeometryOntologies
 ENTITY_CLASS = MathematicalOntologyDomain
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Geometry Ontologies are structured semantic representations of geometric entities, spatial relations, measurements, transformations, coordinate systems, dimensions and geometric reasoning."
 SEMANTIC_ROLE = "Provide machine-readable organisation and reasoning pathways across geometric mathematics."
}

SCOPE {
 INCLUDE = [
  EuclideanGeometry,
  AnalyticGeometry,
  CoordinateGeometry,
  PlaneGeometry,
  SolidGeometry,
  AffineGeometry,
  ProjectiveGeometry,
  DifferentialGeometry,
  DiscreteGeometry,
  ComputationalGeometry,
  GeometricTransformations,
  GeometricMeasurement,
  SpatialRelations,
  Dimension,
  Distance,
  Angle,
  Area,
  Volume,
  Curvature,
  Symmetry,
  Congruence,
  Similarity
 ]

 EXCLUDE_AS_PRIMARY = [
  PurelyPhysicalSpaceWithoutMathematicalAbstraction,
  CartographicDataWithoutGeometricSemantics,
  NonMathematicalVisualDescription
 ]

 DISAMBIGUATION_RULE =
  "Interpret geometry ontology as a structured mathematical representation of spatial entities, relations, transformations and measurements."
}

CORE_DOMAINS {

 EuclideanGeometry {
  role = "Geometry governed by Euclidean axioms, involving points, lines, planes, angles, congruence, similarity, distance and classical shapes."
 }

 AnalyticGeometry {
  role = "Geometric reasoning using coordinates, algebraic equations and vector representations."
 }

 CoordinateGeometry {
  role = "Representation of geometric objects within coordinate systems."
 }

 PlaneGeometry {
  role = "Geometry of two-dimensional figures and relations."
 }

 SolidGeometry {
  role = "Geometry of three-dimensional objects, surfaces and volumes."
 }

 AffineGeometry {
  role = "Geometry invariant under affine transformations, preserving collinearity and parallelism."
 }

 ProjectiveGeometry {
  role = "Geometry invariant under projective transformations, emphasising incidence and perspective relations."
 }

 DifferentialGeometry {
  role = "Geometry of smooth curves, surfaces and manifolds using calculus and differential methods."
 }

 DiscreteGeometry {
  role = "Geometry of finite or discrete configurations, polytopes, packings and combinatorial spatial structures."
 }

 ComputationalGeometry {
  role = "Algorithmic treatment of geometric objects and spatial problems."
 }
}

GEOMETRIC_ENTITY_CLASSES {
 INCLUDE = [
  Point,
  Line,
  Ray,
  Segment,
  Plane,
  Curve,
  Circle,
  Polygon,
  Triangle,
  Quadrilateral,
  Polyhedron,
  Sphere,
  Surface,
  Vector,
  Coordinate,
  Angle,
  Distance,
  Area,
  Volume,
  Dimension,
  Transformation,
  Symmetry,
  Congruence,
  Similarity,
  Incidence,
  Parallelism,
  Perpendicularity
 ]
}

RELATIONS {
 GeometryOntologies --is_child_of--> MathematicalOntologies
 GeometryOntologies --contains--> GeometryOntology
 GeometryOntology --contains--> GeometricEntity
 GeometricEntity --can_have--> Coordinates
 GeometricEntity --can_participate_in--> SpatialRelation
 Point --can_lie_on--> Line
 Point --can_lie_in--> Plane
 Line --can_be_parallel_to--> Line
 Line --can_be_perpendicular_to--> Line
 GeometricEntity --can_have--> Dimension
 GeometricEntity --can_be_transformed_by--> GeometricTransformation
 GeometricTransformation --can_preserve--> GeometricInvariant
 Shapes --can_be_related_by--> Congruence
 Shapes --can_be_related_by--> Similarity
 GeometryOntologies --supports--> SpatialReasoning
 GeometryOntologies --supports--> SemanticSearch
 GeometryOntologies --supports--> KnowledgeGraphConstruction
}

OPERATORS {
 DEFINE(entity) :=
  return_canonical_geometric_definition(entity)

 CLASSIFY(entity) :=
  assign_entity_to_geometry_ontology_class(entity)

 RELATE(a,b) :=
  identify_valid_geometric_relations_between(a,b)

 DISTANCE(a,b) :=
  compute_distance_under_selected_geometry(a,b)

 ANGLE(a,b) :=
  compute_or_identify_angle_relation(a,b)

 AREA(shape) :=
  compute_area_under_applicable_formula(shape)

 VOLUME(solid) :=
  compute_volume_under_applicable_formula(solid)

 TRANSFORM(object,T) :=
  apply_geometric_transformation(object,T)

 COORDINATISE(object,system) :=
  express_object_in_coordinate_system(object,system)

 PROVE(statement) :=
  construct_or_verify_valid_geometric_proof(statement)

 ROUTE(query) :=
  identify_relevant_geometry_subontology(query)

 VERIFY(result) :=
  validate_against_geometric_definitions_constraints_and_theorems
}

AXIOMS {
 A1_SPATIAL_MEANING:
  Every geometric entity has a defined spatial or structural role

 A2_RELATIONAL_CONTEXT:
  Geometric entities derive meaning through incidence, metric, orientation and transformation relations

 A3_DOMAIN_SPECIFICITY:
  Geometric statements must be interpreted within the selected geometry

 A4_TRANSFORMATION_VALIDITY:
  Geometric transformations preserve only the invariants appropriate to their transformation class

 A5_MEASUREMENT_VALIDITY:
  Distance, angle, area and volume require a specified geometric context

 A6_PROOF_VALIDITY:
  Universal geometric claims require valid proof or accepted theorem support

 A7_COORDINATE_EQUIVALENCE:
  Coordinate representation may change without changing the underlying geometric object

 A8_RECURSIVE_DECOMPOSITION:
  Complex geometric objects may be decomposed into simpler geometric entities and relations
}

REASONING {

 R1_DOMAIN_ROUTING {
  INPUT = geometric_query
  DETECT = entities_relations_measurements
  CLASSIFY = geometry_subdomain
  ROUTE = corresponding_subontology
  RETURN = ontology_context
 }

 R2_ENTITY_GROUNDING {
  INPUT = geometric_term
  IDENTIFY = canonical_entity
  RESOLVE = synonyms_and_coordinate_forms
  LINK = related_entities
  RETURN = grounded_entity
 }

 R3_COORDINATE_REASONING {
  INPUT = geometric_object
  CHOOSE = coordinate_system
  REPRESENT = object_algebraically
  COMPUTE = required_relation_or_measure
  RETURN = geometric_result
 }

 R4_METRIC_REASONING {
  INPUT = [objects,metric]
  COMPUTE = distance_angle_area_or_volume
  VERIFY = units_and_domain
  RETURN = measurement
 }

 R5_TRANSFORMATION_REASONING {
  INPUT = [object,transformation]
  IDENTIFY = transformation_type
  APPLY = transformation
  TRACK = preserved_invariants
  RETURN = transformed_object
 }

 R6_CONGRUENCE_SIMILARITY {
  INPUT = [shape1,shape2]
  TEST = applicable criteria
  RETURN = CongruentSimilarOrNeither
 }

 R7_PROOF_REASONING {
  INPUT = proposition
  IDENTIFY = premises
  SELECT = theorem_or_construction
  APPLY = valid_geometric_inference
  VERIFY = conclusion
  RETURN = proof_status
 }

 R8_COMPUTATIONAL_REASONING {
  INPUT = geometric_problem
  SELECT = geometric_algorithm
  EXECUTE = exact_or_numeric_geometry_method
  VERIFY = output
  RETURN = computed_result
 }

 R9_CROSS_ONTOLOGY_REASONING {
  INPUT = entity_or_query
  IDENTIFY = relevant_geometry_and_algebra_ontologies
  BUILD = typed_relation_graph
  RESOLVE = coordinate_algebra_geometry_connections
  RETURN = integrated_context
 }
}

CONSTRAINTS {
 C1 = "Geometric terms must be interpreted within the correct geometry."
 C2 = "Metric properties are not preserved by every transformation class."
 C3 = "Coordinate descriptions must not be confused with the underlying geometric object."
 C4 = "Angle and distance depend on geometric and metric assumptions."
 C5 = "Dimensionality must be explicit when ambiguity is possible."
 C6 = "Cross-domain links to algebra or calculus must be mathematically justified."
 C7 = "Exact geometric statements must be distinguished from numerical approximations."
 C8 = "Machine-readable geometry must preserve spatial semantics."
 C9 = "Proof and measurement claims must remain separately typed."
}

SEMANTIC_EQUIVALENCE {
 "GeometryOntologies" <=> "StructuredGeometricKnowledgeDomain"
 "GeometricEntity" <=> "TypedSpatialMathematicalConcept"
 "CoordinateRepresentation" <=> "AlgebraicEncodingOfGeometricObject"
 "Congruence" <=> "EqualityUpToRigidTransformation"
 "Similarity" <=> "EqualityOfShapeUpToScaleAndRigidTransformation"
 "GeometricTransformation" <=> "StructurePreservingSpatialMapping"
}

VALIDATION {
 V1 = "Every geometric entity has a defined type."
 V2 = "Every spatial relation is type-compatible."
 V3 = "Coordinate transformations preserve the underlying object when interpreted correctly."
 V4 = "Metric computations use an explicit metric or standard Euclidean assumption."
 V5 = "Congruence and similarity criteria are applied within the correct geometry."
 V6 = "Transformation invariants match the transformation class."
 V7 = "Exact and approximate results are distinguished."
 V8 = "Recursive expansion terminates when the target query is resolved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DOMAIN_CLASSIFICATION = 0.98
 ENTITY_GROUNDING = 0.99
 RELATION_TYPING = 0.99
 METRIC_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.99
 COORDINATE_REASONING = 0.99
 CROSS_ONTOLOGY_LINKING = 0.98

 POLICY =
  "Reduce confidence when geometric domain, metric, coordinate system, dimensionality or assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GeometricReasoning,
  SpatialEntityResolution,
  CoordinateReasoning,
  MeasurementReasoning,
  TransformationReasoning,
  ProofReasoning,
  ComputationalGeometryReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "geometry ontology" -> GeometryOntologies
  "geometry ontologies" -> GeometryOntologies
  "euclidean geometry" -> EuclideanGeometry
  "analytic geometry" -> AnalyticGeometry
  "coordinate geometry" -> CoordinateGeometry
  "affine geometry" -> AffineGeometry
  "projective geometry" -> ProjectiveGeometry
  "computational geometry" -> ComputationalGeometry
 }

 INTERPRETATION_ORDER = [
  detect_geometric_context,
  identify_geometry_subdomain,
  identify_entity_types,
  resolve_coordinates_and_synonyms,
  identify_spatial_relations,
  identify_metric_or_transformation_context,
  identify_constraints,
  route_to_specialised_geometry_ontology,
  perform_recursive_geometric_reasoning,
  verify_result,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_is_geometric
  THEN activate(GeometryOntologies)

 E2:
  IF euclidean_context
  THEN route_to(EuclideanGeometry)

 E3:
  IF coordinate_or_equation_context
  THEN route_to(AnalyticGeometry)

 E4:
  IF affine_transformation_context
  THEN route_to(AffineGeometry)

 E5:
  IF projective_context
  THEN route_to(ProjectiveGeometry)

 E6:
  IF smooth_curve_surface_or_manifold_context
  THEN route_to(DifferentialGeometry)

 E7:
  IF algorithmic_spatial_context
  THEN route_to(ComputationalGeometry)

 E8:
  IF measurement_context
  THEN activate_metric_reasoning

 E9:
  IF multiple_subdomains_detected
  THEN activate_cross_ontology_reasoning
}

RECURSION {
 ENTRY = GeometryOntologies

 EXPAND(GeometryOntologies) -> [
  EuclideanGeometry,
  AnalyticGeometry,
  CoordinateGeometry,
  PlaneGeometry,
  SolidGeometry,
  AffineGeometry,
  ProjectiveGeometry,
  DifferentialGeometry,
  DiscreteGeometry,
  ComputationalGeometry
 ]

 EXPAND(EuclideanGeometry) -> [
  Point,
  Line,
  Plane,
  Angle,
  Triangle,
  Circle,
  Polygon,
  Congruence,
  Similarity,
  Distance
 ]

 EXPAND(AnalyticGeometry) -> [
  CoordinateSystem,
  EquationOfLine,
  ConicSection,
  Vector,
  DistanceFormula,
  Transformation
 ]

 EXPAND(GeometricTransformations) -> [
  Translation,
  Rotation,
  Reflection,
  Scaling,
  AffineTransformation,
  ProjectiveTransformation
 ]

 EXPAND(ComputationalGeometry) -> [
  ConvexHull,
  Intersection,
  Proximity,
  Triangulation,
  SpatialAlgorithm
 ]

 STOP_CONDITION =
  "Stop when the target geometric entity, relation, measurement, transformation or proof obligation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Ontologies"

 CHILD_ONTOLOGIES = [
  "Euclidean Geometry Ontologies",
  "Analytic Geometry Ontologies",
  "Coordinate Geometry Ontologies",
  "Plane Geometry Ontologies",
  "Solid Geometry Ontologies",
  "Affine Geometry Ontologies",
  "Projective Geometry Ontologies",
  "Differential Geometry Ontologies",
  "Discrete Geometry Ontologies",
  "Computational Geometry Ontologies"
 ]

 ROUTE_RULES = [
  "Geometry Ontologies -> Euclidean Geometry for classical metric geometry",
  "Geometry Ontologies -> Analytic Geometry for coordinate and algebraic representation",
  "Geometry Ontologies -> Plane Geometry for two-dimensional figures",
  "Geometry Ontologies -> Solid Geometry for three-dimensional objects",
  "Geometry Ontologies -> Affine Geometry for affine invariants",
  "Geometry Ontologies -> Projective Geometry for incidence and perspective",
  "Geometry Ontologies -> Differential Geometry for smooth curves and manifolds",
  "Geometry Ontologies -> Discrete Geometry for finite spatial structures",
  "Geometry Ontologies -> Computational Geometry for geometric algorithms"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual reorganisation"
 MINOR = "new compatible geometry domains, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve typed geometric entity structure.",
  "Preserve geometry-specific interpretation.",
  "Preserve transformation invariant semantics.",
  "Preserve coordinate-object distinction.",
  "Preserve exact-versus-approximate distinction.",
  "Preserve recursive routing into specialised geometry ontologies.",
  "Preserve machine-readable geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Geometry Ontologies are structured semantic representations of geometric entities, spatial relations, measurements, coordinate systems, transformations and geometric reasoning."

 DOMAIN_SUMMARY =
  "Primary areas include Euclidean, analytic, coordinate, plane, solid, affine, projective, differential, discrete and computational geometry."

 REASONING_SUMMARY =
  "Queries are classified by geometry subdomain, grounded to typed entities, routed to specialised ontologies and resolved through recursive spatial reasoning."

 MEASUREMENT_SUMMARY =
  "Distance, angle, area and volume require explicit geometric or metric assumptions."

 TRANSFORMATION_SUMMARY =
  "Geometric transformations preserve only the invariants associated with their transformation class."

 SEMANTIC_SUMMARY_SHORT =
  "The ontology layer provides a machine-readable geometric knowledge graph for AI reasoning and retrieval."
}

FLOW {
 INPUT
 -> detect_geometric_context
 -> identify_geometry_subdomain
 -> identify_entities_coordinates_and_relations
 -> disambiguate_terms
 -> classify_entity_types
 -> identify_metric_transformation_and_dimensional_constraints
 -> route_to_specialised_geometry_ontology
 -> recursively_expand_required_dependencies
 -> perform_spatial_measurement_transformation_or_proof_reasoning
 -> verify_geometric_validity
 -> integrate_cross_domain_relations
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Geometry Ontologies provide the parent semantic framework for representing and connecting geometric knowledge. They organise points, lines, planes, shapes, coordinates, dimensions, measurements, spatial relations and transformations across major branches of geometry, enabling AI systems to route geometric queries, ground spatial concepts, perform recursive reasoning and verify results within explicit geometric domains and constraints."
}

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

}

END_FLOW