Geometric Relationship Ontology

FLOW ALGEBRA
TITLE: Geometric Relationship Ontology
VERSION: 1.0
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/mathematical-relationships/geometric-relationship-ontology.html
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

flow "GeometricRelationshipOntology" v1.0

ENTITY_TYPES:
  GeometricObject: {id: string, dimension: integer, coordinate_system: string?}
  Point: GeometricObject
  Line: GeometricObject
  Segment: GeometricObject
  Ray: GeometricObject
  Plane: GeometricObject
  Angle: GeometricObject
  Polygon: GeometricObject
  Circle: GeometricObject
  Solid: GeometricObject
  Transformation: {id: string, kind: enum[translation,rotation,reflection,scaling,other]}
  Measurement: {value: number, unit: string, uncertainty: number?}
  GeometricAssertion: {subject: string, predicate: string, object: string, context: string?, confidence: number, provenance: string?}

RELATIONSHIP_TYPES:
  incidence: [point_on_line, point_on_plane, line_in_plane, intersects]
  orientation: [parallel_to, perpendicular_to, collinear_with, coplanar_with]
  metric: [distance_from, angle_between, equal_length_to, equal_angle_to]
  topology: [contains, inside, adjacent_to, touches, disjoint_from, overlaps]
  structural: [congruent_to, similar_to, part_of, boundary_of]
  transformation: [maps_to, invariant_under]

TYPE_RULES:
  - parallel_to(Line, Line) requires a shared Euclidean ambient space.
  - perpendicular_to(Line, Line) requires a defined angle/intersection convention.
  - distance_from(Point, Point) yields a nonnegative Measurement.
  - congruent_to and similar_to compare compatible geometric object classes.
  - contains and inside must declare boundary inclusion semantics.

INFERENCE_RULES:
  - parallel_to(A,B) => parallel_to(B,A) where the predicate is well-defined.
  - parallel_to(A,B) AND parallel_to(B,C) => parallel_to(A,C) in a common Euclidean plane, subject to distinct-line conventions.
  - perpendicular_to(A,B) => perpendicular_to(B,A).
  - congruent_to(A,B) => congruent_to(B,A).
  - congruent_to(A,B) AND congruent_to(B,C) => congruent_to(A,C).
  - congruent_to(A,B) => similar_to(A,B).
  - similar_to(A,B) AND similar_to(B,C) => similar_to(A,C).
  - maps_to(T,A,B) AND maps_to(U,B,C) => maps_to(compose(U,T),A,C).
  - point_on_line(P,L) AND line_in_plane(L,H) => point_on_plane(P,H).

SEMANTIC_EQUIVALENCE:
  exact: ["congruent", "same geometric shape and size under an isometry"]
  contextual: ["on", "incident_with"]
  warning: "Similarity is not identity; intersection is not containment; geometric equality depends on object type and frame."

CONFIDENCE:
  range: [0.0, 1.0]
  policy: "Keep asserted, measured, and inferred confidence separate; never promote an uncertain observation to a theorem."

VALIDATION:
  - Check dimensional compatibility, coordinate frame and units.
  - Verify preconditions before applying inference rules.
  - Reject negative distances and invalid angles for the declared convention.
  - Distinguish exact symbolic proofs from tolerance-based numerical tests.
  - Record conflicting claims and do not silently overwrite evidence.

PROVENANCE:
  fields: [source_id, method, timestamp, assumptions, coordinate_system, units]

FLOW_HASH:
  algorithm: SHA-256
  input: "canonical serialisation of version, entities, relations, rules, and assumptions"
  note: "Compute at publication or ingestion; no precomputed hash is asserted here."

UPDATE_MODEL:
  strategy: versioned_append_and_revalidate
  triggers: [new_assertion, changed_measurement, changed_frame, rule_revision]
  actions: [preserve_history, invalidate_affected_inferences, recompute_confidence, recompute_flow_hash]

AI_CONSUMPTION:
  representation: typed_semantic_graph
  outputs: [entities, relationships, derivations, assumptions, confidence, provenance, validation_results]
  constraints: [no_unjustified_inference, preserve_units, preserve_context, distinguish_proof_from_estimate]

END FLOW ALGEBRA