Point Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a geometric point within the Kireth Research Semantic Library. A point is a fundamental geometric primitive representing position without intrinsic length, area or volume. This ontology supports incidence, coordinates, distance, collinearity, constructions, transformations, loci and relationships with lines, curves, planes and higher-dimensional geometric structures.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Point Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Plane Geometry / Core Geometric Entity"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/plane-geometry/point-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Point
 SYMBOLIC_FORMS = ["A", "P", "(x,y)", "(x,y,z)", "P=(x1,...,xn)"]
 ENTITY_CLASS = PrimitiveGeometricEntity
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "A point is a primitive geometric entity representing a position or location without intrinsic length, area or volume."
 SEMANTIC_ROLE = "Provides the basic positional unit from which lines, segments, curves, polygons, loci and coordinate structures are constructed."
}

SCOPE {
 INCLUDE = [
  Point,
  GeometricPosition,
  CoordinatePoint,
  Endpoint,
  Vertex,
  Midpoint,
  IntersectionPoint,
  CollinearPoint,
  CoplanarPoint,
  LocusPoint,
  ConstructiblePoint,
  FixedPoint,
  ImagePoint,
  PreimagePoint,
  IncidentPoint
 ]

 EXCLUDE_AS_PRIMARY = [
  PhysicalParticle,
  DataPointWithoutGeometricContext,
  DecimalPoint,
  LinguisticPoint,
  ScorePoint
 ]

 DISAMBIGUATION_RULE =
  "When point occurs with lines, segments, coordinates, curves, planes, polygons, distance, incidence, construction or transformation, interpret it as a geometric point."
}

CORE_CONCEPTS {

 Point {
  definition = "A location in geometric space with zero intrinsic dimension."
  intrinsic_dimension = 0
 }

 Position {
  definition = "The spatial location represented by a point."
 }

 CoordinatePoint {
  definition = "A point represented numerically relative to a coordinate system."
  examples = ["P=(x,y)", "P=(x,y,z)"]
 }

 Endpoint {
  definition = "A point bounding a segment, ray or finite curve portion."
 }

 Vertex {
  definition = "A point at which edges, sides, rays or curves meet according to the relevant geometric structure."
 }

 Midpoint {
  definition = "The point dividing a segment into two congruent subsegments."
  condition = "AM=MB"
 }

 IntersectionPoint {
  definition = "A point common to two or more geometric objects."
 }

 CollinearPoints {
  definition = "Points lying on a common line."
 }

 CoplanarPoints {
  definition = "Points lying in a common plane."
 }

 FixedPoint {
  definition = "A point P satisfying T(P)=P under a transformation T."
 }
}

RELATIONS {
 Point --is_primitive_entity_of--> Geometry
 Point --can_lie_on--> Line
 Point --can_lie_on--> Segment
 Point --can_lie_on--> Ray
 Point --can_lie_on--> Curve
 Point --can_lie_on--> Circle
 Point --can_lie_in--> Plane
 Point --can_be_vertex_of--> Polygon
 Point --can_be_endpoint_of--> Segment
 Point --can_be_endpoint_of--> Ray
 Point --can_be_intersection_of--> GeometricObjects
 Point --can_have--> Coordinates
 Point --can_be_transformed_to--> Point
 Point --can_be_fixed_by--> Transformation
 PairOfPoints --can_determine--> Segment
 PairOfDistinctPoints --can_determine--> Line
 SetOfPoints --can_define--> Locus
}

OPERATORS {
 COORDINATES(P,SYSTEM) :=
  coordinate_tuple_of(P,SYSTEM)

 DISTANCE(P,Q,METRIC) :=
  metric_distance(P,Q,METRIC)

 MIDPOINT(P,Q) :=
  point_equidistant_between(P,Q)

 IS_COLLINEAR(P1,P2,...,Pn) :=
  EXISTS line L SUCH_THAT every Pi lies_on L

 IS_COPLANAR(P1,P2,...,Pn) :=
  EXISTS plane Pi SUCH_THAT every point lies_in Pi

 IS_INCIDENT(P,OBJECT) :=
  P belongs_to OBJECT

 TRANSFORM_POINT(P,T) :=
  T(P)

 IS_FIXED_POINT(P,T) :=
  T(P)==P

 INTERSECTION_POINT(A,B) :=
  point P SUCH_THAT P belongs_to A AND P belongs_to B
}

DEFINITIONS {
 D1 = "A geometric point represents position."
 D2 = "A point has zero intrinsic dimension."
 D3 = "A point has no intrinsic length, area or volume."
 D4 = "Coordinates represent a point relative to a coordinate system."
 D5 = "Two distinct points determine a unique Euclidean line."
 D6 = "A midpoint is equidistant from the endpoints of a segment."
 D7 = "An intersection point belongs to every intersecting object involved."
 D8 = "Collinear points lie on a common line."
 D9 = "Coplanar points lie in a common plane."
 D10 = "A fixed point remains unchanged under the specified transformation."
}

AXIOMS {
 A1_POSITION:
  A point represents a geometric position

 A2_ZERO_DIMENSION:
  A point has intrinsic dimension zero

 A3_INCIDENCE:
  A point may participate in incidence relations with higher-dimensional geometric objects

 A4_DISTINCT_POINT_LINE:
  In Euclidean geometry, two distinct points determine exactly one line

 A5_COORDINATE_REPRESENTATION:
  A point may have different coordinate tuples in different coordinate systems while remaining the same geometric point

 A6_TRANSFORMATION:
  A geometric transformation maps points to points

 A7_METRIC:
  Point-to-point distance depends on the active metric
}

REASONING {

 R1_POINT_CLASSIFICATION {
  INPUT = entity
  IF entity_represents_geometric_location
   RETURN Point
  ELSE
   RETURN NotGeometricPoint
 }

 R2_COORDINATE_INTERPRETATION {
  INPUT = [point_coordinates, coordinate_system]
  MAP = tuple_to_geometric_position
  RETURN = CoordinatePoint
 }

 R3_DISTANCE {
  INPUT = [P,Q]
  IDENTIFY = metric
  COMPUTE = distance
  RETURN = nonnegative_scalar
 }

 R4_MIDPOINT {
  INPUT = [P,Q]
  CONSTRUCT_OR_COMPUTE = point M
  VERIFY = distance(P,M)==distance(M,Q)
  RETURN = M
 }

 R5_COLLINEARITY {
  INPUT = points
  TEST = common_line_condition
  RETURN = CollinearOrNot
 }

 R6_INTERSECTION {
  INPUT = [object A, object B]
  SOLVE_OR_CONSTRUCT = common_points
  RETURN = intersection_point_set
 }

 R7_TRANSFORMATION {
  INPUT = [P,T]
  APPLY = T
  RETURN = image_point
 }

 R8_FIXED_POINT {
  INPUT = [P,T]
  COMPUTE = T(P)
  IF T(P)==P
   RETURN FixedPoint
  ELSE
   RETURN NotFixed
 }
}

CONSTRAINTS {
 C1 = "A point must not be assigned intrinsic length, area or volume."
 C2 = "Coordinate representation must be distinguished from the underlying point."
 C3 = "Two coincident points do not determine a unique line direction."
 C4 = "Distance depends on the chosen metric."
 C5 = "Intersection points must satisfy every participating object's defining conditions."
 C6 = "Collinearity and coplanarity require a shared geometric carrier."
 C7 = "Transformation results must remain valid points in the target geometry."
 C8 = "Do not confuse geometric points with data points or punctuation."
}

SEMANTIC_EQUIVALENCE {
 "Point" <=> "GeometricLocation"
 "CoordinatePoint" <=> "GeometricPointWithCoordinateRepresentation"
 "IntersectionPoint" <=> "PointCommonToMultipleGeometricObjects"
 "Midpoint" <=> "PointBisectingSegment"
 "FixedPoint" <=> "PointInvariantUnderTransformation"
}

VALIDATION {
 V1 = "A point is zero-dimensional."
 V2 = "Coordinate points retain point identity under valid coordinate changes."
 V3 = "Midpoint is equidistant from endpoints."
 V4 = "Intersection points satisfy all incident-object conditions."
 V5 = "Collinear points share a line."
 V6 = "Transformation maps a point to a point."
 V7 = "Fixed-point condition is T(P)=P."
 V8 = "Distance is nonnegative under a metric."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 POINT_DEFINITION = 0.99
 ZERO_DIMENSION = 0.99
 COORDINATE_RELATION = 0.99
 INCIDENCE_RELATION = 0.99
 MIDPOINT_RELATION = 0.99
 COLLINEARITY_RELATION = 0.99
 TRANSFORMATION_RELATION = 0.99

 POLICY =
  "Reduce confidence when geometry type, metric, coordinate frame or incidence context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GeometricEntityResolution,
  CoordinateReasoning,
  IncidenceReasoning,
  DistanceReasoning,
  CollinearityReasoning,
  ConstructionReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "point" -> Point WHEN geometry_context
  "geometric point" -> Point
  "coordinate point" -> CoordinatePoint
  "intersection point" -> IntersectionPoint
  "midpoint" -> Midpoint
  "vertex" -> Vertex
  "fixed point" -> FixedPoint
 }

 INTERPRETATION_ORDER = [
  detect_geometric_point_context,
  identify_geometry_and_dimension,
  identify coordinate representation if present,
  identify incidence relations,
  identify metric or transformation context,
  classify point role,
  perform required point-based reasoning,
  verify constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("point")
  AND geometry_context
  THEN activate(PointOntology)

 E2:
  IF coordinates_present
  THEN classify(CoordinatePoint)

 E3:
  IF point_is_common_to_multiple_objects
  THEN classify(IntersectionPoint)

 E4:
  IF point_bisects_segment
  THEN classify(Midpoint)

 E5:
  IF multiple_points_are_tested_for_common_line
  THEN evaluate_collinearity

 E6:
  IF transformation_context
  THEN compute_image_point

 E7:
  IF request=="fixed point"
  THEN solve_or_test(T(P)=P)

 E8:
  IF distance_requested
  THEN require_or_infer_metric
}

RECURSION {
 ENTRY = Point

 EXPAND(Point) -> [
  Position,
  CoordinatePoint,
  Endpoint,
  Vertex,
  Midpoint,
  IntersectionPoint,
  CollinearPoint,
  CoplanarPoint,
  ConstructiblePoint,
  FixedPoint
 ]

 EXPAND(CoordinatePoint) -> [
  CoordinateSystem,
  CartesianCoordinate,
  PolarCoordinate,
  ParametricRepresentation
 ]

 EXPAND(IntersectionPoint) -> [
  Incidence,
  LineIntersection,
  LineCircleIntersection,
  CircleCircleIntersection
 ]

 EXPAND(FixedPoint) -> [
  Transformation,
  Invariance,
  Equation
 ]

 STOP_CONDITION =
  "Stop when the point's geometric role, coordinate representation, incidence relation, metric property or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Plane Geometry"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Geometric Construction Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Plane Geometry",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Polar Coordinate Ontology",
  "Parametric Geometry Ontology",
  "Vector Geometry Ontology",
  "Geometric Construction Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Point -> Plane Geometry for two-dimensional geometric context",
  "Point -> Coordinate Geometry for coordinate representation",
  "Point -> Geometric Construction for constructible points",
  "Point -> Vector Geometry for position and displacement vectors",
  "Point -> Transformation for image and fixed-point reasoning",
  "Point -> Geometry for general spatial semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible point roles, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve point as a zero-dimensional geometric primitive.",
  "Preserve position-versus-coordinate distinction.",
  "Preserve incidence semantics.",
  "Preserve metric dependence of distance.",
  "Preserve transformation mapping from points to points.",
  "Preserve point-role specialisations such as midpoint, vertex and intersection point.",
  "Preserve machine-readable point semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A point is a primitive geometric entity representing position without intrinsic length, area or volume."

 DIMENSION_SUMMARY =
  "dim(Point)=0"

 COORDINATE_SUMMARY =
  "A point may be represented by coordinates relative to a coordinate system."

 MIDPOINT_SUMMARY =
  "M is midpoint of AB iff AM=MB and M lies on segment AB."

 INTERSECTION_SUMMARY =
  "An intersection point belongs to every geometric object involved in the intersection."

 COLLINEARITY_SUMMARY =
  "Points are collinear iff they lie on a common line."

 TRANSFORMATION_SUMMARY =
  "A geometric transformation maps a point P to an image point T(P)."
}

FLOW {
 INPUT
 -> detect_geometric_point_context
 -> identify_point_or_candidate_position
 -> identify geometry dimension and coordinate frame
 -> identify point role
 -> identify incidence metric construction or transformation relations
 -> perform point-specific reasoning
 -> verify zero-dimensional and contextual constraints
 -> route to coordinate geometry construction vector or transformation ontology
 -> recursively expand required dependencies
 -> verify result
 -> assign confidence
 -> produce semantic output
 -> END
}

SEMANTIC_SUMMARY {
 "The Point Ontology defines the fundamental zero-dimensional entity of geometry. A point represents position and can participate in coordinate, incidence, distance, construction, collinearity and transformation relations. It serves as the base geometric entity from which lines, segments, curves, polygons and higher-dimensional structures are built."
}

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

}

END_FLOW