Circle Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a circle within the Kireth Research Semantic Library. A circle is a plane geometric locus consisting of all points at a fixed distance from a centre point. This ontology supports radius, diameter, circumference, area, chords, arcs, sectors, segments, tangents, secants, central and inscribed angles, cyclic polygons, coordinate representation, constructions, transformations and theorem-based geometric reasoning.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Circle
 SYMBOLIC_FORMS = ["C(O,r)", "(x-h)^2+(y-k)^2=r^2"]
 ENTITY_CLASS = PlaneLocus
 PRIMARY_CONTEXT = PlaneGeometry
 DEFINITION = "A circle is the set of all points in a plane at a fixed positive distance from a centre point."
 SEMANTIC_ROLE = "Provides the fundamental constant-radius plane locus for arc, chord, tangent, angle, area, symmetry and transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Circle,
  Centre,
  Radius,
  Diameter,
  Circumference,
  Area,
  Chord,
  Arc,
  MinorArc,
  MajorArc,
  Semicircle,
  Sector,
  CircularSegment,
  Tangent,
  Secant,
  CentralAngle,
  InscribedAngle,
  CyclicPolygon,
  Circumcircle,
  Incircle,
  CoordinateCircle
 ]

 EXCLUDE_AS_PRIMARY = [
  Disk,
  Sphere,
  Ellipse,
  CircularArcWithoutParentCircleContext,
  PhysicalRoundObjectWithoutGeometricIdealisation
 ]

 DISAMBIGUATION_RULE =
  "When circle occurs with centre, radius, diameter, circumference, chord, arc, tangent, secant, sector, cyclic angle or coordinate equation, interpret it as a plane geometric circle."
}

CORE_CONCEPTS {

 Circle {
  definition = "The locus of points at fixed distance r>0 from a centre O."
  intrinsic_dimension = 1
  embedded_dimension = 2
 }

 Centre {
  definition = "The point equidistant from every point on the circle."
 }

 Radius {
  definition = "A segment from the centre to a point on the circle."
  length_symbol = "r"
 }

 Diameter {
  definition = "A chord passing through the centre."
  length_relation = "d=2r"
 }

 Circumference {
  definition = "The total boundary length of the circle."
  formula = "C=2*pi*r=pi*d"
 }

 Area {
  definition = "The area of the disk bounded by the circle."
  formula = "A=pi*r^2"
 }

 Chord {
  definition = "A line segment whose endpoints lie on the circle."
 }

 Arc {
  definition = "A connected portion of the circle between two points."
 }

 Sector {
  definition = "The region bounded by two radii and the intercepted arc."
 }

 CircularSegment {
  definition = "The region bounded by a chord and its corresponding arc."
 }

 Tangent {
  definition = "A line meeting the circle at exactly one point in Euclidean plane geometry."
 }

 Secant {
  definition = "A line intersecting the circle at two points."
 }

 CentralAngle {
  definition = "An angle whose vertex is the centre of the circle."
 }

 InscribedAngle {
  definition = "An angle whose vertex lies on the circle and whose sides meet the circle."
 }
}

RELATIONS {
 Circle --is_special_case_of--> PlaneLocus
 Circle --has--> Centre
 Circle --has--> Radius
 Circle --has--> Diameter
 Circle --has--> Circumference
 Circle --bounds--> Disk
 Diameter --is_special_case_of--> Chord
 Chord --has_endpoints_on--> Circle
 Arc --is_subset_of--> Circle
 Radius --connects--> Centre
 Radius --connects--> PointOnCircle
 Tangent --touches--> Circle
 Secant --intersects--> Circle
 CentralAngle --intercepts--> Arc
 InscribedAngle --intercepts--> Arc
 CyclicPolygon --has_vertices_on--> Circle
 Circumcircle --contains_vertices_of--> Polygon
 Incircle --is_tangent_to--> PolygonSides
 Circle --can_be_transformed_to--> CircleOrConicDependingOnTransformation
}

OPERATORS {
 IS_ON_CIRCLE(P,O,r) :=
  distance(P,O)==r

 DIAMETER(r) :=
  2*r

 CIRCUMFERENCE(r) :=
  2*pi*r

 AREA(r) :=
  pi*r^2

 ARC_LENGTH(r,theta_radians) :=
  r*theta_radians

 SECTOR_AREA(r,theta_radians) :=
  (1/2)*r^2*theta_radians

 CHORD_LENGTH(r,theta_radians) :=
  2*r*sin(theta_radians/2)

 CENTRAL_ANGLE_FROM_ARC(s,r) :=
  s/r

 INSCRIBED_ANGLE_FROM_CENTRAL(theta) :=
  theta/2

 TANGENT_AT(P,Circle) :=
  line_through_P_perpendicular_to_radius_OP

 INTERSECT_LINE_CIRCLE(L,C) :=
  solve_line_circle_system(L,C)

 TRANSFORM_CIRCLE(C,T) :=
  image_of_circle_under_transformation(C,T)
}

DEFINITIONS {
 D1 = "A circle is a locus, not the filled disk."
 D2 = "Every point on a circle is the same distance from the centre."
 D3 = "A diameter is twice the radius."
 D4 = "A diameter is the longest chord of a circle."
 D5 = "Circumference equals 2*pi*r."
 D6 = "The disk area bounded by the circle equals pi*r^2."
 D7 = "A tangent is perpendicular to the radius at the point of tangency."
 D8 = "An inscribed angle subtending an arc has half the measure of the corresponding central angle."
 D9 = "Equal chords in the same circle subtend equal central angles."
 D10 = "Perpendicular from the centre to a chord bisects the chord."
}

AXIOMS {
 A1_FIXED_RADIUS:
  Every point P on circle C(O,r) satisfies OP=r

 A2_UNIQUE_CENTRE:
  A nondegenerate Euclidean circle has a unique centre

 A3_DIAMETER:
  Any chord through the centre is a diameter

 A4_TANGENT_RADIUS:
  A tangent to a circle is perpendicular to the radius through the point of tangency

 A5_EQUAL_RADII:
  All radii of the same circle are congruent

 A6_CHORD_SYMMETRY:
  The perpendicular from the centre to a chord bisects that chord

 A7_CENTRAL_INSCRIBED_ANGLE:
  For the same intercepted arc, an inscribed angle is half the corresponding central angle
}

REASONING {

 R1_POINT_MEMBERSHIP {
  INPUT = [P,O,r]
  COMPUTE = distance(P,O)
  IF distance==r
   RETURN PointOnCircle
  ELSE
   RETURN PointNotOnCircle
 }

 R2_RADIUS_DIAMETER {
  INPUT = one_of_radius_or_diameter
  IF radius_known
   RETURN diameter=2r
  ELSE IF diameter_known
   RETURN radius=d/2
 }

 R3_CIRCUMFERENCE_AREA {
  INPUT = radius_or_diameter
  DERIVE = radius_if_needed
  COMPUTE = circumference_and_or_area
  RETURN = requested_measure
 }

 R4_CHORD_REASONING {
  INPUT = [circle,chord]
  IDENTIFY = centre_to_chord_distance_or_central_angle
  APPLY = chord_theorems
  RETURN = chord_properties
 }

 R5_TANGENT_REASONING {
  INPUT = [circle,point_or_line]
  IF point_on_circle
   CONSTRUCT = perpendicular_to_radius_at_point
  ELSE
   ANALYSE = tangent_condition
  RETURN = tangent_relation
 }

 R6_INSCRIBED_ANGLE {
  INPUT = intercepted_arc_or_central_angle
  APPLY = inscribed_angle_theorem
  RETURN = inscribed_angle_measure
 }

 R7_CYCLIC_QUADRILATERAL {
  INPUT = cyclic_quadrilateral
  APPLY = opposite_angles_supplementary
  RETURN = angle_relations
 }

 R8_LINE_CIRCLE_INTERSECTION {
  INPUT = [line,circle]
  SOLVE = coordinate_or_geometric_intersection
  CLASSIFY = zero_one_or_two_intersection_points
  RETURN = secant_tangent_or_disjoint
 }

 R9_COORDINATE_CIRCLE {
  INPUT = "(x-h)^2+(y-k)^2=r^2"
  IDENTIFY = centre(h,k)
  IDENTIFY = radius r
  RETURN = circle_parameters
 }
}

CONSTRAINTS {
 C1 = "Circle must be distinguished from the filled disk."
 C2 = "Radius must be positive for a nondegenerate circle."
 C3 = "Tangent-line theorems require a genuine point of tangency."
 C4 = "Inscribed-angle theorems require the same intercepted arc."
 C5 = "Arc lengths require consistent angle units."
 C6 = "Sector formulas in radian form require radians."
 C7 = "A general affine transformation may map a circle to an ellipse rather than a circle."
 C8 = "Diagram appearance alone does not establish tangency, equality or cyclicity."
}

SEMANTIC_EQUIVALENCE {
 "Circle" <=> "ConstantRadiusPlaneLocus"
 "Diameter" <=> "ChordThroughCentre"
 "Tangent" <=> "LineWithSinglePointOfContact" WHEN Euclidean_context
 "Circumcircle" <=> "CircleThroughPolygonVertices"
 "Incircle" <=> "CircleTangentToPolygonSides"
}

VALIDATION {
 V1 = "Every circle point is radius r from the centre."
 V2 = "Diameter equals 2r."
 V3 = "Circumference equals 2*pi*r."
 V4 = "Disk area equals pi*r^2."
 V5 = "Tangent is perpendicular to radius at tangency."
 V6 = "Inscribed angle equals half corresponding central angle."
 V7 = "Perpendicular from centre to chord bisects chord."
 V8 = "Coordinate-circle equation yields correct centre and radius."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CIRCLE_DEFINITION = 0.99
 RADIUS_DIAMETER_RELATION = 0.99
 CIRCUMFERENCE_RELATION = 0.99
 AREA_RELATION = 0.99
 TANGENT_RELATION = 0.99
 CHORD_RELATION = 0.99
 INSCRIBED_ANGLE_RELATION = 0.99

 POLICY =
  "Reduce confidence when arc orientation, angle units, cyclicity, transformation class or coordinate assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  CircleReasoning,
  LocusReasoning,
  ChordReasoning,
  TangentReasoning,
  ArcReasoning,
  AngleReasoning,
  CoordinateReasoning,
  ConstructionReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "circle" -> Circle
  "centre" -> Centre
  "radius" -> Radius
  "diameter" -> Diameter
  "circumference" -> Circumference
  "chord" -> Chord
  "arc" -> Arc
  "sector" -> Sector
  "tangent" -> Tangent WHEN circle_context
  "secant" -> Secant WHEN circle_context
 }

 INTERPRETATION_ORDER = [
  detect_circle_context,
  identify centre radius diameter or equation,
  identify point chord arc tangent secant or angle relations,
  identify measurement or theorem goal,
  apply circle formulas and theorems,
  verify Euclidean angle and tangency assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("circle")
  AND geometry_context
  THEN activate(CircleOntology)

 E2:
  IF radius_or_diameter_given
  THEN derive_related_basic_measures

 E3:
  IF point_membership_requested
  THEN test distance_to_centre_equals_radius

 E4:
  IF tangent_context
  THEN apply_radius_perpendicular_theorem

 E5:
  IF inscribed_angle_context
  THEN identify_corresponding_central_angle_or_arc

 E6:
  IF chord_context
  THEN apply_equal_chord_and_perpendicular_bisector_theorems

 E7:
  IF line_circle_intersection_requested
  THEN classify_as_secant_tangent_or_disjoint

 E8:
  IF coordinate_equation_present
  THEN extract_or_complete_square_to_find_centre_and_radius
}

RECURSION {
 ENTRY = Circle

 EXPAND(Circle) -> [
  Centre,
  Radius,
  Diameter,
  Circumference,
  Chord,
  Arc,
  Sector,
  CircularSegment,
  Tangent,
  Secant,
  CentralAngle,
  InscribedAngle,
  Circumcircle,
  Incircle
 ]

 EXPAND(Chord) -> [
  Endpoint,
  DistanceFromCentre,
  CentralAngle,
  PerpendicularBisector
 ]

 EXPAND(Arc) -> [
  MinorArc,
  MajorArc,
  Semicircle,
  ArcLength,
  CentralAngle
 ]

 EXPAND(Tangent) -> [
  PointOfTangency,
  Radius,
  Perpendicularity,
  TangentLength
 ]

 EXPAND(CyclicPolygon) -> [
  Circumcircle,
  InscribedAngle,
  OppositeAngleRelation
 ]

 STOP_CONDITION =
  "Stop when the circle's defining data, measurement, chord, arc, tangent, angle, cyclic or coordinate property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Plane Geometry"

 IMMEDIATE_CONTEXTS = [
  "Point Ontology",
  "Line Ontology",
  "Line Segment Ontology",
  "Angle Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Plane Geometry",
  "Point Ontology",
  "Line Ontology",
  "Line Segment Ontology",
  "Angle Ontology",
  "Triangle Ontology",
  "Polygon Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Geometric Construction Ontology",
  "Synthetic Geometry Ontology",
  "Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Circle -> Point for centre and point-on-circle reasoning",
  "Circle -> Line for tangents and secants",
  "Circle -> Line Segment for radii, diameters and chords",
  "Circle -> Angle for central and inscribed-angle reasoning",
  "Circle -> Triangle for circumcircle and incircle relations",
  "Circle -> Coordinate Geometry for circle equations",
  "Circle -> Geometric Construction for circle constructions",
  "Circle -> Synthetic Geometry for theorem-based proofs",
  "Circle -> Transformation for circle-preserving and circle-to-conic mappings"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible circle entities, theorems or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve circle as a constant-radius locus.",
  "Preserve distinction between circle and disk.",
  "Preserve radius, diameter and chord semantics.",
  "Preserve tangent-radius perpendicularity.",
  "Preserve central-versus-inscribed-angle relations.",
  "Preserve angle-unit awareness for arc and sector formulas.",
  "Preserve machine-readable circle semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A circle is the set of all points in a plane at a fixed positive distance from a centre point."

 RADIUS_SUMMARY =
  "OP=r for every point P on the circle."

 DIAMETER_SUMMARY =
  "d=2r"

 CIRCUMFERENCE_SUMMARY =
  "C=2*pi*r"

 AREA_SUMMARY =
  "A=pi*r^2 for the enclosed disk."

 ARC_SUMMARY =
  "s=r*theta for theta in radians."

 TANGENT_SUMMARY =
  "A tangent is perpendicular to the radius at the point of tangency."

 ANGLE_SUMMARY =
  "An inscribed angle is half the corresponding central angle for the same intercepted arc."
}

FLOW {
 INPUT
 -> detect_circle_context
 -> identify_centre_radius_diameter_or_equation
 -> identify point chord arc sector tangent secant or angle relation
 -> identify measurement construction coordinate or proof goal
 -> apply valid circle formulas_and_theorems
 -> verify radius positivity angle units tangency and cyclicity constraints
 -> route to point line segment angle triangle coordinate construction synthetic or transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Circle Ontology defines the constant-radius plane locus and its core structures: centre, radius, diameter, circumference, chords, arcs, sectors, tangents and secants. It supports measurement, coordinate equations, central and inscribed-angle theorems, cyclic figures, geometric constructions and transformation reasoning."
}

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

}

END_FLOW