Sphere Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a sphere within the Kireth Research Semantic Library. A sphere is the set of all points in three-dimensional Euclidean space at a fixed distance from a centre point. This ontology supports radius, diameter, great circles, surface area, enclosed volume, coordinate equations, tangent planes, cross-sections, spherical symmetry, inscribed and circumscribed relations, transformations and links to solid and analytic geometry.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Sphere
 SYMBOLIC_FORMS = [
  "S(O,r)",
  "(x-h)^2+(y-k)^2+(z-l)^2=r^2"
 ]
 ENTITY_CLASS = TwoDimensionalSurfaceInThreeDimensionalSpace
 PRIMARY_CONTEXT = SolidGeometry
 DEFINITION = "A sphere is the set of all points in three-dimensional Euclidean space at a fixed positive distance from a centre point."
 SEMANTIC_ROLE = "Provides the canonical constant-radius surface for three-dimensional metric, coordinate, cross-section, tangent, symmetry and transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Sphere,
  Centre,
  Radius,
  Diameter,
  GreatCircle,
  SmallCircle,
  SurfaceArea,
  EnclosedVolume,
  Ball,
  Hemisphere,
  SphericalCap,
  SphericalZone,
  TangentPlane,
  NormalLine,
  CrossSection,
  CoordinateSphere,
  InscribedSphere,
  CircumscribedSphere,
  SphericalSymmetry
 ]

 EXCLUDE_AS_PRIMARY = [
  Circle,
  Ellipsoid,
  GeneralQuadricSurface,
  BallWhenStrictSurfaceMeaningIsRequired,
  PhysicalBallWithoutMathematicalAbstraction
 ]

 DISAMBIGUATION_RULE =
  "When sphere occurs with centre, radius, surface area, volume, great circle, tangent plane, coordinate equation or three-dimensional distance, interpret it as a geometric sphere."
}

CORE_CONCEPTS {

 Sphere {
  definition = "The set of points P in R^3 satisfying distance(P,O)=r for fixed centre O and radius r>0."
  intrinsic_dimension = 2
  embedded_dimension = 3
 }

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

 Radius {
  symbol = "r"
  definition = "The fixed distance from the centre to every point on the sphere."
 }

 Diameter {
  definition = "A line segment through the centre with endpoints on the sphere."
  length_relation = "d=2r"
 }

 Ball {
  definition = "The three-dimensional region consisting of points whose distance from the centre is less than or equal to r."
 }

 GreatCircle {
  definition = "A circle obtained by intersecting the sphere with a plane passing through the centre."
  radius = "r"
 }

 SmallCircle {
  definition = "A circular cross-section of the sphere by a plane not passing through the centre."
 }

 Hemisphere {
  definition = "One of the two equal halves of the sphere determined by a great-circle plane."
 }

 TangentPlane {
  definition = "A plane touching the sphere at exactly one point and perpendicular there to the radius."
 }

 NormalLine {
  definition = "The line through the centre and point of tangency, perpendicular to the tangent plane."
 }

 SurfaceArea {
  formula = "4*pi*r^2"
 }

 EnclosedVolume {
  formula = "(4/3)*pi*r^3"
 }
}

RELATIONS {
 Sphere --is_special_case_of--> QuadricSurface
 Sphere --is_surface_boundary_of--> Ball
 Sphere --has--> Centre
 Sphere --has--> Radius
 Sphere --has--> Diameter
 Sphere --has--> GreatCircle
 Sphere --can_have--> SmallCircle
 Sphere --can_have--> TangentPlane
 Sphere --can_have--> CrossSection
 Sphere --can_be--> InscribedSphere
 Sphere --can_be--> CircumscribedSphere
 GreatCircle --is_intersection_of--> SphereAndCentralPlane
 SmallCircle --is_intersection_of--> SphereAndOffsetPlane
 TangentPlane --is_perpendicular_to--> RadiusAtTangency
 Sphere --has_symmetry--> FullRotationalSymmetryAboutCentre
}

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

 IS_INSIDE_BALL(P,O,r) :=
  distance(P,O)<=r

 DIAMETER(r) :=
  2*r

 SURFACE_AREA(r) :=
  4*pi*r^2

 VOLUME(r) :=
  (4/3)*pi*r^3

 STANDARD_EQUATION(h,k,l,r) :=
  (x-h)^2+(y-k)^2+(z-l)^2=r^2

 CROSS_SECTION_RADIUS(r,d) :=
  sqrt(r^2-d^2)
  WHERE d=perpendicular_distance_from_centre_to_plane
  AND 0<=d<=r

 TANGENT_PLANE_AT(P,O) :=
  plane_through_P_with_normal_vector(P-O)

 NORMAL_LINE_AT(P,O) :=
  line_through_O_and_P

 GREAT_CIRCLE(plane,Sphere) :=
  intersection_when_plane_passes_through_centre

 TRANSFORM_SPHERE(S,T) :=
  image_of_sphere_under_3D_transformation(S,T)
}

DEFINITIONS {
 D1 = "A sphere is a surface, not the filled ball."
 D2 = "Every point on a sphere lies the same distance from the centre."
 D3 = "The diameter is twice the radius."
 D4 = "A great circle has the same centre and radius as the sphere."
 D5 = "Every plane cross-section of a sphere is a circle, a point, or empty."
 D6 = "A tangent plane is perpendicular to the radius at the point of tangency."
 D7 = "Surface area equals 4*pi*r^2."
 D8 = "The enclosed ball volume equals (4/3)*pi*r^3."
 D9 = "A sphere is invariant under every rotation about its centre."
 D10 = "A sphere is centrally symmetric about its centre."
}

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

 A2_UNIQUE_CENTRE:
  A nondegenerate Euclidean sphere has a unique centre

 A3_SURFACE_DIMENSION:
  A sphere is intrinsically two-dimensional and embedded in three-dimensional space

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

 A5_CENTRAL_SECTION:
  A plane through the centre intersects the sphere in a great circle

 A6_CROSS_SECTION:
  A plane at perpendicular distance d<r from the centre intersects the sphere in a circle of radius sqrt(r^2-d^2)

 A7_ROTATIONAL_SYMMETRY:
  Rotations about the centre map the sphere to itself

 A8_RIGID_INVARIANCE:
  Rigid motions map spheres to congruent spheres
}

REASONING {

 R1_POINT_MEMBERSHIP {
  INPUT = [P,O,r]
  COMPUTE = distance(P,O)
  IF distance==r
   RETURN PointOnSphere
  ELSE IF distance<r
   RETURN PointInsideBall
  ELSE
   RETURN PointOutsideBall
 }

 R2_STANDARD_EQUATION {
  INPUT = "(x-h)^2+(y-k)^2+(z-l)^2=r^2"
  IDENTIFY = centre(h,k,l)
  IDENTIFY = radius r
  RETURN = sphere_parameters
 }

 R3_SURFACE_AREA {
  INPUT = r
  REQUIRE = r>0
  COMPUTE = 4*pi*r^2
  RETURN = surface_area
 }

 R4_VOLUME {
  INPUT = r
  REQUIRE = r>0
  COMPUTE = (4/3)*pi*r^3
  RETURN = enclosed_volume
 }

 R5_CROSS_SECTION {
  INPUT = [sphere,plane]
  COMPUTE = d=distance(centre,plane)
  IF d>r
   RETURN EmptyIntersection
  ELSE IF d==r
   RETURN TangencyPoint
  ELSE
   COMPUTE = rho=sqrt(r^2-d^2)
   RETURN CircleCrossSection(radius=rho)
 }

 R6_GREAT_CIRCLE {
  INPUT = [sphere,plane]
  IF plane_contains_centre
   RETURN GreatCircle(radius=r)
  ELSE
   ROUTE = SmallCircleReasoning
 }

 R7_TANGENT_PLANE {
  INPUT = [sphere,point P]
  VERIFY = P lies_on_sphere
  SET_NORMAL = P-O
  RETURN = plane_through_P_perpendicular_to_radius
 }

 R8_SPHERE_SPHERE_INTERSECTION {
  INPUT = [S1,S2]
  COMPARE = centre_distance_with_radii
  CLASSIFY = disjoint_tangent_circle_intersection_coincident
  RETURN = intersection_type
 }

 R9_TRANSFORMATION {
  INPUT = [sphere,T]
  IF rigid_or_similarity_transformation
   RETURN sphere_with_transformed_centre_and_scaled_radius
  ELSE IF general_affine_transformation
   RETURN ellipsoid_or_degenerate_quadric_in_general
 }
}

CONSTRAINTS {
 C1 = "Sphere must be distinguished from the enclosed ball."
 C2 = "Radius must be positive for a nondegenerate sphere."
 C3 = "Great circles require section planes through the centre."
 C4 = "Cross-section radius formula requires perpendicular centre-to-plane distance."
 C5 = "A tangent plane intersects a sphere at exactly one point in Euclidean geometry."
 C6 = "General affine transformations need not preserve spherical shape."
 C7 = "Surface area and enclosed volume must not be conflated."
 C8 = "Coordinate sphere equations assume Euclidean Cartesian coordinates unless otherwise stated."
}

SEMANTIC_EQUIVALENCE {
 "Sphere" <=> "ConstantRadiusSurfaceInR3"
 "Ball" <=> "ClosedThreeDimensionalRegionInsideSphere"
 "GreatCircle" <=> "CentralPlaneSectionOfSphere"
 "TangentPlane" <=> "PlanePerpendicularToRadiusAtContactPoint"
 "CircumscribedSphere" <=> "SpherePassingThroughSpecifiedVerticesOrBoundaryPoints"
}

VALIDATION {
 V1 = "Every sphere point is radius r from the centre."
 V2 = "Diameter equals 2r."
 V3 = "Surface area equals 4*pi*r^2."
 V4 = "Enclosed volume equals (4/3)*pi*r^3."
 V5 = "Central plane section is a great circle."
 V6 = "Offset plane section radius satisfies rho^2=r^2-d^2."
 V7 = "Tangent plane normal is parallel to the radius at tangency."
 V8 = "Rigid motions preserve sphere radius."
 V9 = "General affine image is not assumed to remain a sphere."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SPHERE_DEFINITION = 0.99
 CENTRE_RADIUS_RELATION = 0.99
 SURFACE_AREA_REASONING = 0.99
 VOLUME_REASONING = 0.99
 CROSS_SECTION_REASONING = 0.99
 TANGENT_PLANE_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when sphere-versus-ball usage, coordinate metric, cross-section plane, radius or transformation class are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SphereReasoning,
  SolidGeometryReasoning,
  SurfaceAreaReasoning,
  VolumeReasoning,
  CrossSectionReasoning,
  TangentPlaneReasoning,
  CoordinateReasoning,
  SymmetryReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "sphere" -> Sphere
  "geometric sphere" -> Sphere
  "ball" -> Ball WHEN solid_region_context
  "great circle" -> GreatCircle
  "small circle" -> SmallCircle
  "sphere surface area" -> SurfaceArea
  "sphere volume" -> EnclosedVolume
  "tangent plane" -> TangentPlane WHEN sphere_context
 }

 INTERPRETATION_ORDER = [
  detect_sphere_context,
  identify centre radius diameter or coordinate equation,
  distinguish sphere_surface from ball_region when relevant,
  identify point cross_section great_circle tangent_plane volume surface_area or transformation goal,
  apply sphere_specific geometry,
  verify metric dimensional radius and transformation constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("sphere")
  AND geometry_context
  THEN activate(SphereOntology)

 E2:
  IF coordinate_equation_present
  THEN extract centre_and_radius

 E3:
  IF point_membership_requested
  THEN compare distance_to_centre_with_radius

 E4:
  IF surface_area_requested
  THEN compute 4*pi*r^2

 E5:
  IF volume_requested
  THEN clarify_or_interpret_as_enclosed_ball_volume
  AND compute (4/3)*pi*r^3

 E6:
  IF cross_section_requested
  THEN compute centre_to_plane_distance
  AND classify empty_point_or_circle intersection

 E7:
  IF tangent_plane_requested
  THEN verify contact_point_on_sphere
  AND use radius_as_plane_normal

 E8:
  IF transformation_context
  THEN classify rigid_similarity_or_affine map
}

RECURSION {
 ENTRY = Sphere

 EXPAND(Sphere) -> [
  Centre,
  Radius,
  Diameter,
  Ball,
  GreatCircle,
  SmallCircle,
  Hemisphere,
  SphericalCap,
  TangentPlane,
  CrossSection,
  SurfaceArea,
  EnclosedVolume,
  Symmetry
 ]

 EXPAND(CrossSection) -> [
  Plane,
  CentreToPlaneDistance,
  Circle,
  TangencyPoint,
  EmptyIntersection
 ]

 EXPAND(TangentPlane) -> [
  PointOfTangency,
  Radius,
  NormalVector,
  PlaneEquation
 ]

 EXPAND(Symmetry) -> [
  RotationalSymmetry,
  ReflectionSymmetry,
  CentralSymmetry
 ]

 STOP_CONDITION =
  "Stop when the sphere's defining data, point relation, surface area, enclosed volume, cross-section, tangent plane, symmetry or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Solid Geometry"

 IMMEDIATE_CONTEXTS = [
  "Solid Geometry Ontology",
  "Circle Ontology",
  "Coordinate Geometry Ontology",
  "Vector Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Solid Geometry Ontology",
  "Circle Ontology",
  "Plane Geometry",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Vector Geometry Ontology",
  "Analytic Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology",
  "Dimension Ontology"
 ]

 ROUTE_RULES = [
  "Sphere -> Solid Geometry Ontology for general three-dimensional semantics",
  "Sphere -> Circle for planar cross-sections",
  "Sphere -> Coordinate Geometry for Cartesian equations",
  "Sphere -> Vector Geometry for normals and tangent planes",
  "Sphere -> Analytic Geometry for quadric representation",
  "Sphere -> Transformation for rigid, similarity and affine images",
  "Sphere -> Dimension for intrinsic-versus-embedded dimensional reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve sphere as a constant-radius surface.",
  "Preserve distinction between sphere and ball.",
  "Preserve intrinsic dimension 2 and embedded dimension 3.",
  "Preserve great-circle and cross-section semantics.",
  "Preserve tangent-plane perpendicularity to radius.",
  "Preserve surface-area and enclosed-volume distinction.",
  "Preserve machine-readable sphere semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A sphere is the set of all points in three-dimensional Euclidean space at a fixed positive distance from a centre point."

 DIMENSION_SUMMARY =
  "Intrinsic dimension=2; embedded dimension=3."

 DIAMETER_SUMMARY =
  "d=2r"

 SURFACE_AREA_SUMMARY =
  "A=4*pi*r^2"

 VOLUME_SUMMARY =
  "V=(4/3)*pi*r^3 for the enclosed ball."

 COORDINATE_SUMMARY =
  "(x-h)^2+(y-k)^2+(z-l)^2=r^2"

 GREAT_CIRCLE_SUMMARY =
  "A plane through the centre cuts the sphere in a great circle of radius r."

 TANGENT_SUMMARY =
  "The tangent plane at P is perpendicular to OP."
}

FLOW {
 INPUT
 -> detect_sphere_context
 -> identify centre_radius_diameter_or_coordinate_equation
 -> distinguish sphere_surface_and_enclosed_ball when relevant
 -> identify point_membership surface_area volume cross_section tangent_plane symmetry or transformation goal
 -> apply sphere_specific formulas_and_theorems
 -> verify Euclidean_metric positive_radius dimensional and transformation constraints
 -> route to solid_geometry circle coordinate vector analytic or transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Sphere Ontology defines the constant-radius surface in three-dimensional Euclidean space. It supports exact reasoning about centre, radius, diameter, great circles, planar cross-sections, tangent planes, surface area, enclosed volume, coordinate equations, symmetry and transformations while preserving the distinction between the sphere surface and the enclosed ball."
}

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

}

END_FLOW