Hemisphere Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a hemisphere within the Kireth Research Semantic Library. A hemisphere is one half of a sphere determined by a plane through the sphere's centre. This ontology supports the bounding great circle, curved surface area, total surface area, enclosed half-ball volume, symmetry, cross-sections, coordinate representation, transformations and relationships with spheres, circles and solid geometry.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Hemisphere 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/hemisphere-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Hemisphere
 SYMBOLIC_FORMS = [
  "Hemisphere(S,O,r)",
  "z>=0 AND x^2+y^2+z^2=r^2",
  "z>=0 AND x^2+y^2+z^2<=r^2"
 ]
 ENTITY_CLASS = HalfSphereOrHalfBallStructure
 PRIMARY_CONTEXT = SolidGeometry
 DEFINITION = "A hemisphere is one of the two equal halves of a sphere, or of the enclosed ball, determined by a plane passing through the sphere's centre."
 SEMANTIC_ROLE = "Provides the half-spherical framework for curved surface, base circle, total surface area, half-ball volume, symmetry, coordinate and cross-section reasoning."
}

SCOPE {
 INCLUDE = [
  Hemisphere,
  Sphere,
  HalfSphereSurface,
  HalfBall,
  Centre,
  Radius,
  GreatCircle,
  BaseCircle,
  EquatorialPlane,
  CurvedSurface,
  CurvedSurfaceArea,
  TotalSurfaceArea,
  EnclosedVolume,
  CrossSection,
  Symmetry,
  CoordinateHemisphere,
  SolidHemisphere,
  OpenHemisphere,
  ClosedHemisphere
 ]

 EXCLUDE_AS_PRIMARY = [
  Semicircle,
  SphericalCapSmallerThanHemisphere,
  GeneralHalfEllipsoid,
  PhysicalDomeWithoutMathematicalIdealisation
 ]

 DISAMBIGUATION_RULE =
  "When hemisphere occurs with sphere, radius, great circle, curved surface area, base circle, half-volume or a central cutting plane, interpret it as a spherical hemisphere."
}

CORE_CONCEPTS {

 Hemisphere {
  definition = "One of two equal portions of a sphere determined by a plane through the centre."
 }

 HalfSphereSurface {
  definition = "One half of the spherical surface bounded by a great circle."
  curved_surface_area = "2*pi*r^2"
 }

 HalfBall {
  definition = "One half of the solid ball bounded by a hemisphere and its circular base."
  volume = "(2/3)*pi*r^3"
 }

 BaseCircle {
  definition = "The great circle forming the planar boundary of a hemisphere."
  radius = "r"
  area = "pi*r^2"
 }

 EquatorialPlane {
  definition = "A plane through the sphere centre whose intersection with the sphere is the bounding great circle."
 }

 CurvedSurfaceArea {
  formula = "2*pi*r^2"
 }

 TotalSurfaceArea {
  definition = "Curved hemisphere area plus circular base area for a closed solid hemisphere."
  formula = "3*pi*r^2"
 }

 EnclosedVolume {
  definition = "Volume of the associated half-ball."
  formula = "(2/3)*pi*r^3"
 }

 OpenHemisphere {
  definition = "A hemisphere excluding its bounding great circle in topological usage."
 }

 ClosedHemisphere {
  definition = "A hemisphere including its bounding great circle."
 }
}

RELATIONS {
 Hemisphere --is_half_of--> Sphere
 HalfBall --is_half_of--> Ball
 Hemisphere --has_boundary--> GreatCircle
 GreatCircle --lies_in--> EquatorialPlane
 BaseCircle --is--> GreatCircle
 Hemisphere --has--> Radius
 Hemisphere --shares_centre_with--> Sphere
 Hemisphere --has--> CurvedSurface
 SolidHemisphere --has--> CircularBase
 SolidHemisphere --has_measure--> EnclosedVolume
 Hemisphere --can_have--> CrossSection
 Hemisphere --has_symmetry_about--> AxisPerpendicularToBase
 Hemisphere --can_be_transformed_by--> ThreeDimensionalTransformation
}

OPERATORS {
 CURVED_SURFACE_AREA(r) :=
  2*pi*r^2

 BASE_AREA(r) :=
  pi*r^2

 TOTAL_SURFACE_AREA(r) :=
  3*pi*r^2

 VOLUME(r) :=
  (2/3)*pi*r^3

 DIAMETER(r) :=
  2*r

 IS_ON_HEMISPHERE_SURFACE(P,O,r,side_condition) :=
  distance(P,O)==r
  AND satisfies_half_space_condition(P,side_condition)

 IS_IN_HALF_BALL(P,O,r,side_condition) :=
  distance(P,O)<=r
  AND satisfies_half_space_condition(P,side_condition)

 BASE_CIRCLE(HEMISPHERE) :=
  sphere_intersection_with_equatorial_plane

 CROSS_SECTION(H,PLANE) :=
  geometric_intersection(H,PLANE)

 TRANSFORM_HEMISPHERE(H,T) :=
  image_of_hemisphere_under_3D_transformation(H,T)
}

DEFINITIONS {
 D1 = "A hemisphere is one half of a sphere determined by a central plane."
 D2 = "The boundary of a spherical hemisphere is a great circle."
 D3 = "The curved surface area of a hemisphere is 2*pi*r^2."
 D4 = "The circular base area is pi*r^2."
 D5 = "The total surface area of a closed solid hemisphere is 3*pi*r^2."
 D6 = "The volume of a half-ball is (2/3)*pi*r^3."
 D7 = "A hemisphere and its opposite hemisphere together form the full sphere."
 D8 = "A half-ball and its opposite half-ball together form the full ball."
 D9 = "Open and closed hemisphere terminology depends on whether the equatorial boundary is included."
 D10 = "A semicircle is two-dimensional and must not be confused with a hemisphere."
}

AXIOMS {
 A1_CENTRAL_DIVISION:
  A plane through the centre divides a sphere into two congruent hemispheres

 A2_EQUAL_HALVES:
  The two hemispheres have equal curved surface area

 A3_BOUNDARY:
  The boundary between the hemispheres is a great circle

 A4_CURVED_AREA:
  Each hemisphere has half the surface area of the sphere

 A5_HALF_VOLUME:
  Each half-ball has half the volume of the full ball

 A6_ROTATIONAL_SYMMETRY:
  A hemisphere is rotationally symmetric about the axis through the sphere centre perpendicular to its base plane

 A7_RIGID_INVARIANCE:
  Rigid motions map hemispheres to congruent hemispheres
}

REASONING {

 R1_HEMISPHERE_IDENTIFICATION {
  INPUT = [sphere,cutting_plane]
  VERIFY = cutting_plane_contains_centre
  IF true
   RETURN TwoCongruentHemispheres
  ELSE
   RETURN SphericalCapsNotHemispheres
 }

 R2_CURVED_SURFACE_AREA {
  INPUT = r
  REQUIRE = r>0
  COMPUTE = half_of(4*pi*r^2)
  RETURN = 2*pi*r^2
 }

 R3_TOTAL_SURFACE_AREA {
  INPUT = r
  COMPUTE = curved_area + base_circle_area
  RETURN = 3*pi*r^2
 }

 R4_VOLUME {
  INPUT = r
  REQUIRE = r>0
  COMPUTE = half_of((4/3)*pi*r^3)
  RETURN = (2/3)*pi*r^3
 }

 R5_POINT_CLASSIFICATION {
  INPUT = [P,O,r,half_space]
  COMPUTE = d=distance(P,O)
  TEST = half_space_condition
  IF d==r AND condition
   RETURN PointOnHemisphereSurface
  ELSE IF d<r AND condition
   RETURN PointInsideHalfBall
  ELSE
   RETURN OutsideSelectedHemisphereOrHalfBall
 }

 R6_BASE_RELATION {
  INPUT = hemisphere
  IDENTIFY = equatorial_plane
  INTERSECT = sphere_with_plane
  RETURN = great_circle_base
 }

 R7_CROSS_SECTION {
  INPUT = [hemisphere,plane]
  INTERSECT = selected_half_sphere_or_half_ball_with_plane
  CLASSIFY = circle_semicircle_segment_or_other_planar_section
  RETURN = cross_section
 }

 R8_TRANSFORMATION {
  INPUT = [hemisphere,T]
  IF rigid_or_similarity_transformation
   RETURN hemisphere_with_transformed_centre_radius_and_orientation
  ELSE IF general_affine_transformation
   RETURN half_ellipsoidal_image_in_general
 }
}

CONSTRAINTS {
 C1 = "A hemisphere must be distinguished from a semicircle."
 C2 = "The dividing plane must pass through the sphere centre."
 C3 = "Curved surface area must be distinguished from total surface area."
 C4 = "Volume refers to the associated half-ball, not the two-dimensional spherical surface itself."
 C5 = "Open versus closed hemisphere usage must be explicit in topological contexts."
 C6 = "A noncentral plane cut produces a spherical cap, not a hemisphere."
 C7 = "General affine transformations do not preserve hemispherical shape."
 C8 = "Radius must be positive for a nondegenerate hemisphere."
}

SEMANTIC_EQUIVALENCE {
 "Hemisphere" <=> "HalfOfSphereByCentralPlane"
 "HalfBall" <=> "HalfOfClosedBallByCentralPlane"
 "BaseCircle" <=> "BoundingGreatCircle"
 "CurvedSurfaceArea" <=> "2*pi*r^2"
 "TotalSurfaceArea" <=> "3*pi*r^2" WHEN closed_solid_hemisphere
}

VALIDATION {
 V1 = "Dividing plane passes through sphere centre."
 V2 = "Curved surface area equals 2*pi*r^2."
 V3 = "Base area equals pi*r^2."
 V4 = "Total closed-surface area equals 3*pi*r^2."
 V5 = "Half-ball volume equals (2/3)*pi*r^3."
 V6 = "Base circle is a great circle."
 V7 = "Opposite hemispheres reconstruct the sphere."
 V8 = "Rigid motions preserve radius and hemisphere structure."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 HEMISPHERE_DEFINITION = 0.99
 CENTRAL_PLANE_RELATION = 0.99
 CURVED_AREA_REASONING = 0.99
 TOTAL_AREA_REASONING = 0.99
 VOLUME_REASONING = 0.99
 BASE_CIRCLE_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when sphere-versus-ball usage, open-versus-closed convention, cutting-plane position, radius or transformation class are underspecified."
}

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

 NORMALISE {
  "hemisphere" -> Hemisphere
  "half sphere" -> Hemisphere
  "half-sphere" -> Hemisphere
  "solid hemisphere" -> HalfBall WHEN volume_context
  "hemispherical surface" -> HalfSphereSurface
  "hemisphere base" -> BaseCircle
  "hemisphere volume" -> EnclosedVolume
 }

 INTERPRETATION_ORDER = [
  detect_hemisphere_context,
  identify parent sphere centre radius and dividing plane,
  distinguish spherical_surface from half_ball when relevant,
  identify curved_area total_area volume base cross_section symmetry or transformation goal,
  apply hemisphere_specific formulas_and_theorems,
  verify central_plane positive_radius and convention constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("hemisphere")
  AND geometry_context
  THEN activate(HemisphereOntology)

 E2:
  IF cutting_plane_given
  THEN verify plane_passes_through_sphere_centre

 E3:
  IF curved_surface_area_requested
  THEN compute 2*pi*r^2

 E4:
  IF total_surface_area_requested
  THEN compute 3*pi*r^2 for closed solid hemisphere

 E5:
  IF volume_requested
  THEN interpret as half-ball volume
  AND compute (2/3)*pi*r^3

 E6:
  IF base_requested
  THEN return great_circle_of_radius_r

 E7:
  IF noncentral_plane_cut_detected
  THEN route_to_spherical_cap_reasoning

 E8:
  IF transformation_context
  THEN classify rigid_similarity_or_affine map
}

RECURSION {
 ENTRY = Hemisphere

 EXPAND(Hemisphere) -> [
  Sphere,
  Centre,
  Radius,
  GreatCircle,
  BaseCircle,
  EquatorialPlane,
  HalfSphereSurface,
  HalfBall,
  CurvedSurfaceArea,
  TotalSurfaceArea,
  EnclosedVolume,
  CrossSection,
  Symmetry
 ]

 EXPAND(BaseCircle) -> [
  Circle,
  Radius,
  Diameter,
  Area,
  EquatorialPlane
 ]

 EXPAND(HalfBall) -> [
  CurvedSurface,
  CircularBase,
  Volume,
  Boundary
 ]

 EXPAND(CrossSection) -> [
  Plane,
  Circle,
  Semicircle,
  Segment,
  SphericalCapRelation
 ]

 STOP_CONDITION =
  "Stop when the hemisphere's defining relation, surface measure, enclosed volume, base, cross-section, symmetry or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Solid Geometry"

 IMMEDIATE_CONTEXTS = [
  "Solid Geometry Ontology",
  "Sphere Ontology",
  "Circle Ontology"
 ]

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

 ROUTE_RULES = [
  "Hemisphere -> Sphere for parent spherical geometry",
  "Hemisphere -> Circle for base great-circle semantics",
  "Hemisphere -> Solid Geometry Ontology for volume and surface reasoning",
  "Hemisphere -> Coordinate Geometry for half-space coordinate representations",
  "Hemisphere -> Vector Geometry for base-plane normals and spatial orientation",
  "Hemisphere -> Transformation for rigid, similarity and affine images",
  "Hemisphere -> Dimension for surface-versus-solid dimensional interpretation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve hemisphere as half of a sphere by a central plane.",
  "Preserve distinction between hemispherical surface and half-ball.",
  "Preserve curved-versus-total surface area distinction.",
  "Preserve great-circle boundary semantics.",
  "Preserve half-volume relation.",
  "Preserve open-versus-closed topology when relevant.",
  "Preserve machine-readable hemisphere semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A hemisphere is one of the two equal halves of a sphere determined by a plane through its centre."

 CURVED_AREA_SUMMARY =
  "A_curved=2*pi*r^2"

 BASE_AREA_SUMMARY =
  "A_base=pi*r^2"

 TOTAL_AREA_SUMMARY =
  "A_total=3*pi*r^2 for a closed solid hemisphere."

 VOLUME_SUMMARY =
  "V=(2/3)*pi*r^3 for the enclosed half-ball."

 BASE_SUMMARY =
  "The bounding base is a great circle of radius r."

 SYMMETRY_SUMMARY =
  "The hemisphere is rotationally symmetric about the axis through the centre perpendicular to its base plane."
}

FLOW {
 INPUT
 -> detect_hemisphere_context
 -> identify parent_sphere centre radius and dividing_plane
 -> verify central_plane condition
 -> distinguish hemisphere_surface_and_half_ball when relevant
 -> identify curved_area total_area volume base cross_section symmetry or transformation goal
 -> apply hemisphere_specific geometry
 -> verify positive_radius boundary convention and transformation constraints
 -> route to sphere circle solid_geometry coordinate vector or transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Hemisphere Ontology defines one half of a sphere or enclosed ball produced by a plane through the sphere centre. It supports exact reasoning about the bounding great circle, curved and total surface area, half-ball volume, symmetry, coordinate representation, cross-sections and transformations while preserving the distinction between spherical surface and solid region."
}

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

}

END_FLOW