Cone Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a cone within the Kireth Research Semantic Library. A cone is a three-dimensional solid formed by joining every point of a plane base region to a single apex. This ontology supports right and oblique cones, circular cones, axis, radius, height, slant height, lateral surface area, total surface area, volume, cross-sections, coordinate equations, symmetry, transformations and relationships with cylinders, pyramids and conic sections.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Cone
 SYMBOLIC_FORMS = [
  "V=(1/3)Ah",
  "x^2+y^2=(r^2/h^2)z^2"
 ]
 ENTITY_CLASS = ApexBasedSolid
 PRIMARY_CONTEXT = SolidGeometry
 DEFINITION = "A cone is a three-dimensional solid formed by joining every point of a plane base region to a single point called the apex."
 SEMANTIC_ROLE = "Provides the canonical apex-based solid framework for volume, slant geometry, cross-sections, conic sections, coordinate representation and transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Cone,
  RightCone,
  ObliqueCone,
  CircularCone,
  RightCircularCone,
  Apex,
  Base,
  Axis,
  Radius,
  Height,
  SlantHeight,
  Generator,
  LateralSurface,
  CurvedSurfaceArea,
  TotalSurfaceArea,
  Volume,
  CrossSection,
  CoordinateCone,
  DoubleCone,
  SolidOfRevolution,
  Symmetry
 ]

 EXCLUDE_AS_PRIMARY = [
  Cylinder,
  Pyramid,
  GeneralRuledSurfaceWithoutCommonApex,
  FrustumUnlessExplicitlyDerived,
  PhysicalConeWithoutMathematicalIdealisation
 ]

 DISAMBIGUATION_RULE =
  "When cone occurs with apex, base, radius, height, slant height, curved surface, volume, cross-sections or conic sections, interpret it as a geometric cone."
}

CORE_CONCEPTS {

 Cone {
  definition = "A solid generated by line segments joining a base region to a common apex."
 }

 Apex {
  definition = "The common endpoint of all generators of the cone."
 }

 Base {
  definition = "The plane region opposite the apex."
 }

 RightCone {
  definition = "A cone whose axis is perpendicular to the base plane."
 }

 ObliqueCone {
  definition = "A cone whose axis is not perpendicular to the base plane."
 }

 CircularCone {
  definition = "A cone with a circular base."
 }

 RightCircularCone {
  definition = "A circular cone whose axis is perpendicular to the base plane and passes through the base centre."
 }

 Axis {
  definition = "The line joining the apex to the centre of the circular base."
 }

 Radius {
  symbol = "r"
  definition = "The radius of a circular base."
 }

 Height {
  symbol = "h"
  definition = "The perpendicular distance from the apex to the base plane."
 }

 SlantHeight {
  symbol = "l"
  definition = "For a right circular cone, the distance from the apex to a point on the base circumference along a generator."
  relation = "l=sqrt(r^2+h^2)"
 }

 Generator {
  definition = "A segment joining the apex to a boundary point of the base."
 }

 LateralSurface {
  definition = "The boundary surface formed by the generators, excluding the base."
 }

 CurvedSurfaceArea {
  right_circular_formula = "pi*r*l"
 }

 TotalSurfaceArea {
  right_circular_formula = "pi*r*l+pi*r^2"
 }

 Volume {
  formula = "(1/3)*base_area*h"
  right_circular_formula = "(1/3)*pi*r^2*h"
 }
}

RELATIONS {
 Cone --is_special_case_of--> Solid
 RightCone --is_special_case_of--> Cone
 ObliqueCone --is_special_case_of--> Cone
 CircularCone --is_special_case_of--> Cone
 RightCircularCone --is_special_case_of--> RightCone
 RightCircularCone --is_special_case_of--> CircularCone
 Cone --has--> Apex
 Cone --has--> Base
 Cone --has--> Height
 Cone --has--> Generator
 CircularCone --has--> Radius
 CircularCone --has--> Axis
 RightCircularCone --has--> SlantHeight
 Cone --can_have--> CrossSection
 DoubleCone --contains--> TwoOppositelyDirectedConicalSheets
 Plane --can_intersect--> DoubleCone
 PlaneIntersection --can_produce--> ConicSection
}

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

 VOLUME(base_area,h) :=
  (1/3)*base_area*h

 VOLUME_RIGHT_CIRCULAR(r,h) :=
  (1/3)*pi*r^2*h

 SLANT_HEIGHT(r,h) :=
  sqrt(r^2+h^2)

 CURVED_SURFACE_AREA_RIGHT_CIRCULAR(r,l) :=
  pi*r*l

 TOTAL_SURFACE_AREA_RIGHT_CIRCULAR(r,l) :=
  pi*r*l+pi*r^2

 AXIAL_CROSS_SECTION_RIGHT_CIRCULAR(r,h) :=
  isosceles_triangle(base=2*r,height=h)

 TRANSVERSE_CROSS_SECTION_RIGHT_CIRCULAR(r,z,h) :=
  similar_circle_with_radius_depending_linearly_on_height(z)

 STANDARD_DOUBLE_CONE_Z_AXIS(k) :=
  x^2+y^2=k^2*z^2

 TRANSFORM_CONE(C,T) :=
  image_of_cone_under_3D_transformation(C,T)
}

DEFINITIONS {
 D1 = "A cone has a single apex."
 D2 = "Every generator joins the apex to the base boundary."
 D3 = "Cone height is the perpendicular distance from apex to base plane."
 D4 = "A right cone has axis perpendicular to the base."
 D5 = "A circular cone has a circular base."
 D6 = "A right circular cone has volume (1/3)*pi*r^2*h."
 D7 = "Its slant height satisfies l^2=r^2+h^2."
 D8 = "Its curved surface area is pi*r*l."
 D9 = "Its total surface area is pi*r*l+pi*r^2."
 D10 = "Plane sections of a double cone generate classical conic sections."
}

AXIOMS {
 A1_COMMON_APEX:
  All generators of a cone meet at one apex

 A2_PLANAR_BASE:
  The base lies in a plane not containing the apex

 A3_HEIGHT:
  Height is perpendicular distance from apex to base plane

 A4_VOLUME:
  Cone volume equals one third of base_area times perpendicular_height

 A5_RIGHT_CIRCULAR_SLANT:
  For a right circular cone l^2=r^2+h^2

 A6_RIGHT_CIRCULAR_LATERAL_AREA:
  Lateral area equals pi*r*l

 A7_CONIC_SECTION:
  Suitable plane intersections of a double cone produce circle ellipse parabola or hyperbola

 A8_RIGID_INVARIANCE:
  Rigid motions preserve cone metric structure
}

REASONING {

 R1_CONE_CLASSIFICATION {
  INPUT = solid_properties
  IDENTIFY = base_shape
  IDENTIFY = apex
  IDENTIFY = axis_and_height_relation
  CLASSIFY = right_oblique_circular_or_right_circular
  RETURN = cone_type
 }

 R2_VOLUME {
  INPUT = [base_area,h]
  REQUIRE = h>=0
  COMPUTE = (1/3)*base_area*h
  RETURN = volume
 }

 R3_SLANT_HEIGHT {
  INPUT = [r,h]
  REQUIRE = right_circular_context
  APPLY = Pythagorean_theorem
  RETURN = sqrt(r^2+h^2)
 }

 R4_SURFACE_AREA {
  INPUT = [r,l]
  REQUIRE = right_circular_context
  COMPUTE = curved_area=pi*r*l
  COMPUTE = base_area=pi*r^2
  RETURN = requested_area
 }

 R5_AXIAL_SECTION {
  INPUT = right_circular_cone
  SELECT = plane_through_axis
  INTERSECT = solid_with_plane
  RETURN = isosceles_triangle(base=2r,height=h)
 }

 R6_PARALLEL_SECTION {
  INPUT = [cone,plane_parallel_to_base]
  INTERSECT = cone_with_plane
  APPLY = similarity
  RETURN = similar_scaled_base_section
 }

 R7_CONIC_SECTION_REASONING {
  INPUT = [double_cone,plane]
  CLASSIFY = plane_orientation_relative_to_axis_and_generators
  RETURN = circle_ellipse_parabola_hyperbola_or_degenerate_section
 }

 R8_COORDINATE_CONE {
  INPUT = quadratic_equation
  IDENTIFY = axis_orientation
  IDENTIFY = opening_rate
  DETERMINE = single_or_double_cone_context
  RETURN = cone_parameters
 }

 R9_TRANSFORMATION {
  INPUT = [cone,T]
  IF rigid_or_similarity
   RETURN corresponding_cone
  ELSE IF general_affine
   RETURN affine_cone_with_base_and_metric_properties_changed
 }
}

CONSTRAINTS {
 C1 = "Cone height is perpendicular distance, not slant height."
 C2 = "The formula l=sqrt(r^2+h^2) is specific to right circular cones."
 C3 = "Curved surface area pi*r*l is specific to right circular cones."
 C4 = "Volume uses perpendicular height for both right and oblique cones."
 C5 = "Coordinate equations may describe infinite double cones rather than bounded solid cones."
 C6 = "A frustum is not a full cone."
 C7 = "General affine transformations need not preserve circular bases or angles."
 C8 = "Radius must be positive for a nondegenerate circular cone."
}

SEMANTIC_EQUIVALENCE {
 "Cone" <=> "ApexJoinedToPlanarBaseRegion"
 "RightCircularCone" <=> "CircularConeWithAxisPerpendicularToBase"
 "ConeVolume" <=> "OneThirdBaseAreaTimesHeight"
 "SlantHeight" <=> "GeneratorLengthFromApexToBaseCircumference" WHEN right_circular_context
 "DoubleCone" <=> "TwoOppositelyDirectedConicalSheetsSharingApex"
}

VALIDATION {
 V1 = "Cone has a unique apex."
 V2 = "Base is planar."
 V3 = "Height is perpendicular to base plane."
 V4 = "Volume equals (1/3)*base_area*h."
 V5 = "Right circular slant height satisfies l^2=r^2+h^2."
 V6 = "Right circular lateral area equals pi*r*l."
 V7 = "Axial section of a right circular cone is an isosceles triangle."
 V8 = "Parallel-to-base sections are similar to the base."
 V9 = "Conic-section classification depends on plane orientation."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONE_DEFINITION = 0.99
 CLASSIFICATION = 0.99
 VOLUME_REASONING = 0.99
 SLANT_HEIGHT_REASONING = 0.99
 SURFACE_AREA_REASONING = 0.99
 CROSS_SECTION_REASONING = 0.99
 CONIC_SECTION_REASONING = 0.99

 POLICY =
  "Reduce confidence when right-versus-oblique status, base shape, height, boundedness, plane orientation or transformation class are underspecified."
}

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

 NORMALISE {
  "cone" -> Cone
  "right cone" -> RightCone
  "oblique cone" -> ObliqueCone
  "circular cone" -> CircularCone
  "right circular cone" -> RightCircularCone
  "slant height" -> SlantHeight
  "double cone" -> DoubleCone
  "conical surface" -> LateralSurface
 }

 INTERPRETATION_ORDER = [
  detect_cone_context,
  identify apex base_shape axis radius height and slant_height_when_applicable,
  classify right oblique circular or right_circular,
  identify volume surface_area cross_section conic_section coordinate symmetry or transformation goal,
  apply cone_specific formulas_and_theorems,
  verify perpendicular_height right_cone and boundedness constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("cone")
  AND geometry_context
  THEN activate(ConeOntology)

 E2:
  IF circular_base_and_axis_perpendicular
  THEN classify(RightCircularCone)

 E3:
  IF volume_requested
  THEN compute (1/3)*base_area*perpendicular_height

 E4:
  IF slant_height_requested
  AND right_circular_context
  THEN compute sqrt(r^2+h^2)

 E5:
  IF curved_surface_area_requested
  AND right_circular_context
  THEN compute pi*r*l

 E6:
  IF total_surface_area_requested
  AND right_circular_context
  THEN compute pi*r*l+pi*r^2

 E7:
  IF plane_section_of_double_cone_requested
  THEN classify conic_section by plane_orientation

 E8:
  IF coordinate_equation_is_unbounded
  THEN interpret as conical_surface_or_double_cone rather than finite solid
}

RECURSION {
 ENTRY = Cone

 EXPAND(Cone) -> [
  Apex,
  Base,
  Axis,
  Height,
  Generator,
  LateralSurface,
  RightCone,
  ObliqueCone,
  CircularCone,
  RightCircularCone,
  CrossSection,
  SurfaceArea,
  Volume,
  DoubleCone
 ]

 EXPAND(RightCircularCone) -> [
  Radius,
  Diameter,
  SlantHeight,
  BaseCircle,
  CurvedSurfaceArea,
  TotalSurfaceArea,
  AxialSection,
  ParallelSection,
  Net
 ]

 EXPAND(ConicSection) -> [
  Circle,
  Ellipse,
  Parabola,
  Hyperbola,
  DegenerateConic
 ]

 EXPAND(CoordinateCone) -> [
  AxisDirection,
  OpeningAngle,
  QuadraticEquation,
  SingleConeOrDoubleCone
 ]

 STOP_CONDITION =
  "Stop when the cone's type, dimensions, surface measure, volume, cross-section, conic-section relation, coordinate representation or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Solid Geometry"

 IMMEDIATE_CONTEXTS = [
  "Solid Geometry Ontology",
  "Cylinder Ontology",
  "Circle Ontology",
  "Conic Section Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Solid Geometry Ontology",
  "Cylinder Ontology",
  "Circle Ontology",
  "Conic Section Ontology",
  "Plane Geometry",
  "Triangle Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Vector Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Cone -> Solid Geometry Ontology for general three-dimensional semantics",
  "Cone -> Circle for circular-base reasoning",
  "Cone -> Triangle for axial-section reasoning",
  "Cone -> Conic Section for plane sections of the double cone",
  "Cone -> Coordinate Geometry for axis-aligned equations",
  "Cone -> Analytic Geometry for quadratic-surface representation",
  "Cone -> Vector Geometry for axis and plane-section reasoning",
  "Cone -> Transformation for rigid, similarity and affine images",
  "Cone -> Cylinder for comparative curved-solid reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve apex-base generator semantics.",
  "Preserve distinction between height and slant height.",
  "Preserve right-versus-oblique cone distinction.",
  "Preserve right-circular surface formulas.",
  "Preserve double-cone relation to conic sections.",
  "Preserve bounded-solid versus unbounded-coordinate-surface distinction.",
  "Preserve machine-readable cone semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A cone is a three-dimensional solid formed by joining every point of a plane base region to a common apex."

 VOLUME_SUMMARY =
  "V=(1/3)*base_area*h"

 RIGHT_CIRCULAR_VOLUME_SUMMARY =
  "V=(1/3)*pi*r^2*h"

 SLANT_HEIGHT_SUMMARY =
  "l=sqrt(r^2+h^2) for a right circular cone."

 CURVED_AREA_SUMMARY =
  "A_lateral=pi*r*l for a right circular cone."

 TOTAL_AREA_SUMMARY =
  "A_total=pi*r*l+pi*r^2 for a closed right circular cone."

 AXIAL_SECTION_SUMMARY =
  "A plane through the axis of a right circular cone gives an isosceles triangle."

 CONIC_SECTION_SUMMARY =
  "Plane intersections of a double cone generate the classical conic sections."
}

FLOW {
 INPUT
 -> detect_cone_context
 -> identify apex_base_axis_height_radius_and_slant_height_when_applicable
 -> classify right_oblique_circular_or_right_circular cone
 -> identify volume_surface_area_cross_section_conic_section_coordinate_symmetry_or_transformation goal
 -> apply cone_specific geometry
 -> verify perpendicular_height right_cone boundedness and surface_or_solid constraints
 -> route to solid_geometry circle triangle conic_section coordinate analytic vector cylinder_or_transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Cone Ontology defines apex-based solids formed by joining a plane base to a common apex. It supports exact reasoning about right and oblique cones, circular bases, height, slant height, surface area, volume, axial and parallel cross-sections, double-cone conic sections, coordinate equations and transformations."
}

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

}

END_FLOW