Conic Section Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of conic sections within the Kireth Research Semantic Library. Conic sections are plane curves obtained by intersecting a plane with a double cone, or equivalently by quadratic equations and focus-directrix relations. This ontology unifies the circle, ellipse, parabola and hyperbola and supports classification, eccentricity, focal structure, coordinate equations, transformations, degeneracy analysis and comparative conic reasoning.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ConicSection
 SYMBOLIC_FORMS = [
  "Ax^2+Bxy+Cy^2+Dx+Ey+F=0",
  "PF=e*PD"
 ]
 ENTITY_CLASS = QuadraticPlaneCurveFamily
 PRIMARY_CONTEXT = PlaneGeometry
 DEFINITION = "A conic section is a plane curve obtained from the intersection of a plane with a cone or represented algebraically by a second-degree equation in two variables."
 SEMANTIC_ROLE = "Provides the unified ontology for circles, ellipses, parabolas and hyperbolas through geometric, focal and algebraic representations."
}

SCOPE {
 INCLUDE = [
  ConicSection,
  Circle,
  Ellipse,
  Parabola,
  Hyperbola,
  DegenerateConic,
  Focus,
  Directrix,
  Eccentricity,
  Centre,
  Vertex,
  Axis,
  Asymptote,
  QuadraticEquation,
  StandardForm,
  GeneralForm,
  RotatedConic,
  TranslatedConic,
  Tangent,
  Normal,
  ParametricRepresentation
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralAlgebraicCurveOfDegreeGreaterThanTwo,
  QuadricSurface,
  ArbitraryOval,
  NonplanarCurve
 ]

 DISAMBIGUATION_RULE =
  "When conic occurs with quadratic equations, circle, ellipse, parabola, hyperbola, eccentricity, focus-directrix relations or plane-cone intersections, interpret it as a conic section."
}

CORE_CONCEPTS {

 ConicSection {
  definition = "A member of the classical family of second-degree plane curves."
  principal_types = [
   Circle,
   Ellipse,
   Parabola,
   Hyperbola
  ]
 }

 Circle {
  definition = "A conic with eccentricity e=0 in the standard focal classification and equal principal semi-axes."
 }

 Ellipse {
  definition = "A conic with eccentricity 0<e<1."
 }

 Parabola {
  definition = "A conic with eccentricity e=1."
 }

 Hyperbola {
  definition = "A conic with eccentricity e>1."
 }

 DegenerateConic {
  definition = "A second-degree equation whose geometric locus collapses into lower-dimensional components or the empty set."
  examples = [
   PairOfLines,
   RepeatedLine,
   Point,
   EmptySet
  ]
 }

 Eccentricity {
  symbol = "e"
  definition = "The constant ratio PF/PD in a focus-directrix representation."
 }

 Focus {
  definition = "A fixed point used in a focal definition of a conic."
 }

 Directrix {
  definition = "A fixed line paired with a focus in a focus-directrix definition."
 }

 GeneralQuadraticEquation {
  form = "Ax^2+Bxy+Cy^2+Dx+Ey+F=0"
 }

 Discriminant {
  definition = "The quantity B^2-4AC used in affine classification of nondegenerate real conics."
 }
}

RELATIONS {
 ConicSection --contains_type--> Circle
 ConicSection --contains_type--> Ellipse
 ConicSection --contains_type--> Parabola
 ConicSection --contains_type--> Hyperbola
 ConicSection --can_be--> DegenerateConic
 ConicSection --can_have--> Focus
 ConicSection --can_have--> Directrix
 ConicSection --can_have--> Eccentricity
 ConicSection --can_have--> Tangent
 ConicSection --can_have--> Normal
 Circle --is_special_case_of--> Ellipse
 Ellipse --has_eccentricity_range--> "0<=e<1"
 Parabola --has_eccentricity--> 1
 Hyperbola --has_eccentricity_range--> "e>1"
 QuadraticEquation --can_represent--> ConicSection
 CoordinateTransformation --can_reduce--> GeneralConicToStandardForm
}

OPERATORS {
 CLASSIFY_BY_ECCENTRICITY(e) :=
  IF e==0 THEN Circle
  ELSE IF 0<e<1 THEN Ellipse
  ELSE IF e==1 THEN Parabola
  ELSE IF e>1 THEN Hyperbola

 CLASSIFY_BY_DISCRIMINANT(A,B,C,degeneracy_status) :=
  IF degeneracy_status==degenerate
   THEN DegenerateConic
  ELSE IF B^2-4AC<0
   THEN EllipseFamily
  ELSE IF B^2-4AC==0
   THEN ParabolaFamily
  ELSE
   THEN HyperbolaFamily

 FOCUS_DIRECTRIX_LOCUS(F,D,e) :=
  set_of_points_P_such_that_distance(P,F)=e*distance(P,D)

 ROTATE_AXES(theta,E) :=
  remove_or_reduce_xy_cross_term_when_possible(E)

 TRANSLATE_AXES(h,k,E) :=
  shift_conic_centre_or_vertex_to_origin(E)

 STANDARDISE(E) :=
  apply_rotation_translation_scaling_to_canonical_form(E)

 VERIFY_CONIC_POINT(P,E) :=
  substitute_coordinates(P,E)

 TANGENT_AT(P,E) :=
  derive_tangent_from_gradient_or_conic_specific_rule(P,E)
}

DEFINITIONS {
 D1 = "A conic section is a second-degree plane curve."
 D2 = "Classical nondegenerate conics are circle, ellipse, parabola and hyperbola."
 D3 = "The quantity B^2-4AC helps classify a general quadratic conic after degeneracy is checked."
 D4 = "Negative discriminant corresponds to ellipse-family conics."
 D5 = "Zero discriminant corresponds to parabola-family conics."
 D6 = "Positive discriminant corresponds to hyperbola-family conics."
 D7 = "Eccentricity classifies conics through focus-directrix geometry."
 D8 = "Coordinate rotation can remove an xy cross-term under suitable conditions."
 D9 = "Translation can move a conic centre or vertex to a standard location."
 D10 = "Degenerate conics must be separated from nondegenerate conics."
}

AXIOMS {
 A1_QUADRATIC_FORM:
  Every algebraic conic is represented by a polynomial equation of total degree at most two with nonzero quadratic part

 A2_CLASSICAL_TYPES:
  Every nondegenerate real conic belongs to the circle, ellipse, parabola or hyperbola family under Euclidean affine classification

 A3_ECCENTRICITY:
  Focus-directrix eccentricity distinguishes ellipse, parabola and hyperbola families

 A4_DISCRIMINANT:
  B^2-4AC determines the quadratic-form family after degeneracy and reality conditions are handled

 A5_COORDINATE_EQUIVALENCE:
  Euclidean coordinate translation and rotation preserve the underlying conic type

 A6_DEGENERACY:
  Degenerate conics require separate classification from genuine conic curves

 A7_TANGENCY:
  Tangent lines arise as first-order contact lines at regular conic points
}

REASONING {

 R1_GENERAL_CLASSIFICATION {
  INPUT = "Ax^2+Bxy+Cy^2+Dx+Ey+F=0"
  COMPUTE = Delta=B^2-4AC
  CHECK = degeneracy
  CHECK = real_locus
  CLASSIFY = ellipse_parabola_hyperbola_or_degenerate
  REFINE = circle_when_quadratic_form_is_isotropic_after_rotation_and_scaling_conditions
  RETURN = conic_type
 }

 R2_ECCENTRICITY_CLASSIFICATION {
  INPUT = e
  REQUIRE = e>=0
  RETURN = type_from_eccentricity
 }

 R3_STANDARDISATION {
  INPUT = general_conic_equation
  IF B!=0
   ROTATE = axes_to_remove_cross_term_when_possible
  TRANSLATE = eliminate_linear_terms
  NORMALISE = coefficients
  RETURN = standard_form_and_type
 }

 R4_FOCUS_DIRECTRIX {
  INPUT = [focus F,directrix D,e]
  FORM = distance(P,F)=e*distance(P,D)
  SQUARE_AND_SIMPLIFY = coordinate_equation
  RETURN = conic_equation
 }

 R5_CONIC_COMPARISON {
  INPUT = conic_type
  RETURN = [
   eccentricity_range,
   focal_structure,
   centre_or_vertex_structure,
   asymptotic_structure,
   standard_equation_family
  ]
 }

 R6_TANGENT_REASONING {
  INPUT = [point P,conic equation E]
  VERIFY = P lies_on E
  COMPUTE = gradient_or_implicit_derivative
  DERIVE = tangent_line
  RETURN = tangent
 }

 R7_DEGENERACY_ANALYSIS {
  INPUT = general_quadratic_equation
  ANALYSE = augmented_quadratic_matrix_or_factorisation
  CLASSIFY = pair_of_lines_repeated_line_point_empty_or_nondegenerate
  RETURN = degeneracy_status
 }

 R8_TRANSFORMATION_REASONING {
  INPUT = [conic,T]
  CLASSIFY = transformation_type
  IF rigid_or_similarity
   preserve_conic_type_and_metric_relations_as_applicable
  ELSE IF nondegenerate_affine
   preserve_general_conic_family_with_circle_may_map_to_ellipse
  RETURN = transformed_conic
 }
}

CONSTRAINTS {
 C1 = "Discriminant classification requires degeneracy checks."
 C2 = "Circle and ellipse both lie in the negative-discriminant family."
 C3 = "Eccentricity-based classification assumes a valid focus-directrix representation."
 C4 = "Rotated conics may contain an xy cross-term."
 C5 = "Affine transformations do not preserve every metric property."
 C6 = "Circle may map to ellipse under general affine transformation."
 C7 = "Degenerate conics must not be treated as genuine ellipse, parabola or hyperbola curves."
 C8 = "Real and empty loci must be distinguished."
}

SEMANTIC_EQUIVALENCE {
 "ConicSection" <=> "QuadraticPlaneCurve"
 "EllipseFamily" <=> "NegativeQuadraticDiscriminantFamily"
 "ParabolaFamily" <=> "ZeroQuadraticDiscriminantFamily"
 "HyperbolaFamily" <=> "PositiveQuadraticDiscriminantFamily"
 "DegenerateConic" <=> "CollapsedSecondDegreeLocus"
}

VALIDATION {
 V1 = "General conic equation has nonzero quadratic part."
 V2 = "Discriminant classification is applied after degeneracy check."
 V3 = "Circle is recognised as a special ellipse-family case."
 V4 = "Eccentricity classification gives ellipse for e<1, parabola for e=1, hyperbola for e>1."
 V5 = "Standardisation preserves the underlying conic type under Euclidean coordinate changes."
 V6 = "Tangent calculation is performed only at regular points."
 V7 = "Degenerate loci are separately typed."
 V8 = "Coordinate-transformed equations describe the same geometric locus under invertible coordinate changes."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONIC_DEFINITION = 0.99
 DISCRIMINANT_CLASSIFICATION = 0.99
 ECCENTRICITY_CLASSIFICATION = 0.99
 DEGENERACY_ANALYSIS = 0.98
 STANDARDISATION_REASONING = 0.98
 TANGENT_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when degeneracy, reality conditions, coordinate frame, focus-directrix data or transformation type are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ConicClassification,
  QuadraticCurveReasoning,
  EccentricityReasoning,
  FocusDirectrixReasoning,
  CoordinateStandardisation,
  TangentReasoning,
  DegeneracyReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "conic section" -> ConicSection
  "conic" -> ConicSection WHEN geometry_context
  "quadratic curve" -> ConicSection
  "circle" -> Circle
  "ellipse" -> Ellipse
  "parabola" -> Parabola
  "hyperbola" -> Hyperbola
  "degenerate conic" -> DegenerateConic
 }

 INTERPRETATION_ORDER = [
  detect_conic_context,
  identify general_standard_focus_directrix_or_parametric representation,
  test degeneracy and real-locus conditions,
  compute discriminant or eccentricity,
  classify conic family,
  standardise coordinates when needed,
  perform focal tangent parametric or transformation reasoning,
  verify constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("conic")
  OR query_mentions_specific_conic_type
  THEN activate(ConicSectionOntology)

 E2:
  IF general_quadratic_equation_present
  THEN compute B^2-4AC
  AND test_degeneracy

 E3:
  IF eccentricity_present
  THEN classify_by_eccentricity

 E4:
  IF xy_cross_term_present
  THEN consider_rotation_of_axes

 E5:
  IF linear_terms_present
  THEN consider_translation_to_standard_form

 E6:
  IF tangent_requested
  THEN verify_regular_point
  AND derive_tangent

 E7:
  IF degenerate_case_detected
  THEN classify_as_degenerate_conic

 E8:
  IF specific_conic_type_resolved
  THEN route_to_circle_ellipse_parabola_or_hyperbola_ontology
}

RECURSION {
 ENTRY = ConicSection

 EXPAND(ConicSection) -> [
  Circle,
  Ellipse,
  Parabola,
  Hyperbola,
  DegenerateConic,
  Focus,
  Directrix,
  Eccentricity,
  QuadraticEquation,
  StandardForm,
  Tangent,
  Transformation
 ]

 EXPAND(QuadraticEquation) -> [
  QuadraticPart,
  Discriminant,
  CrossTerm,
  LinearTerms,
  ConstantTerm,
  Degeneracy
 ]

 EXPAND(Eccentricity) -> [
  CircleCase,
  EllipseCase,
  ParabolaCase,
  HyperbolaCase
 ]

 EXPAND(Standardisation) -> [
  RotationOfAxes,
  TranslationOfAxes,
  Scaling,
  CanonicalEquation
 ]

 STOP_CONDITION =
  "Stop when the conic's type, degeneracy, standard form, focal structure, tangent or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Plane Geometry"

 IMMEDIATE_CONTEXTS = [
  "Circle Ontology",
  "Ellipse Ontology",
  "Parabola Ontology",
  "Hyperbola Ontology",
  "Analytic Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Plane Geometry",
  "Circle Ontology",
  "Ellipse Ontology",
  "Parabola Ontology",
  "Hyperbola Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Parametric Geometry Ontology",
  "Vector Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Conic Section -> Circle for equal-axis closed conics",
  "Conic Section -> Ellipse for eccentricity below one",
  "Conic Section -> Parabola for eccentricity one",
  "Conic Section -> Hyperbola for eccentricity above one",
  "Conic Section -> Analytic Geometry for quadratic-equation classification",
  "Conic Section -> Coordinate Geometry for standard coordinate forms",
  "Conic Section -> Parametric Geometry for parameterisations",
  "Conic Section -> Transformation for rotated translated and affine images"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible conic classes, invariants, representations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve unification of circle, ellipse, parabola and hyperbola.",
  "Preserve discriminant-based family classification.",
  "Preserve eccentricity-based focal classification.",
  "Preserve degeneracy checks.",
  "Preserve rotated and translated conic handling.",
  "Preserve circle-as-special-ellipse relation.",
  "Preserve machine-readable conic semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A conic section is a plane curve represented by a second-degree equation or obtained by intersecting a plane with a cone."

 GENERAL_EQUATION_SUMMARY =
  "Ax^2+Bxy+Cy^2+Dx+Ey+F=0"

 DISCRIMINANT_SUMMARY =
  "B^2-4AC<0: ellipse family; =0: parabola family; >0: hyperbola family, subject to degeneracy checks."

 ECCENTRICITY_SUMMARY =
  "0<=e<1: ellipse family; e=1: parabola; e>1: hyperbola."

 CIRCLE_SUMMARY =
  "A circle is the equal-axis special case of the ellipse family."

 STANDARDISATION_SUMMARY =
  "Rotation and translation of axes reduce a general conic equation toward canonical form."

 DEGENERACY_SUMMARY =
  "Second-degree equations may degenerate into lines, a point or the empty set."
}

FLOW {
 INPUT
 -> detect_conic_section_context
 -> identify_quadratic_focus_directrix_standard_or_parametric_representation
 -> inspect_quadratic_coefficients_and_cross_term
 -> test_degeneracy_and_real_locus
 -> compute_discriminant_or_eccentricity
 -> classify_circle_ellipse_parabola_hyperbola_or_degenerate
 -> standardise_coordinates_when_required
 -> perform focal tangent parametric or transformation reasoning
 -> verify classification_and_constraints
 -> route_to_specific_conic_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Conic Section Ontology unifies the classical circle, ellipse, parabola and hyperbola as second-degree plane curves. It supports classification by quadratic discriminant and eccentricity, focus-directrix definitions, standardisation by coordinate transformation, degeneracy analysis, tangents and comparative conic reasoning."
}

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

}

END_FLOW