Analytic Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of analytic geometry within the Kireth Research Semantic Library. Analytic geometry studies geometric entities and relations by representing them with coordinates, algebraic equations, vectors and functions. It provides a systematic framework for lines, curves, conic sections, surfaces, intersections, loci and transformations, linking geometry directly with algebra. This record is designed for semantic retrieval, recursive mathematical reasoning, ontology linking, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AnalyticGeometry
 SYMBOLIC_FORMS = ["F(x,y)=0", "F(x,y,z)=0", "y=f(x)", "Ax+By+C=0"]
 ENTITY_CLASS = AlgebraicGeometricFramework
 PRIMARY_CONTEXT = Geometry
 SECONDARY_CONTEXT = Algebra
 DEFINITION = "Analytic geometry is the study of geometric objects and relations through coordinate systems, algebraic equations, vectors and functions."
 SEMANTIC_ROLE = "Translates geometric structure into algebraic form so that geometric problems can be analysed through symbolic and computational methods."
}

SCOPE {
 INCLUDE = [
  AnalyticGeometry,
  CartesianCoordinates,
  CoordinateSystem,
  AlgebraicCurve,
  AlgebraicSurface,
  LineEquation,
  CircleEquation,
  ConicSection,
  QuadraticEquation,
  Intersection,
  Locus,
  Distance,
  Gradient,
  VectorEquation,
  ParametricEquation,
  ImplicitEquation,
  Transformation,
  CoordinateProof,
  SurfaceEquation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureSyntheticGeometry,
  PureAxiomaticGeometryWithoutCoordinateRepresentation,
  NumericalPlottingWithoutAnalyticStructure
 ]

 DISAMBIGUATION_RULE =
  "When geometric entities are represented by equations, coordinate functions, vectors, parametric forms or implicit relations, interpret the context as analytic geometry."
}

CORE_CONCEPTS {

 AnalyticGeometry {
  definition = "Geometry expressed and analysed using algebraic and coordinate methods."
  methodological_role = "Bridge between geometric objects and equations."
 }

 CoordinateSystem {
  definition = "A reference framework assigning coordinates to points in geometric space."
 }

 AlgebraicCurve {
  definition = "A curve represented by an algebraic equation in coordinate variables."
  general_form = "F(x,y)=0"
 }

 AlgebraicSurface {
  definition = "A surface represented by an algebraic equation in three or more coordinates."
  general_form = "F(x,y,z)=0"
 }

 LineEquation {
  common_forms = [
   "y=mx+c",
   "Ax+By+C=0",
   "r=a+t*d"
  ]
 }

 CircleEquation {
  standard_form = "(x-h)^2+(y-k)^2=r^2"
 }

 ConicSection {
  definition = "A plane curve represented by a second-degree equation."
  general_form = "Ax^2+Bxy+Cy^2+Dx+Ey+F=0"
  types = [Circle, Ellipse, Parabola, Hyperbola]
 }

 ParametricEquation {
  definition = "A representation of a geometric object using one or more parameters."
  planar_form = ["x=x(t)", "y=y(t)"]
 }

 ImplicitEquation {
  definition = "A relation defining a geometric object without solving explicitly for one coordinate."
  form = "F(x,y)=0"
 }

 Locus {
  definition = "The set of all points satisfying a specified geometric or algebraic condition."
 }

 Intersection {
  definition = "The set of points satisfying multiple geometric equations simultaneously."
 }
}

RELATIONS {
 AnalyticGeometry --is_subdomain_of--> CoordinateAnalyticGeometry
 AnalyticGeometry --uses--> Algebra
 AnalyticGeometry --uses--> CoordinateSystem
 GeometricEntity --can_be_represented_by--> Equation
 GeometricEntity --can_be_represented_by--> ParametricEquation
 Equation --can_define--> Locus
 Line --can_be_represented_by--> LineEquation
 Circle --can_be_represented_by--> CircleEquation
 ConicSection --can_be_represented_by--> QuadraticEquation
 AlgebraicCurve --can_intersect--> AlgebraicCurve
 Surface --can_be_represented_by--> SurfaceEquation
 AnalyticGeometry --supports--> TransformationReasoning
 AnalyticGeometry --supports--> CoordinateProof
 AnalyticGeometry --supports--> AlgebraicClassification
}

OPERATORS {
 REPRESENT(entity,coordinates) :=
  derive_equation_or_parametric_form(entity,coordinates)

 INTERSECT(E1,E2) :=
  solve_simultaneous_equations(E1,E2)

 CLASSIFY_CONIC(E) :=
  analyse_quadratic_coefficients(E)

 LOCUS(condition) :=
  translate_geometric_condition_to_equation(condition)

 PARAMETRISE(entity) :=
  construct_parametric_representation(entity)

 IMPLICITISE(parametric_form) :=
  eliminate_parameters_to_obtain_implicit_equation

 TRANSFORM_EQUATION(E,T) :=
  apply_coordinate_transformation_to_equation(E,T)

 DISTANCE(P,Q) :=
  apply_coordinate_metric(P,Q)

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

 VERIFY_EQUIVALENT_LOCI(E1,E2) :=
  determine_whether_solution_sets_are_equal
}

DEFINITIONS {
 D1 = "Analytic geometry expresses geometric objects through algebraic equations."
 D2 = "An implicit equation defines a set of coordinate points."
 D3 = "A parametric equation defines a geometric object through parameter-dependent coordinates."
 D4 = "Intersections correspond to common solutions of equations."
 D5 = "Conic sections arise from quadratic equations in two variables."
 D6 = "Coordinate transformations can simplify or classify geometric equations."
 D7 = "Equivalent equations may represent the same locus."
 D8 = "Geometric classification can be derived from algebraic coefficients and invariants."
 D9 = "Analytic geometry does not identify an object solely with one chosen equation form."
}

AXIOMS {
 A1_ALGEBRA_GEOMETRY_CORRESPONDENCE:
  Valid equations encode geometric loci in a chosen coordinate system

 A2_INTERSECTION:
  A point lies on multiple geometric objects iff its coordinates satisfy all corresponding equations

 A3_COORDINATE_REPRESENTATION:
  Coordinate changes may alter equations while preserving the underlying geometry

 A4_CLASSIFICATION:
  Algebraic invariants can determine geometric type under suitable coordinate assumptions

 A5_PARAMETRIC_IMPLICIT_RELATION:
  Parametric and implicit forms may describe the same geometric object

 A6_PROOF_VALIDITY:
  Analytic derivations must preserve both algebraic correctness and geometric interpretation
}

REASONING {

 R1_EQUATION_FROM_GEOMETRY {
  INPUT = geometric_definition
  TRANSLATE = positions_distances_angles_or_incidence_into_coordinates
  SIMPLIFY = algebraic_relation
  RETURN = defining_equation
 }

 R2_INTERSECTION {
  INPUT = [E1,E2]
  SOLVE = simultaneous_system
  VERIFY = candidate_solutions
  RETURN = geometric_intersection_set
 }

 R3_CONIC_CLASSIFICATION {
  INPUT = "Ax^2+Bxy+Cy^2+Dx+Ey+F=0"
  COMPUTE = discriminant "B^2-4AC"
  CONSIDER = degeneracy_and_coordinate_rotation
  RETURN = conic_classification
 }

 R4_PARAMETRIC_TO_IMPLICIT {
  INPUT = parametric_equations
  ELIMINATE = parameters
  RETURN = implicit_relation
 }

 R5_IMPLICIT_TO_PARAMETRIC {
  INPUT = geometric_or_algebraic_curve
  SEEK = suitable_parameter
  CONSTRUCT = coordinate_functions
  RETURN = parametric_representation
 }

 R6_COORDINATE_TRANSFORMATION {
  INPUT = [equation E, transformation T]
  SUBSTITUTE = inverse_or_direct_coordinate_change_as_required
  SIMPLIFY = transformed_equation
  IDENTIFY = preserved_geometric_structure
  RETURN = new_representation
 }

 R7_LOCUS_REASONING {
  INPUT = geometric_condition
  EXPRESS = condition_algebraically
  DERIVE = equation
  VERIFY = necessity_and_sufficiency
  RETURN = locus_equation
 }

 R8_ANALYTIC_PROOF {
  INPUT = geometric_proposition
  ASSIGN = coordinates
  CONVERT = geometry_to_algebra
  DERIVE = target_relation
  INTERPRET = result_geometrically
  RETURN = proof_status
 }
}

CONSTRAINTS {
 C1 = "Coordinate system and metric must be known when formulas depend on them."
 C2 = "Equivalent algebraic equations may represent the same locus."
 C3 = "Algebraic solutions must satisfy all geometric restrictions."
 C4 = "Degenerate conics must be distinguished from nondegenerate conics."
 C5 = "Coordinate changes may alter coefficients without changing intrinsic geometry."
 C6 = "Parametric representations may fail to be one-to-one."
 C7 = "Implicit equations may contain multiple geometric components."
 C8 = "Algebraic simplification must not introduce or remove geometric solution branches incorrectly."
}

SEMANTIC_EQUIVALENCE {
 "AnalyticGeometry" <=> "GeometryViaAlgebraicRepresentation"
 "ImplicitEquation" <=> "CoordinateRelationDefiningLocus"
 "ParametricEquation" <=> "ParameterBasedGeometricRepresentation"
 "IntersectionPoint" <=> "CommonSolutionOfDefiningEquations"
 "ConicSection" <=> "SecondDegreePlaneCurve"
}

VALIDATION {
 V1 = "Defining equations contain all intended geometric points."
 V2 = "Intersection points satisfy all relevant equations."
 V3 = "Conic classification accounts for degenerate cases."
 V4 = "Coordinate transformations preserve intended geometry."
 V5 = "Parametric and implicit forms are cross-verified where both are available."
 V6 = "Locus derivations prove both necessity and sufficiency when required."
 V7 = "Geometric conclusions follow from valid algebraic reasoning."
 V8 = "Extraneous algebraic solutions are rejected when geometrically invalid."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ANALYTIC_GEOMETRY_DEFINITION = 0.99
 EQUATION_GEOMETRY_CORRESPONDENCE = 0.99
 INTERSECTION_REASONING = 0.99
 CONIC_CLASSIFICATION = 0.98
 PARAMETRIC_REASONING = 0.98
 LOCUS_REASONING = 0.99
 COORDINATE_TRANSFORMATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when coordinate system, equation domain, degeneracy conditions, parameter ranges or metric assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AnalyticGeometryReasoning,
  EquationSolving,
  CurveClassification,
  SurfaceReasoning,
  LocusReasoning,
  CoordinateTransformationReasoning,
  GeometricProofReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "analytic geometry" -> AnalyticGeometry
  "equation of curve" -> AlgebraicCurve
  "implicit equation" -> ImplicitEquation
  "parametric equation" -> ParametricEquation
  "conic equation" -> ConicSection
  "analytic proof" -> CoordinateProof
 }

 INTERPRETATION_ORDER = [
  detect_analytic_geometry_context,
  identify_coordinate_system,
  identify_geometric_object,
  identify_equation_or_parametric_representation,
  classify_algebraic_structure,
  solve_or_transform_equations,
  interpret algebraic result geometrically,
  verify constraints and degeneracy,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("analytic geometry")
  THEN activate(AnalyticGeometryOntology)

 E2:
  IF curve_equation_present
  THEN classify_curve_type_when_possible

 E3:
  IF two_or_more_equations_present
  AND intersection_requested
  THEN solve_simultaneously

 E4:
  IF conic_equation_present
  THEN analyse_quadratic_invariants

 E5:
  IF parametric_form_present
  AND implicit_form_requested
  THEN eliminate_parameters

 E6:
  IF implicit_form_present
  AND parametric_form_requested
  THEN seek_valid_parameterisation

 E7:
  IF locus_requested
  THEN derive_equation_and_verify_both_directions

 E8:
  IF coordinate_proof_requested
  THEN translate_geometry_to_algebra
  AND verify_target_relation
}

RECURSION {
 ENTRY = AnalyticGeometry

 EXPAND(AnalyticGeometry) -> [
  CoordinateSystem,
  AlgebraicCurve,
  AlgebraicSurface,
  LineEquation,
  CircleEquation,
  ConicSection,
  ParametricEquation,
  ImplicitEquation,
  Intersection,
  Locus,
  Transformation
 ]

 EXPAND(ConicSection) -> [
  Circle,
  Ellipse,
  Parabola,
  Hyperbola,
  QuadraticEquation,
  Discriminant,
  RotationOfAxes
 ]

 EXPAND(AlgebraicCurve) -> [
  Polynomial,
  Root,
  Intersection,
  Tangency,
  SingularPoint
 ]

 EXPAND(CoordinateTransformation) -> [
  Translation,
  Rotation,
  Scaling,
  AffineTransformation,
  ChangeOfBasis
 ]

 STOP_CONDITION =
  "Stop when the requested equation, classification, intersection, locus, transformation or proof is resolved and geometrically verified."
}

ROUTING {
 PARENT_ONTOLOGY = "Coordinate / Analytic Geometry"

 IMMEDIATE_CONTEXTS = [
  "Coordinate Geometry Ontology",
  "Geometry Ontology",
  "Synthetic Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Algebra Ontologies",
  "Polynomial Ontology",
  "Linear Equation Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology",
  "Vector Space Ontology"
 ]

 ROUTE_RULES = [
  "Analytic Geometry -> Coordinate Geometry for coordinate-formula reasoning",
  "Analytic Geometry -> Algebra for symbolic equation manipulation",
  "Analytic Geometry -> Polynomial for algebraic curves and conics",
  "Analytic Geometry -> Transformation for coordinate changes",
  "Analytic Geometry -> Synthetic Geometry when translating algebraic results into intrinsic geometric statements",
  "Analytic Geometry -> Geometry for core entity semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve geometry-algebra correspondence.",
  "Preserve coordinate-system awareness.",
  "Preserve object-versus-representation distinction.",
  "Preserve conic degeneracy handling.",
  "Preserve parametric-versus-implicit distinctions.",
  "Preserve geometric interpretation of algebraic solutions.",
  "Preserve machine-readable analytic-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Analytic geometry studies geometric objects and relations using coordinates, algebraic equations, vectors and functions."

 CURVE_SUMMARY =
  "A plane curve may be represented implicitly by F(x,y)=0 or parametrically by x=x(t), y=y(t)."

 INTERSECTION_SUMMARY =
  "Geometric intersections are common solutions of defining equations."

 CONIC_SUMMARY =
  "Conic sections are represented by second-degree equations in two variables."

 LOCUS_SUMMARY =
  "A locus is represented by the equation satisfied by exactly the points meeting its defining condition."

 TRANSFORMATION_SUMMARY =
  "Coordinate transformations change algebraic representation while preserving specified geometric structure."

 PROOF_SUMMARY =
  "Analytic proofs convert geometric conditions into algebraic relations and derive the target statement."
}

FLOW {
 INPUT
 -> detect_analytic_geometry_context
 -> identify_coordinate_system
 -> identify_geometric_object
 -> identify_implicit_parametric_or_vector_representation
 -> classify_equation_structure
 -> perform_algebraic_reasoning
 -> solve_intersections_or_transformations_when_required
 -> interpret_algebraic_result_geometrically
 -> verify_domain_parameter_and_degeneracy_constraints
 -> route_to_related_geometry_or_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Analytic Geometry represents geometric objects through equations, vectors and coordinate functions, allowing geometric problems to be solved algebraically. It supports reasoning about lines, curves, conics, surfaces, intersections, loci and transformations while preserving the distinction between a geometric object and any particular algebraic representation of it."
}

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

}

END_FLOW