Coordinate & Analytic Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of coordinate and analytic geometry within the Kireth Research Semantic Library. Coordinate / analytic geometry represents geometric entities using coordinates, algebraic equations, vectors and transformations, allowing geometric problems to be translated into algebraic form and solved systematically. 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 = "Coordinate / Analytic Geometry"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Coordinate Geometry / Analytic Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/coordinate-analytic-geometry.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = CoordinateAnalyticGeometry
 SYMBOLIC_FORMS = ["(x,y)", "(x,y,z)", "Ax+By+C=0", "y=mx+c", "F(x,y)=0"]
 ENTITY_CLASS = AlgebraicGeometricFramework
 PRIMARY_CONTEXT = Geometry
 SECONDARY_CONTEXT = Algebra
 DEFINITION = "Coordinate or analytic geometry studies geometric objects by representing them with coordinates, equations, vectors and algebraic relations."
 SEMANTIC_ROLE = "Provides a bridge between geometric structure and algebraic computation."
}

SCOPE {
 INCLUDE = [
  CoordinateGeometry,
  AnalyticGeometry,
  CartesianCoordinates,
  PolarCoordinates,
  ThreeDimensionalCoordinates,
  PointCoordinates,
  DistanceFormula,
  MidpointFormula,
  Gradient,
  LineEquation,
  CircleEquation,
  ConicSection,
  VectorRepresentation,
  Intersection,
  Transformation,
  Locus,
  CoordinateProof,
  CoordinateChange
 ]

 EXCLUDE_AS_PRIMARY = [
  PureSyntheticGeometryWithoutCoordinates,
  PureTopologyWithoutCoordinateStructure,
  NumericalPlottingWithoutGeometricSemantics
 ]

 DISAMBIGUATION_RULE =
  "When geometry is described through coordinates, equations, slopes, vectors, intersections or algebraic loci, interpret it as coordinate or analytic geometry."
}

CORE_CONCEPTS {

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

 CartesianCoordinates {
  definition = "Coordinates based on perpendicular axes."
  forms = ["(x,y)", "(x,y,z)"]
 }

 PolarCoordinates {
  definition = "Planar coordinates described by radius and angle."
  form = "(r,theta)"
 }

 Point {
  coordinate_form = "P=(x,y)"
  role = "Basic located geometric entity."
 }

 Distance {
  planar_formula = "d=sqrt((x2-x1)^2+(y2-y1)^2)"
  spatial_formula = "d=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)"
 }

 Midpoint {
  planar_formula = "M=((x1+x2)/2,(y1+y2)/2)"
 }

 Gradient {
  definition = "Rate of vertical change relative to horizontal change for a nonvertical line."
  formula = "m=(y2-y1)/(x2-x1)"
 }

 LineEquation {
  common_forms = [
   "y=mx+c",
   "y-y1=m(x-x1)",
   "Ax+By+C=0"
  ]
 }

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

 Locus {
  definition = "A set of points satisfying a coordinate equation or geometric condition."
 }

 ConicSection {
  types = [Circle, Ellipse, Parabola, Hyperbola]
  role = "Curves represented by second-degree equations."
 }
}

RELATIONS {
 CoordinateAnalyticGeometry --is_subdomain_of--> Geometry
 CoordinateAnalyticGeometry --uses--> Algebra
 CoordinateAnalyticGeometry --uses--> CoordinateSystem
 GeometricEntity --can_be_represented_by--> Coordinates
 Coordinates --can_satisfy--> Equation
 Equation --can_define--> Locus
 Line --can_be_represented_by--> LineEquation
 Circle --can_be_represented_by--> CircleEquation
 ConicSection --can_be_represented_by--> QuadraticEquation
 CoordinateGeometry --supports--> IntersectionReasoning
 CoordinateGeometry --supports--> DistanceReasoning
 CoordinateGeometry --supports--> TransformationReasoning
 CoordinateGeometry --supports--> Proof
}

OPERATORS {
 DISTANCE(P1,P2) :=
  euclidean_coordinate_distance(P1,P2)

 MIDPOINT(P1,P2) :=
  coordinate_average(P1,P2)

 GRADIENT(P1,P2) :=
  (y2-y1)/(x2-x1) WHEN x2!=x1

 LINE_THROUGH(P1,P2) :=
  derive_line_equation(P1,P2)

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

 LOCUS(condition) :=
  derive_coordinate_equation(condition)

 TRANSFORM_POINT(P,T) :=
  apply_coordinate_transformation(P,T)

 CHANGE_COORDINATES(P,S1,S2) :=
  convert_representation_between_coordinate_systems(P,S1,S2)

 VERIFY_POINT_ON_LOCUS(P,E) :=
  substitute_coordinates(P,E) AND test_truth
}

DEFINITIONS {
 D1 = "Coordinate geometry represents geometric objects numerically."
 D2 = "Analytic geometry translates geometric relations into algebraic equations."
 D3 = "A coordinate equation may define a geometric locus."
 D4 = "Distance in Cartesian coordinates follows the Pythagorean metric."
 D5 = "Gradient characterises line direction where defined."
 D6 = "Intersections are found by solving simultaneous equations."
 D7 = "Coordinate transformations may change representation without changing underlying geometry."
 D8 = "Conic sections can be classified through quadratic equations."
 D9 = "Coordinate proofs establish geometric properties algebraically."
}

AXIOMS {
 A1_COORDINATE_REPRESENTATION:
  Geometric points may be represented by coordinate tuples within a chosen system

 A2_REPRESENTATION_INDEPENDENCE:
  Coordinate changes do not necessarily change the underlying geometric object

 A3_ALGEBRA_GEOMETRY_BRIDGE:
  Valid coordinate equations encode geometric relations

 A4_DISTANCE_METRIC:
  Standard Cartesian distance obeys Euclidean metric structure

 A5_INTERSECTION:
  A point belongs to the intersection of geometric loci iff its coordinates satisfy all defining equations

 A6_TRANSFORMATION_VALIDITY:
  Coordinate transformation formulas must preserve the declared geometric relation or invariant

 A7_PROOF_VALIDITY:
  Coordinate derivations must preserve algebraic correctness and geometric interpretation
}

REASONING {

 R1_POINT_RELATION {
  INPUT = point_and_equation
  SUBSTITUTE = point_coordinates
  IF equation_true
   RETURN PointOnLocus
  ELSE
   RETURN PointNotOnLocus
 }

 R2_LINE_FROM_POINTS {
  INPUT = [P1,P2]
  IF x1!=x2
   COMPUTE = gradient
   DERIVE = point_slope_equation
  ELSE
   RETURN "x=x1"
 }

 R3_INTERSECTION {
  INPUT = [equation1,equation2]
  SOLVE = simultaneous_equations
  MAP = solutions_to_points
  RETURN = intersection_set
 }

 R4_DISTANCE {
  INPUT = [P1,P2]
  APPLY = coordinate_distance_formula
  RETURN = metric_distance
 }

 R5_LOCUS {
  INPUT = geometric_condition
  EXPRESS = condition_in_coordinates
  SIMPLIFY = algebraic_equation
  RETURN = locus_equation
 }

 R6_CONIC_CLASSIFICATION {
  INPUT = quadratic_equation
  IDENTIFY = coefficients
  ANALYSE = discriminant_or_canonical_form
  RETURN = conic_type
 }

 R7_COORDINATE_PROOF {
  INPUT = geometric_proposition
  CHOOSE = convenient_coordinate_system
  ASSIGN = coordinates
  DERIVE = algebraic_relations
  VERIFY = target_property
  RETURN = proof_status
 }

 R8_COORDINATE_TRANSFORMATION {
  INPUT = [object,T]
  APPLY = transformation_formula
  TRACK = invariant_properties
  RETURN = transformed_coordinates_and_geometry
 }
}

CONSTRAINTS {
 C1 = "Coordinate systems must be explicitly identified when formulas depend on them."
 C2 = "Gradient is undefined for vertical lines in slope form."
 C3 = "Algebraic solutions must be interpreted geometrically."
 C4 = "Extraneous algebraic solutions must be rejected when they violate geometric constraints."
 C5 = "Coordinate changes must preserve the underlying geometric object when used representationally."
 C6 = "Distance formulas depend on the active metric."
 C7 = "Equation form may vary while defining the same locus."
 C8 = "Synthetic and coordinate proofs may establish the same theorem through different representations."
}

SEMANTIC_EQUIVALENCE {
 "CoordinateGeometry" <=> "GeometryUsingCoordinates"
 "AnalyticGeometry" <=> "GeometryUsingAlgebraicEquations"
 "PointOnLocus" <=> "CoordinatesSatisfyEquation"
 "IntersectionPoint" <=> "SimultaneousEquationSolution"
 "LineEquation" <=> "CoordinateRepresentationOfLine"
}

VALIDATION {
 V1 = "Distance formula matches Euclidean metric."
 V2 = "Midpoint coordinates are componentwise averages."
 V3 = "Line equations contain their defining points."
 V4 = "Intersection solutions satisfy all defining equations."
 V5 = "Coordinate transformations preserve declared invariants."
 V6 = "Conic classification matches equation structure."
 V7 = "Coordinate proof conclusions follow from valid algebra."
 V8 = "Equivalent equations may define the same geometric locus."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COORDINATE_GEOMETRY_DEFINITION = 0.99
 DISTANCE_RELATION = 0.99
 LINE_EQUATION_RELATION = 0.99
 INTERSECTION_REASONING = 0.99
 LOCUS_REASONING = 0.99
 CONIC_CLASSIFICATION = 0.98
 COORDINATE_PROOF = 0.99

 POLICY =
  "Reduce confidence when the coordinate system, metric, equation domain or geometric constraints are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  CoordinateReasoning,
  AnalyticGeometryReasoning,
  EquationSolving,
  LocusReasoning,
  ConicReasoning,
  TransformationReasoning,
  GeometricProofReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "coordinate geometry" -> CoordinateGeometry
  "analytic geometry" -> AnalyticGeometry
  "Cartesian geometry" -> CartesianCoordinates
  "coordinate proof" -> CoordinateProof
  "equation of line" -> LineEquation
  "equation of circle" -> CircleEquation
  "locus equation" -> Locus
 }

 INTERPRETATION_ORDER = [
  detect_coordinate_geometry_context,
  identify_coordinate_system,
  identify_geometric_entities,
  identify_equations_and_relations,
  translate_geometry_to_algebra,
  solve_or_transform_algebraically,
  translate_result_back_to_geometry,
  verify_geometric_constraints,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("coordinate geometry")
  OR query_mentions("analytic geometry")
  THEN activate(CoordinateAnalyticGeometryOntology)

 E2:
  IF two_points_and_line_requested
  THEN derive_line_equation

 E3:
  IF intersection_requested
  THEN solve_simultaneous_equations

 E4:
  IF locus_requested
  THEN translate_condition_to_equation

 E5:
  IF conic_equation_present
  THEN classify_conic

 E6:
  IF coordinate_proof_requested
  THEN select_convenient_coordinate_frame
  AND derive_target_property_algebraically

 E7:
  IF transformation_context
  THEN apply_coordinate_transformation_and_track_invariants

 E8:
  IF metric_not_standard
  THEN require_explicit_metric
}

RECURSION {
 ENTRY = CoordinateAnalyticGeometry

 EXPAND(CoordinateAnalyticGeometry) -> [
  CoordinateSystem,
  CartesianCoordinates,
  PolarCoordinates,
  Point,
  Line,
  Circle,
  ConicSection,
  Distance,
  Gradient,
  Locus,
  Transformation,
  CoordinateProof
 ]

 EXPAND(Line) -> [
  Gradient,
  Intercept,
  PointSlopeForm,
  GeneralForm,
  Parallelism,
  Perpendicularity
 ]

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

 EXPAND(CoordinateProof) -> [
  CoordinateAssignment,
  AlgebraicDerivation,
  Distance,
  Gradient,
  Midpoint,
  Intersection
 ]

 STOP_CONDITION =
  "Stop when the requested geometric relation, equation, locus, transformation or proof is resolved and geometrically validated."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

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

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

 ROUTE_RULES = [
  "Coordinate / Analytic Geometry -> Geometry for core spatial semantics",
  "Coordinate / Analytic Geometry -> Algebra for equation manipulation",
  "Coordinate / Analytic Geometry -> Linear Equation for line systems",
  "Coordinate / Analytic Geometry -> Polynomial for conic equations",
  "Coordinate / Analytic Geometry -> Transformation for coordinate mappings",
  "Coordinate / Analytic Geometry -> Synthetic Geometry when converting algebraic results to intrinsic geometric interpretation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve geometry-algebra translation semantics.",
  "Preserve coordinate-system awareness.",
  "Preserve representation-versus-object distinction.",
  "Preserve metric assumptions.",
  "Preserve interpretation of algebraic solutions as geometric objects.",
  "Preserve transformation-specific invariants.",
  "Preserve machine-readable coordinate-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Coordinate or analytic geometry studies geometric objects using coordinates, vectors and algebraic equations."

 DISTANCE_SUMMARY =
  "d=sqrt((x2-x1)^2+(y2-y1)^2) in the Euclidean plane."

 LINE_SUMMARY =
  "Lines may be represented by y=mx+c, point-slope form, or Ax+By+C=0."

 INTERSECTION_SUMMARY =
  "Geometric intersections correspond to simultaneous solutions of defining equations."

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

 TRANSFORMATION_SUMMARY =
  "Coordinate transformations alter representation or position according to explicit algebraic rules."

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

FLOW {
 INPUT
 -> detect_coordinate_or_analytic_geometry_context
 -> identify_coordinate_system
 -> identify_geometric_entities
 -> identify_equations_coordinates_and_vectors
 -> translate_geometric_conditions_to_algebra
 -> perform_algebraic_computation
 -> interpret_solutions_geometrically
 -> verify_metric_domain_and_geometric_constraints
 -> route_to_specialised_geometry_or_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Coordinate and analytic geometry connect geometry with algebra by assigning coordinates to geometric objects and representing relations through equations. This framework enables exact reasoning about distance, lines, circles, conics, intersections, loci and transformations, while preserving the distinction between a geometric object and its chosen coordinate representation."
}

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

}

END_FLOW