Parametric Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of parametric geometry within the Kireth Research Semantic Library. Parametric geometry represents geometric objects using one or more parameters that generate coordinates, vectors or points along curves, surfaces and higher-dimensional geometric structures. This ontology supports curve tracing, surface representation, tangent and velocity reasoning, coordinate conversion, intersections, transformations and links between parametric, implicit and explicit forms.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ParametricGeometry
 SYMBOLIC_FORMS = ["x=x(t)", "y=y(t)", "z=z(t)", "r(t)", "S(u,v)"]
 ENTITY_CLASS = ParametricGeometricFramework
 PRIMARY_CONTEXT = AnalyticGeometry
 SECONDARY_CONTEXTS = [CoordinateGeometry, VectorGeometry]
 DEFINITION = "Parametric geometry represents geometric objects by expressing their coordinates or position vectors as functions of one or more parameters."
 SEMANTIC_ROLE = "Provides an ordered generative representation of curves, surfaces and geometric motion through parameter-dependent coordinates."
}

SCOPE {
 INCLUDE = [
  ParametricGeometry,
  Parameter,
  ParametricCurve,
  ParametricLine,
  ParametricCircle,
  ParametricConic,
  ParametricSurface,
  VectorValuedFunction,
  ParameterDomain,
  Orientation,
  CurveTracing,
  TangentVector,
  VelocityVector,
  ArcLength,
  Intersection,
  Implicitisation,
  Reparameterisation,
  CoordinateTransformation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureImplicitGeometryWithoutParameters,
  PureSyntheticGeometry,
  NumericalAnimationWithoutMathematicalParametricStructure
 ]

 DISAMBIGUATION_RULE =
  "When geometric objects are described by coordinates or vectors depending on one or more parameters, interpret the context as parametric geometry."
}

CORE_CONCEPTS {

 ParametricGeometry {
  definition = "Geometric representation in which points are generated from parameter values."
 }

 Parameter {
  definition = "An independent variable controlling position on a geometric object."
  common_symbols = ["t", "u", "v", "s"]
 }

 ParameterDomain {
  definition = "The set of admissible parameter values."
  role = "Determines which part of the geometric object is generated and how often it may be traced."
 }

 ParametricCurve {
  definition = "A curve represented by coordinate functions of a single parameter."
  planar_form = ["x=x(t)", "y=y(t)"]
  spatial_form = ["x=x(t)", "y=y(t)", "z=z(t)"]
  vector_form = "r(t)"
 }

 ParametricLine {
  definition = "A straight line represented by a point plus a scalar multiple of a direction vector."
  form = "r(t)=a+t*d"
 }

 ParametricCircle {
  standard_form = [
   "x=h+r*cos(t)",
   "y=k+r*sin(t)"
  ]
 }

 ParametricSurface {
  definition = "A surface represented by coordinates or position vector depending on two parameters."
  form = "S(u,v)"
 }

 Orientation {
  definition = "The direction in which increasing parameter values traverse a parametric object."
 }

 Reparameterisation {
  definition = "Replacement of one parameter by another mapping that preserves the underlying geometric image under suitable conditions."
 }

 Implicitisation {
  definition = "Elimination of parameters to obtain an implicit equation for the same geometric locus where possible."
 }
}

RELATIONS {
 ParametricGeometry --is_subdomain_of--> AnalyticGeometry
 ParametricGeometry --uses--> Parameter
 Parameter --ranges_over--> ParameterDomain
 ParametricCurve --is_generated_by--> Parameter
 ParametricSurface --is_generated_by--> ParameterPair
 ParametricCurve --can_be_represented_by--> VectorValuedFunction
 ParametricCurve --can_have--> Orientation
 ParametricCurve --can_have--> TangentVector
 ParametricCurve --can_have--> ArcLength
 ParametricRepresentation --can_be_converted_to--> ImplicitRepresentation
 ParametricRepresentation --can_be_reparameterised_by--> ParameterTransformation
 ParametricGeometry --supports--> CurveTracing
 ParametricGeometry --supports--> IntersectionReasoning
 ParametricGeometry --supports--> TransformationReasoning
}

OPERATORS {
 POINT_ON_CURVE(t,C) :=
  evaluate_coordinate_functions(C,t)

 POINT_ON_SURFACE(u,v,S) :=
  evaluate_surface_parameterisation(S,u,v)

 TANGENT_VECTOR(C,t) :=
  derivative_of_position_vector(C,t)

 SPEED(C,t) :=
  norm(TANGENT_VECTOR(C,t))

 ARC_LENGTH(C,a,b) :=
  integral_from_a_to_b(norm(r_prime(t))) dt

 REPARAMETERISE(C,t=phi(s)) :=
  compose_parameter_functions(C,phi)

 IMPLICITISE(C) :=
  eliminate_parameters_from_coordinate_equations(C)

 INTERSECT_PARAMETRIC(C1,C2) :=
  solve_parameter_values_where_positions_equal(C1,C2)

 TRANSFORM_PARAMETRIC(C,T) :=
  apply_geometric_transformation_to_position_vector(C,T)

 TRACE(C,DOMAIN) :=
  ordered_set_of_points_generated_over_parameter_domain
}

DEFINITIONS {
 D1 = "A parametric curve is generated by one parameter."
 D2 = "A parametric surface is typically generated by two parameters."
 D3 = "The parameter is not itself necessarily a geometric coordinate."
 D4 = "Different parameterisations can describe the same geometric locus."
 D5 = "Parameter domain affects coverage, multiplicity and orientation."
 D6 = "The derivative of a regular parametric curve gives a tangent direction."
 D7 = "Arc length depends on the norm of the derivative of the position vector."
 D8 = "Implicitisation removes parameters to produce an equation of the locus where possible."
 D9 = "Reparameterisation may change speed and orientation without changing the geometric image."
 D10 = "A parameterisation may fail to be injective and may trace points multiple times."
}

AXIOMS {
 A1_PARAMETER_GENERATION:
  Every point in a parametric representation is generated by admissible parameter values

 A2_DOMAIN_DEPENDENCE:
  The represented geometric image depends on the parameter domain

 A3_REPRESENTATION_NONUNIQUENESS:
  Different parameterisations may represent the same geometric object

 A4_TANGENT_REGULARITY:
  A nonzero derivative determines a tangent direction for a regular parametric curve

 A5_REPARAMETERISATION:
  Suitable parameter transformations preserve the geometric image

 A6_TRANSFORMATION_VALIDITY:
  Applying a valid geometric transformation to a parameterisation transforms the underlying geometric object consistently

 A7_GEOMETRIC_INTERPRETATION:
  Parameter values must be interpreted through the generated coordinates or position vectors
}

REASONING {

 R1_CURVE_EVALUATION {
  INPUT = [curve C, parameter t0]
  SUBSTITUTE = t=t0
  COMPUTE = coordinates
  RETURN = point_on_curve
 }

 R2_DOMAIN_ANALYSIS {
  INPUT = [parameterisation, domain]
  DETERMINE = coverage
  DETERMINE = endpoints_or_periodicity
  DETECT = repeated_tracing
  RETURN = represented_geometric_image
 }

 R3_TANGENT_REASONING {
  INPUT = [curve r(t), t0]
  DIFFERENTIATE = r'(t)
  EVALUATE = r'(t0)
  IF r'(t0)!=0
   RETURN = tangent_direction
  ELSE
   RETURN = singular_or_nonregular_case
 }

 R4_ARC_LENGTH {
  INPUT = [curve r(t), interval [a,b]]
  COMPUTE = norm(r'(t))
  INTEGRATE = from a to b
  RETURN = arc_length
 }

 R5_REPARAMETERISATION {
  INPUT = [curve C(t), t=phi(s)]
  SUBSTITUTE = phi(s)
  CHECK = parameter_mapping_domain
  COMPARE = geometric_image
  RETURN = reparameterised_curve
 }

 R6_IMPLICITISATION {
  INPUT = coordinate_functions
  ELIMINATE = parameter
  RETURN = implicit_equation_when_possible
 }

 R7_INTERSECTION {
  INPUT = [C1(t),C2(s)]
  SOLVE = C1(t)=C2(s)
  VERIFY = parameter_domains
  RETURN = intersection_points
 }

 R8_PARAMETRIC_TRANSFORMATION {
  INPUT = [C,T]
  APPLY = T to each generated point or vector form
  RETURN = transformed_parameterisation
 }

 R9_SURFACE_REASONING {
  INPUT = S(u,v)
  IDENTIFY = parameter_domain
  COMPUTE = tangent_vectors S_u and S_v when differentiable
  ANALYSE = local_surface_structure
  RETURN = surface_information
 }
}

CONSTRAINTS {
 C1 = "Parameter domains must be explicit when coverage matters."
 C2 = "Different parameterisations may trace the same point multiple times."
 C3 = "A zero derivative can invalidate ordinary tangent-vector reasoning."
 C4 = "Arc-length formulas require suitable differentiability."
 C5 = "Implicitisation may introduce extraneous algebraic components and must be verified."
 C6 = "Reparameterisation must preserve admissible domain mapping."
 C7 = "Orientation can reverse under decreasing reparameterisations."
 C8 = "The parameter must not be confused automatically with arc length."
 C9 = "Do not confuse a parametric representation with the geometric object itself."
}

SEMANTIC_EQUIVALENCE {
 "ParametricGeometry" <=> "GeometryUsingParameterDependentCoordinates"
 "ParametricCurve" <=> "ImageOfOneParameterMap"
 "ParametricSurface" <=> "ImageOfTwoParameterMap"
 "Reparameterisation" <=> "ChangeOfParameterPreservingGeometricImage"
 "Implicitisation" <=> "ParameterEliminationToLocusEquation"
}

VALIDATION {
 V1 = "Evaluated points satisfy the parameterisation."
 V2 = "Parameter domains match the intended geometric coverage."
 V3 = "Tangent vectors come from derivatives where regularity holds."
 V4 = "Reparameterised curves represent the same image when the parameter map is appropriate."
 V5 = "Implicit equations are checked against the original parameterisation."
 V6 = "Intersections satisfy both parameterisations."
 V7 = "Arc length is nonnegative."
 V8 = "Transformations act consistently on all generated points."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PARAMETRIC_GEOMETRY_DEFINITION = 0.99
 PARAMETER_DOMAIN_REASONING = 0.99
 TANGENT_REASONING = 0.99
 REPARAMETERISATION_REASONING = 0.98
 IMPLICITISATION_REASONING = 0.98
 INTERSECTION_REASONING = 0.99
 SURFACE_REASONING = 0.98

 POLICY =
  "Reduce confidence when parameter domains, differentiability, orientation, multiplicity or reparameterisation conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ParametricGeometryReasoning,
  CurveReasoning,
  SurfaceReasoning,
  TangentReasoning,
  ArcLengthReasoning,
  ReparameterisationReasoning,
  Implicitisation,
  IntersectionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "parametric geometry" -> ParametricGeometry
  "parametric curve" -> ParametricCurve
  "parametric line" -> ParametricLine
  "parametric circle" -> ParametricCircle
  "parametric surface" -> ParametricSurface
  "reparameterise" -> Reparameterisation
  "eliminate parameter" -> Implicitisation
 }

 INTERPRETATION_ORDER = [
  detect_parametric_geometry_context,
  identify_parameter_or_parameters,
  identify_parameter_domain,
  identify_coordinate_or_vector functions,
  determine represented geometric image,
  apply tangent_arc_length_intersection_or_transformation reasoning,
  reparameterise or implicitise when requested,
  verify domain and regularity constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("parametric geometry")
  OR query_mentions("parametric curve")
  THEN activate(ParametricGeometryOntology)

 E2:
  IF parameter_value_given
  THEN evaluate_parameterisation

 E3:
  IF tangent_requested
  THEN differentiate_position_vector
  AND check_nonzero_derivative

 E4:
  IF arc_length_requested
  THEN integrate_norm_of_derivative

 E5:
  IF implicit_equation_requested
  THEN eliminate_parameters
  AND verify resulting locus

 E6:
  IF reparameterisation_requested
  THEN substitute_parameter_mapping
  AND verify geometric image

 E7:
  IF intersection_requested
  THEN equate_position_vectors
  AND solve_parameter_values

 E8:
  IF surface_context
  THEN activate_two_parameter_surface_reasoning
}

RECURSION {
 ENTRY = ParametricGeometry

 EXPAND(ParametricGeometry) -> [
  Parameter,
  ParameterDomain,
  ParametricCurve,
  ParametricLine,
  ParametricCircle,
  ParametricSurface,
  VectorValuedFunction,
  Reparameterisation,
  Implicitisation,
  Intersection
 ]

 EXPAND(ParametricCurve) -> [
  PositionVector,
  TangentVector,
  Speed,
  ArcLength,
  Orientation,
  Curvature
 ]

 EXPAND(ParametricSurface) -> [
  ParameterPair,
  TangentPlane,
  SurfaceNormal,
  CoordinatePatch
 ]

 EXPAND(Reparameterisation) -> [
  ParameterMapping,
  Orientation,
  SpeedChange,
  DomainMapping
 ]

 STOP_CONDITION =
  "Stop when the parameterisation, geometric image, tangent, length, intersection, transformation or representation conversion is resolved and parameter constraints are explicit."
}

ROUTING {
 PARENT_ONTOLOGY = "Coordinate / Analytic Geometry"

 IMMEDIATE_CONTEXTS = [
  "Analytic Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Polar Coordinate Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Analytic Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Polar Coordinate Ontology",
  "Geometry Ontology",
  "Transformation Ontology",
  "Vector Space Ontology",
  "Calculus & Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Parametric Geometry -> Analytic Geometry for equation and curve reasoning",
  "Parametric Geometry -> Coordinate Geometry for coordinate evaluation",
  "Parametric Geometry -> Cartesian Coordinate for standard tuple representation",
  "Parametric Geometry -> Polar Coordinate for radial-angle parameterisations",
  "Parametric Geometry -> Calculus for derivatives, tangent vectors and arc length",
  "Parametric Geometry -> Transformation for transformed parameterisations",
  "Parametric Geometry -> Geometry for underlying spatial semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible parameterisation types, operators or relations"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve parameter-dependent representation semantics.",
  "Preserve explicit parameter-domain reasoning.",
  "Preserve representation-versus-geometric-image distinction.",
  "Preserve regularity conditions for tangent reasoning.",
  "Preserve reparameterisation invariance of geometric image.",
  "Preserve verification after implicitisation.",
  "Preserve machine-readable parametric-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Parametric geometry represents geometric objects by expressing coordinates or position vectors as functions of one or more parameters."

 CURVE_SUMMARY =
  "C(t)=(x(t),y(t)) or r(t)"

 SURFACE_SUMMARY =
  "S(u,v)=(x(u,v),y(u,v),z(u,v))"

 TANGENT_SUMMARY =
  "For a regular curve, r'(t) gives a tangent direction."

 ARC_LENGTH_SUMMARY =
  "L=integral ||r'(t)|| dt over the parameter interval."

 REPARAMETERISATION_SUMMARY =
  "A suitable change of parameter can alter speed or orientation without changing the geometric image."

 IMPLICITISATION_SUMMARY =
  "Parameters may sometimes be eliminated to obtain an implicit equation of the same locus."
}

FLOW {
 INPUT
 -> detect_parametric_geometry_context
 -> identify_parameter_variables
 -> identify_parameter_domains
 -> parse_coordinate_or_vector_functions
 -> determine_geometric_image
 -> identify_requested evaluation tangent length intersection transformation or conversion
 -> apply parametric operator
 -> verify domain regularity orientation and multiplicity constraints
 -> route_to analytic_geometry coordinate_geometry calculus_or_transformation
 -> recursively_expand_required_dependencies
 -> verify result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Parametric Geometry Ontology represents curves, surfaces and other geometric objects through parameter-dependent coordinates or vectors. It supports ordered traversal, tangent and arc-length reasoning, intersections, transformations, reparameterisation and conversion to implicit forms while preserving the distinction between a parameterisation and the geometric image it generates."
}

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

}

END_FLOW