Coordinate Measurement Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of coordinate measurement within the Kireth Research Semantic Library. Coordinate measurement derives geometric quantities from numerical coordinates in Cartesian, polar, cylindrical, spherical and other coordinate systems. This ontology supports coordinate differences, distances, midpoints, slopes, direction vectors, lengths, angles, areas, volumes, coordinate transformations, scale, units, dimensional consistency, exact and approximate values and relationships with coordinate geometry, vector geometry and geometric measurement.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = CoordinateMeasurement
 SYMBOLIC_FORMS = [
  "(x,y)",
  "(x,y,z)",
  "(r,theta)",
  "Delta x",
  "Delta y",
  "Delta z"
 ]
 ENTITY_CLASS = CoordinateDerivedGeometricMeasure
 PRIMARY_CONTEXT = GeometricMeasurement
 DEFINITION = "Coordinate measurement is the computation of geometric quantities from coordinate representations of points, vectors, curves, regions or solids."
 SEMANTIC_ROLE = "Provides the quantitative bridge between coordinate systems and geometric measurements such as distance, midpoint, slope, angle, length, area and volume."
}

SCOPE {
 INCLUDE = [
  CoordinateMeasurement,
  CoordinateSystem,
  CartesianCoordinates,
  PolarCoordinates,
  CylindricalCoordinates,
  SphericalCoordinates,
  CoordinateDifference,
  EuclideanDistance,
  ManhattanDistance,
  Midpoint,
  SectionPoint,
  Slope,
  DirectionVector,
  VectorMagnitude,
  DotProductAngle,
  CoordinatePerimeter,
  CoordinateArea,
  ShoelaceArea,
  DeterminantArea,
  CoordinateVolume,
  Jacobian,
  CoordinateTransformation,
  ScaleFactor,
  Unit,
  UnitConversion,
  ExactMeasurement,
  ApproximateMeasurement,
  MeasurementPrecision,
  ErrorBound
 ]

 EXCLUDE_AS_PRIMARY = [
  PureCoordinateRepresentationWithoutMeasurementGoal,
  GeodeticCoordinateSystemsUnlessExplicit,
  NumericalDataWithoutGeometricSemantics,
  StatisticalCoordinateAnalysis
 ]

 DISAMBIGUATION_RULE =
  "When geometric quantities are computed directly from point or vector coordinates, interpret the context as coordinate measurement."
}

CORE_CONCEPTS {

 CoordinateMeasurement {
  definition = "A geometric measurement derived from coordinates."
 }

 CartesianCoordinates {
  definition = "Coordinates expressed relative to orthogonal axes, such as (x,y) or (x,y,z)."
 }

 PolarCoordinates {
  definition = "Planar coordinates represented by radial distance and angle."
 }

 CylindricalCoordinates {
  definition = "Three-dimensional coordinates represented by radial distance, azimuth and height."
 }

 SphericalCoordinates {
  definition = "Three-dimensional coordinates represented by radius and two angles."
 }

 CoordinateDifference {
  definition = "The componentwise displacement between coordinates."
 }

 Midpoint {
  definition = "The point halfway between two coordinate points."
 }

 Slope {
  definition = "The ratio of vertical to horizontal coordinate change for a nonvertical planar line."
 }

 DirectionVector {
  definition = "A vector representing the direction from one coordinate point to another."
 }

 CoordinateArea {
  definition = "Area computed from point coordinates, determinants or coordinate integration."
 }

 CoordinateVolume {
  definition = "Volume computed from coordinate representations, determinants or multiple integrals."
 }
}

RELATIONS {
 CoordinateMeasurement --is_special_case_of--> GeometricMeasurement
 CoordinateMeasurement --uses--> CoordinateSystem
 CartesianCoordinates --support--> EuclideanDistance
 CartesianCoordinates --support--> Midpoint
 CartesianCoordinates --support--> Slope
 CoordinateDifference --defines--> DirectionVector
 DirectionVector --supports--> VectorMagnitude
 DirectionVector --supports--> AngleMeasurement
 DotProduct --supports--> DotProductAngle
 PolygonCoordinates --support--> ShoelaceArea
 CoordinateTransformation --maps_between--> CoordinateSystems
 Jacobian --adjusts--> DifferentialMeasure
 ScaleFactor --transforms--> CoordinateMeasurement
 UnitConversion --preserves--> MeasurementMagnitude
}

OPERATORS {
 COORDINATE_DIFFERENCE(P,Q) :=
  Q-P

 MIDPOINT_2D((x1,y1),(x2,y2)) :=
  ((x1+x2)/2,(y1+y2)/2)

 MIDPOINT_3D((x1,y1,z1),(x2,y2,z2)) :=
  ((x1+x2)/2,(y1+y2)/2,(z1+z2)/2)

 EUCLIDEAN_DISTANCE_2D(P,Q) :=
  sqrt((x_Q-x_P)^2+(y_Q-y_P)^2)

 EUCLIDEAN_DISTANCE_3D(P,Q) :=
  sqrt((x_Q-x_P)^2+(y_Q-y_P)^2+(z_Q-z_P)^2)

 MANHATTAN_DISTANCE(P,Q) :=
  sum_i abs(Q_i-P_i)

 SLOPE(P,Q) :=
  (y_Q-y_P)/(x_Q-x_P)
  WHEN x_Q!=x_P

 DIRECTION_VECTOR(P,Q) :=
  Q-P

 VECTOR_MAGNITUDE(v) :=
  sqrt(sum_i v_i^2)

 VECTOR_ANGLE(u,v) :=
  arccos((u dot v)/(norm(u)*norm(v)))
  WHEN u!=0 AND v!=0

 TRIANGLE_AREA_COORDINATES(A,B,C) :=
  0.5*abs(det([[x_B-x_A,y_B-y_A],[x_C-x_A,y_C-y_A]]))

 POLYGON_AREA_SHOELACE(P_1,...,P_n) :=
  0.5*abs(sum_i(x_i*y_{i+1}-y_i*x_{i+1}))
  WITH P_{n+1}=P_1

 TETRAHEDRON_VOLUME(A,B,C,D) :=
  abs(det([B-A,C-A,D-A]))/6

 CARTESIAN_TO_POLAR(x,y) :=
  (sqrt(x^2+y^2),atan2(y,x))

 POLAR_TO_CARTESIAN(r,theta) :=
  (r*cos(theta),r*sin(theta))

 CARTESIAN_TO_CYLINDRICAL(x,y,z) :=
  (sqrt(x^2+y^2),atan2(y,x),z)

 CYLINDRICAL_TO_CARTESIAN(r,theta,z) :=
  (r*cos(theta),r*sin(theta),z)

 SPHERICAL_TO_CARTESIAN(r,theta,phi) :=
  coordinate_convention_specific_mapping

 JACOBIAN_SCALE(F,u) :=
  abs(det(DF(u)))
  WHEN square_full_dimensional_transformation
}

DEFINITIONS {
 D1 = "Coordinate measurement depends on the selected coordinate system."
 D2 = "Coordinate differences define displacement."
 D3 = "Euclidean distance is invariant under rigid Cartesian transformations."
 D4 = "Slope is undefined for vertical lines."
 D5 = "Midpoint is computed componentwise in affine Cartesian coordinates."
 D6 = "Shoelace area requires ordered planar polygon vertices."
 D7 = "Determinants encode oriented area and volume."
 D8 = "Coordinate transformations may require Jacobian factors for area or volume."
 D9 = "Angular coordinates require declared units and conventions."
 D10 = "Coordinates are representations; geometric measurements are invariant only under transformations preserving the relevant structure."
}

AXIOMS {
 A1_COMPONENT_DIFFERENCE:
  Coordinate displacement is computed componentwise

 A2_EUCLIDEAN_DISTANCE:
  Cartesian Euclidean distance is norm(Q-P)

 A3_MIDPOINT:
  The affine midpoint is (P+Q)/2

 A4_VECTOR_ANGLE:
  For nonzero Euclidean vectors u dot v=||u||||v||cos(theta)

 A5_DETERMINANT_AREA:
  Absolute determinant gives parallelogram area in two dimensions

 A6_DETERMINANT_VOLUME:
  Absolute determinant gives parallelepiped volume in three dimensions

 A7_JACOBIAN:
  Coordinate changes scale differential area or volume by the appropriate Jacobian determinant magnitude

 A8_UNIT_INVARIANCE:
  Correct unit conversion preserves the represented geometric magnitude
}

REASONING {

 R1_COORDINATE_SYSTEM {
  INPUT = coordinate_data
  IDENTIFY = Cartesian_polar_cylindrical_spherical_or_other
  IDENTIFY = dimension_and_angle_convention
  RETURN = coordinate_context
 }

 R2_DISTANCE {
  INPUT = points_and_metric
  IF Cartesian_Euclidean
   APPLY = norm_of_coordinate_difference
  ELSE IF Manhattan
   APPLY = sum_absolute_differences
  ELSE
   ROUTE = metric_specific reasoning
  RETURN = distance
 }

 R3_MIDPOINT_SECTION {
  INPUT = endpoints_and_ratio
  IF midpoint
   COMPUTE = componentwise_average
  ELSE
   COMPUTE = affine_section_formula
  RETURN = point
 }

 R4_SLOPE_DIRECTION {
  INPUT = two_planar_points
  COMPUTE = delta_x_delta_y
  IF slope_requested
   IF delta_x=0
    RETURN VerticalUndefinedSlope
   ELSE
    RETURN delta_y/delta_x
  ELSE
   RETURN = direction_vector
 }

 R5_ANGLE {
  INPUT = direction_vectors_or_lines
  COMPUTE = dot_product_and_norms
  APPLY = arccos_or_atan2_method
  RETURN = angle
 }

 R6_POLYGON_MEASUREMENT {
  INPUT = ordered_polygon_vertices
  COMPUTE = edge_lengths
  IF perimeter_requested
   SUM = edge_lengths
  IF area_requested
   APPLY = shoelace
  RETURN = requested_measurement
 }

 R7_SIMPLEX_MEASUREMENT {
  INPUT = coordinate_vertices
  IF triangle
   APPLY = determinant_area
  ELSE IF tetrahedron
   APPLY = determinant_volume
  RETURN = measure
 }

 R8_COORDINATE_TRANSFORMATION {
  INPUT = coordinates_source_system_target_system
  APPLY = transformation_formula
  VERIFY = angle_units_and_convention
  RETURN = transformed_coordinates
 }

 R9_JACOBIAN_MEASUREMENT {
  INPUT = coordinate_change_and_region
  COMPUTE = Jacobian_determinant
  TRANSFORM = differential_measure
  INTEGRATE = over_transformed_region
  RETURN = area_or_volume
 }
}

CONSTRAINTS {
 C1 = "Coordinate system and dimensionality must be identified."
 C2 = "Angular conventions must be explicit for polar, cylindrical and spherical coordinates."
 C3 = "Slope is undefined for vertical lines."
 C4 = "Vector-angle formulas require nonzero vectors."
 C5 = "Shoelace formula requires cyclically ordered polygon vertices."
 C6 = "Coordinate transformations can change component values while preserving geometry."
 C7 = "Jacobian determinant must correspond to the correct transformation direction."
 C8 = "Distance metric must not be inferred as Euclidean when another metric is specified."
 C9 = "Approximate coordinates imply approximate derived measurements."
}

SEMANTIC_EQUIVALENCE {
 "CoordinateMeasurement" <=> "GeometryMeasuredFromCoordinates"
 "CoordinateDifference" <=> "DisplacementVector"
 "EuclideanCoordinateDistance" <=> "NormOfCoordinateDifference"
 "CoordinateArea" <=> "AreaDerivedFromCoordinates"
 "CoordinateVolume" <=> "VolumeDerivedFromCoordinates"
 "CoordinateTransformation" <=> "ChangeOfCoordinateRepresentation"
}

VALIDATION {
 V1 = "Coordinate dimension is consistent."
 V2 = "Coordinate system is identified."
 V3 = "Angle units and conventions are consistent."
 V4 = "Distance metric matches requested geometry."
 V5 = "Polygon vertex order is valid."
 V6 = "Determinant dimension matches target measure."
 V7 = "Jacobian factor uses absolute determinant where ordinary measure is intended."
 V8 = "Output units match measurement dimension."
 V9 = "Approximation status is preserved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COORDINATE_MEASUREMENT_DEFINITION = 0.99
 DISTANCE_REASONING = 0.99
 MIDPOINT_REASONING = 0.99
 SLOPE_DIRECTION_REASONING = 0.99
 ANGLE_REASONING = 0.99
 POLYGON_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.98
 JACOBIAN_REASONING = 0.98

 POLICY =
  "Reduce confidence when coordinate system, dimension, angle convention, metric, vertex ordering, transformation direction or unit assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  CoordinateMeasurementReasoning,
  GeometricMeasurementReasoning,
  CoordinateGeometryReasoning,
  DistanceReasoning,
  MidpointReasoning,
  SlopeReasoning,
  VectorReasoning,
  AngleReasoning,
  PolygonMeasurementReasoning,
  CoordinateTransformationReasoning,
  JacobianReasoning,
  UnitConversionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "coordinate measurement" -> CoordinateMeasurement
  "coordinate distance" -> CoordinateDistance
  "midpoint" -> Midpoint
  "slope" -> Slope
  "direction vector" -> DirectionVector
  "coordinate area" -> CoordinateArea
  "coordinate volume" -> CoordinateVolume
  "shoelace formula" -> ShoelaceArea
  "coordinate transformation" -> CoordinateTransformation
  "Jacobian" -> Jacobian
 }

 INTERPRETATION_ORDER = [
  detect_coordinate_measurement_context,
  identify coordinate_system_dimension_metric_units_angle_convention_and_precision,
  identify distance_midpoint_slope_direction_angle_perimeter_area_volume_or_transformation goal,
  select coordinate_formula_determinant_or_Jacobian pathway,
  compute requested quantity,
  verify coordinate_metric_orientation_dimension_unit_and_precision constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("coordinate measurement")
  OR coordinate_geometry_context_with_measurement_goal
  THEN activate(CoordinateMeasurementOntology)

 E2:
  IF Cartesian_point_distance_requested
  THEN use declared_metric

 E3:
  IF midpoint_requested
  THEN use componentwise_affine_average

 E4:
  IF slope_requested
  THEN detect vertical_line_before division

 E5:
  IF vector_angle_requested
  THEN require nonzero_vectors

 E6:
  IF polygon_coordinates_and_area_requested
  THEN use shoelace_formula with ordered_vertices

 E7:
  IF simplex_volume_requested
  THEN use determinant_with_correct factorial divisor

 E8:
  IF coordinate_conversion_requested
  THEN identify source_target_conventions before mapping

 E9:
  IF change_of_variables_measurement
  THEN apply absolute_Jacobian determinant
}

RECURSION {
 ENTRY = CoordinateMeasurement

 EXPAND(CoordinateMeasurement) -> [
  CoordinateSystem,
  CartesianCoordinates,
  PolarCoordinates,
  CylindricalCoordinates,
  SphericalCoordinates,
  CoordinateDifference,
  EuclideanDistance,
  ManhattanDistance,
  Midpoint,
  SectionPoint,
  Slope,
  DirectionVector,
  VectorMagnitude,
  VectorAngle,
  CoordinatePerimeter,
  CoordinateArea,
  CoordinateVolume,
  CoordinateTransformation,
  Jacobian,
  ScaleFactor,
  Unit,
  UnitConversion,
  ErrorBound
 ]

 EXPAND(CoordinateArea) -> [
  TriangleDeterminantArea,
  ShoelaceArea,
  SignedArea,
  PolygonArea
 ]

 EXPAND(CoordinateVolume) -> [
  TetrahedronVolume,
  ParallelepipedVolume,
  TripleIntegral,
  Jacobian
 ]

 EXPAND(CoordinateTransformation) -> [
  CartesianToPolar,
  PolarToCartesian,
  CartesianToCylindrical,
  CylindricalToCartesian,
  CartesianToSpherical,
  SphericalToCartesian
 ]

 STOP_CONDITION =
  "Stop when the coordinate system, measurement type, formula, metric, orientation, units, transformation and precision status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometric Measurement"

 IMMEDIATE_CONTEXTS = [
  "Geometric Measurement",
  "Coordinate Geometry Ontology",
  "Distance Ontology",
  "Angle Measurement Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometric Measurement",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Polar Coordinate Ontology",
  "Vector Geometry Ontology",
  "Distance Ontology",
  "Length Ontology",
  "Perimeter Ontology",
  "Area Ontology",
  "Volume Ontology",
  "Angle Measurement Ontology",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies",
  "Measurement & Units Ontologies"
 ]

 ROUTE_RULES = [
  "Coordinate Measurement -> Geometric Measurement for parent quantitative semantics",
  "Coordinate Measurement -> Coordinate Geometry for coordinate-object semantics",
  "Coordinate Measurement -> Cartesian and Polar ontologies for coordinate systems",
  "Coordinate Measurement -> Vector Geometry for displacement, magnitude and angle",
  "Coordinate Measurement -> Distance for metric separation",
  "Coordinate Measurement -> Area and Volume for determinant and integral measures",
  "Coordinate Measurement -> Linear Algebra for matrix and determinant reasoning",
  "Coordinate Measurement -> Calculus for Jacobian-based change of variables",
  "Coordinate Measurement -> Measurement & Units for unit conversion"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible coordinate systems, measurement formulas or transformation pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve coordinate-derived measurement semantics.",
  "Preserve coordinate-system awareness.",
  "Preserve metric dependence.",
  "Preserve determinant-based area and volume semantics.",
  "Preserve Jacobian measure transformation.",
  "Preserve angle convention awareness.",
  "Preserve unit and precision semantics.",
  "Preserve machine-readable coordinate-measurement semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Coordinate measurement computes geometric quantities directly from coordinate representations."

 DIFFERENCE_SUMMARY =
  "Coordinate displacement is Q-P."

 DISTANCE_SUMMARY =
  "Euclidean coordinate distance is ||Q-P||."

 MIDPOINT_SUMMARY =
  "The midpoint of P and Q is (P+Q)/2."

 SLOPE_SUMMARY =
  "For nonvertical planar points, slope=(y2-y1)/(x2-x1)."

 AREA_SUMMARY =
  "Planar coordinate area may be computed using determinants or the shoelace formula."

 VOLUME_SUMMARY =
  "Three-dimensional simplex volume may be computed from a determinant."

 TRANSFORMATION_SUMMARY =
  "Coordinate transformations change representation while preserving the underlying geometry when interpreted correctly."

 JACOBIAN_SUMMARY =
  "Change-of-variables area and volume use the absolute value of the appropriate Jacobian determinant."
}

FLOW {
 INPUT
 -> detect_coordinate_measurement_context
 -> identify coordinate_system_dimension_metric_units_angle_convention_scale_and_precision
 -> identify distance_midpoint_section_slope_direction_angle_perimeter_area_volume_or_transformation goal
 -> select coordinate_formula_vector_determinant_or_Jacobian method
 -> compute requested_measurement
 -> apply coordinate_conversion_scale_or_unit_conversion when required
 -> verify dimension_metric_orientation_ordering_Jacobian_unit_and_precision constraints
 -> route to geometric_measurement coordinate_geometry vector distance length perimeter area volume angle linear_algebra calculus_or_units ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Coordinate Measurement Ontology defines geometric measurement derived from coordinate representations. It supports Cartesian, polar, cylindrical and spherical systems; coordinate differences; distances; midpoints; slopes; direction vectors; vector angles; coordinate perimeters, areas and volumes; determinants; coordinate transformations; Jacobian scaling; units and precision-aware reasoning."
}

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

}

END_FLOW