Geometric Measurement

AI INTERPRETATION NOTE
This Flow Algebra defines the Geometric Measurement layer within the Kireth Research Semantic Library. Geometric measurement concerns quantitative properties of geometric objects, including length, distance, angle, perimeter, area, surface area, volume, radius, diameter, arc length, scale, coordinate-based measurement, dimensional consistency and unit conversion. This ontology layer provides a semantic bridge between geometry, mathematical measurement, units, coordinate systems and computational reasoning.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = GeometricMeasurement
 ENTITY_CLASS = QuantitativeGeometryFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Geometric measurement is the quantitative determination of spatial properties such as length, distance, angle, perimeter, area, surface area and volume."
 SEMANTIC_ROLE = "Provides the quantitative framework connecting geometric objects to measurable magnitudes, units, formulas, coordinate computations, scale relationships and dimensional reasoning."
}

SCOPE {
 INCLUDE = [
  GeometricMeasurement,
  Length,
  Distance,
  AngleMeasure,
  Perimeter,
  Circumference,
  Area,
  SurfaceArea,
  Volume,
  Radius,
  Diameter,
  ArcLength,
  SectorArea,
  ScaleFactor,
  SimilarityScale,
  CoordinateDistance,
  EuclideanDistance,
  ManhattanDistance,
  Metric,
  Unit,
  UnitConversion,
  DimensionalAnalysis,
  MeasurementPrecision,
  Approximation,
  ExactMeasurement,
  ErrorBound
 ]

 EXCLUDE_AS_PRIMARY = [
  PhysicalMeasurementWithoutGeometricQuantity,
  StatisticalMeasurement,
  InstrumentCalibrationUnlessExplicit,
  NonGeometricUnitsWithoutSpatialContext
 ]

 DISAMBIGUATION_RULE =
  "When the task concerns quantitative geometric properties such as distance, length, angle, perimeter, area, surface area, volume, radius or scale, interpret the context as geometric measurement."
}

CORE_CONCEPTS {

 GeometricMeasurement {
  definition = "The assignment or computation of quantitative magnitudes to geometric properties."
 }

 Length {
  definition = "A one-dimensional measure of extent."
  dimension = "L"
 }

 Distance {
  definition = "A measure of separation between points or sets under a specified metric."
  dimension = "L"
 }

 AngleMeasure {
  definition = "A quantitative measure of rotation or separation between rays or directions."
  common_units = ["radian","degree"]
 }

 Perimeter {
  definition = "The total boundary length of a planar figure."
  dimension = "L"
 }

 Circumference {
  definition = "The perimeter of a circle."
  relation = "C=2*pi*r=pi*d"
 }

 Area {
  definition = "A two-dimensional measure of region size."
  dimension = "L^2"
 }

 SurfaceArea {
  definition = "The two-dimensional measure of the boundary surface of a three-dimensional object."
  dimension = "L^2"
 }

 Volume {
  definition = "A three-dimensional measure of occupied space."
  dimension = "L^3"
 }

 Radius {
  definition = "The distance from a centre to a point on a circle or sphere."
 }

 Diameter {
  definition = "Twice the radius in a circle or sphere."
  relation = "d=2r"
 }

 ArcLength {
  definition = "The length measured along a curve or circular arc."
 }

 ScaleFactor {
  definition = "A multiplicative factor relating corresponding lengths in similar or scaled objects."
 }
}

RELATIONS {
 GeometricMeasurement --is_subdomain_of--> Geometry
 Length --measures--> OneDimensionalExtent
 Distance --is_special_case_of--> MetricMeasurement
 Perimeter --is_sum_of--> BoundaryLengths
 Circumference --is_special_case_of--> Perimeter
 Area --measures--> PlanarRegion
 SurfaceArea --measures--> Surface
 Volume --measures--> SolidRegion
 Radius --determines--> Diameter
 Radius --determines--> Circumference
 Radius --determines--> CircleArea
 ScaleFactor --transforms--> Length
 ScaleFactor --transforms--> Area
 ScaleFactor --transforms--> Volume
 UnitConversion --preserves--> PhysicalMagnitude
 CoordinateGeometry --supports--> CoordinateDistance
 DimensionalAnalysis --validates--> MeasurementFormula
}

OPERATORS {
 SEGMENT_LENGTH(A,B) :=
  EuclideanDistance(A,B)

 EUCLIDEAN_DISTANCE_2D((x1,y1),(x2,y2)) :=
  sqrt((x2-x1)^2+(y2-y1)^2)

 EUCLIDEAN_DISTANCE_3D((x1,y1,z1),(x2,y2,z2)) :=
  sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)

 MANHATTAN_DISTANCE(x,y) :=
  sum_i abs(x_i-y_i)

 PERIMETER_POLYGON(vertices) :=
  sum_of_edge_lengths

 CIRCUMFERENCE(r) :=
  2*pi*r

 CIRCLE_AREA(r) :=
  pi*r^2

 RECTANGLE_AREA(w,h) :=
  w*h

 TRIANGLE_AREA_BASE_HEIGHT(b,h) :=
  0.5*b*h

 TRIANGLE_AREA_SIDES(a,b,c) :=
  sqrt(s*(s-a)*(s-b)*(s-c))
  WHERE s=(a+b+c)/2

 ARC_LENGTH_CIRCLE(r,theta_radians) :=
  r*theta_radians

 SECTOR_AREA(r,theta_radians) :=
  0.5*r^2*theta_radians

 BOX_VOLUME(l,w,h) :=
  l*w*h

 SPHERE_VOLUME(r) :=
  (4/3)*pi*r^3

 SPHERE_SURFACE_AREA(r) :=
  4*pi*r^2

 CYLINDER_VOLUME(r,h) :=
  pi*r^2*h

 CONE_VOLUME(r,h) :=
  (1/3)*pi*r^2*h

 SCALE_LENGTH(L,k) :=
  k*L

 SCALE_AREA(A,k) :=
  k^2*A

 SCALE_VOLUME(V,k) :=
  k^3*V
}

DEFINITIONS {
 D1 = "Lengths and distances have dimension L."
 D2 = "Areas have dimension L^2."
 D3 = "Volumes have dimension L^3."
 D4 = "Angle measure may be expressed in radians or degrees."
 D5 = "Perimeter is a boundary-length measure."
 D6 = "Surface area is not the same as volume."
 D7 = "Scale factor k multiplies lengths by k, areas by k^2 and volumes by k^3."
 D8 = "Unit conversion changes representation, not underlying magnitude."
 D9 = "Exact and approximate measurements must be distinguished."
 D10 = "A geometric formula must be dimensionally consistent."
}

AXIOMS {
 A1_NONNEGATIVITY:
  Geometric magnitudes such as length, area and volume are nonnegative

 A2_DISTANCE_IDENTITY:
  d(x,y)=0 iff x=y for a metric distance

 A3_DISTANCE_SYMMETRY:
  d(x,y)=d(y,x)

 A4_TRIANGLE_INEQUALITY:
  d(x,z)<=d(x,y)+d(y,z)

 A5_DIMENSIONAL_CONSISTENCY:
  Quantities added or equated must have compatible dimensions

 A6_UNIT_INVARIANCE:
  Unit conversion preserves represented magnitude

 A7_SIMILARITY_SCALING:
  Similarity scale k transforms one-, two- and three-dimensional measures by k, k^2 and k^3 respectively

 A8_MEASURE_ADDIVITY:
  Disjoint geometric regions combine by additive measure under suitable conditions
}

REASONING {

 R1_QUANTITY_CLASSIFICATION {
  INPUT = geometric_measurement_problem
  IDENTIFY = object_type
  IDENTIFY = requested_quantity
  CLASSIFY = length_distance_angle_perimeter_area_surface_area_volume_scale_or_unit
  RETURN = quantity_type
 }

 R2_FORMULA_SELECTION {
  INPUT = object_type_and_known_parameters
  MATCH = valid_geometric_formula
  VERIFY = required_inputs_available
  RETURN = formula
 }

 R3_DIMENSION_CHECK {
  INPUT = formula_or_result
  COMPUTE = dimensions_of_terms
  VERIFY = compatibility
  RETURN = DimensionallyValidOrInvalid
 }

 R4_UNIT_CONVERSION {
  INPUT = quantity_value_source_unit_target_unit
  VERIFY = same_dimension
  APPLY = conversion_factor
  RETURN = converted_quantity
 }

 R5_SCALE_REASONING {
  INPUT = original_measure_and_scale_factor_k
  IDENTIFY = dimensional_order_n
  COMPUTE = scaled_measure=k^n*original_measure
  RETURN = scaled_result
 }

 R6_COORDINATE_DISTANCE {
  INPUT = coordinate_points_and_metric
  IF Euclidean
   APPLY = square_root_of_sum_of_squared_coordinate_differences
  ELSE IF Manhattan
   APPLY = sum_of_absolute_coordinate_differences
  ELSE
   ROUTE = metric_specific reasoning
  RETURN = distance
 }

 R7_AREA_REASONING {
  INPUT = planar_object
  SELECT = exact_formula_decomposition_coordinate_or_integration_method
  COMPUTE = area
  RETURN = area
 }

 R8_VOLUME_REASONING {
  INPUT = solid_object
  SELECT = exact_formula_decomposition_or_integration_method
  COMPUTE = volume
  RETURN = volume
 }

 R9_ERROR_REASONING {
  INPUT = approximate_measurement
  IDENTIFY = precision_or_error_bound
  PROPAGATE = through_formula_when_required
  RETURN = approximate_result_with_error_context
 }
}

CONSTRAINTS {
 C1 = "Units must be compatible before arithmetic combination."
 C2 = "Degrees and radians must not be mixed without conversion."
 C3 = "Area units are squared length units."
 C4 = "Volume units are cubed length units."
 C5 = "Scale factor effects depend on dimensional order."
 C6 = "Coordinate distance depends on the selected metric."
 C7 = "Approximate values must not be presented as exact."
 C8 = "Negative length, area or volume is invalid unless representing oriented or signed variants explicitly."
 C9 = "Formula choice must match the geometric object and required assumptions."
}

SEMANTIC_EQUIVALENCE {
 "GeometricMeasurement" <=> "QuantitativeGeometry"
 "Length" <=> "OneDimensionalMeasure"
 "Area" <=> "TwoDimensionalMeasure"
 "Volume" <=> "ThreeDimensionalMeasure"
 "Circumference" <=> "CirclePerimeter"
 "ScaleFactor" <=> "SimilarityRatio"
 "CoordinateDistance" <=> "MetricDistanceBetweenCoordinatePoints"
}

VALIDATION {
 V1 = "Requested quantity matches object dimension."
 V2 = "Formula assumptions are satisfied."
 V3 = "Units are dimensionally compatible."
 V4 = "Angle units are identified."
 V5 = "Scale exponent matches dimensional order."
 V6 = "Distance metric is specified or correctly inferred."
 V7 = "Exact versus approximate status is preserved."
 V8 = "Result has correct dimensional unit."
 V9 = "Nonnegativity holds where required."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 GEOMETRIC_MEASUREMENT_DEFINITION = 0.99
 FORMULA_SELECTION_REASONING = 0.99
 DIMENSIONAL_REASONING = 0.99
 UNIT_CONVERSION_REASONING = 0.99
 SCALE_REASONING = 0.99
 AREA_REASONING = 0.99
 VOLUME_REASONING = 0.99
 ERROR_REASONING = 0.98

 POLICY =
  "Reduce confidence when units, angle convention, scale factor, geometric assumptions, metric choice or approximation precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GeometricMeasurementReasoning,
  LengthReasoning,
  DistanceReasoning,
  AngleMeasurementReasoning,
  PerimeterReasoning,
  AreaReasoning,
  SurfaceAreaReasoning,
  VolumeReasoning,
  ScaleReasoning,
  UnitConversionReasoning,
  DimensionalAnalysis,
  CoordinateReasoning,
  ErrorReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "geometric measurement" -> GeometricMeasurement
  "length" -> Length WHEN geometry_context
  "distance" -> Distance WHEN geometry_context
  "perimeter" -> Perimeter
  "circumference" -> Circumference
  "area" -> Area WHEN geometry_context
  "surface area" -> SurfaceArea
  "volume" -> Volume WHEN geometry_context
  "scale factor" -> ScaleFactor
  "unit conversion" -> UnitConversion
 }

 INTERPRETATION_ORDER = [
  detect_geometric_measurement_context,
  identify geometric_object_and_requested_quantity,
  identify known_parameters_units_and_angle_convention,
  select measurement_formula_or_metric,
  perform dimensional_and_unit validation,
  compute exact_or_approximate result,
  apply scale_or_error reasoning_when_required,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("geometric measurement")
  OR geometry_context_with_length_area_volume_distance_or_scale
  THEN activate(GeometricMeasurement)

 E2:
  IF formula_based_measurement
  THEN verify object_type_and_required_dimensions

 E3:
  IF coordinate_distance_requested
  THEN identify metric_before_computation

 E4:
  IF angle_in_degrees_but_formula_requires_radians
  THEN convert_before_use

 E5:
  IF similarity_scale_factor_k
  THEN apply k^1_to_length k^2_to_area k^3_to_volume

 E6:
  IF unit_conversion_requested
  THEN ensure source_and_target units_have_same_dimension

 E7:
  IF approximate_inputs
  THEN preserve approximation_and_error context

 E8:
  IF dimensional_mismatch_detected
  THEN reject_or_correct formula
}

RECURSION {
 ENTRY = GeometricMeasurement

 EXPAND(GeometricMeasurement) -> [
  Length,
  Distance,
  AngleMeasure,
  Perimeter,
  Circumference,
  Area,
  SurfaceArea,
  Volume,
  Radius,
  Diameter,
  ArcLength,
  SectorArea,
  ScaleFactor,
  CoordinateDistance,
  Metric,
  Unit,
  UnitConversion,
  DimensionalAnalysis,
  MeasurementPrecision,
  ErrorBound
 ]

 EXPAND(Length) -> [
  SegmentLength,
  ArcLength,
  Perimeter,
  Circumference,
  Distance
 ]

 EXPAND(Area) -> [
  TriangleArea,
  RectangleArea,
  CircleArea,
  PolygonArea,
  SurfaceArea
 ]

 EXPAND(Volume) -> [
  CubeVolume,
  CuboidVolume,
  CylinderVolume,
  ConeVolume,
  SphereVolume,
  PolyhedronVolume
 ]

 EXPAND(ScaleFactor) -> [
  LengthScale,
  AreaScale,
  VolumeScale,
  Similarity
 ]

 STOP_CONDITION =
  "Stop when the requested geometric magnitude, units, formula, metric, scaling relationship, precision and dimensional validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Plane Geometry",
  "Solid Geometry"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Plane Geometry",
  "Solid Geometry",
  "Circle Ontology",
  "Triangle Ontology",
  "Polygon Ontology",
  "Polyhedron Ontology",
  "Scientific Measurement Ontologies",
  "Measurement & Units Ontologies",
  "Calculus & Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Geometric Measurement -> Geometry Ontology for object semantics",
  "Geometric Measurement -> Coordinate Geometry for coordinate distances and areas",
  "Geometric Measurement -> Plane Geometry for perimeter and area",
  "Geometric Measurement -> Solid Geometry for surface area and volume",
  "Geometric Measurement -> Circle for radius, circumference, arc and sector measurement",
  "Geometric Measurement -> Triangle for side, angle and area reasoning",
  "Geometric Measurement -> Measurement & Units for units and conversion",
  "Geometric Measurement -> Calculus for arc length, surface area and volume by integration"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible measurement quantities, formulas, metrics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve quantitative-geometry semantics.",
  "Preserve dimensional consistency.",
  "Preserve unit-conversion invariance.",
  "Preserve distance-metric distinction.",
  "Preserve scale-factor dimensional exponents.",
  "Preserve exact-versus-approximate distinction.",
  "Preserve formula-assumption validation.",
  "Preserve machine-readable geometric-measurement semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Geometric measurement quantifies spatial properties such as length, distance, angle, perimeter, area, surface area and volume."

 LENGTH_SUMMARY =
  "Length has dimension L."

 AREA_SUMMARY =
  "Area has dimension L^2."

 VOLUME_SUMMARY =
  "Volume has dimension L^3."

 SCALE_SUMMARY =
  "A scale factor k multiplies lengths by k, areas by k^2 and volumes by k^3."

 UNIT_SUMMARY =
  "Unit conversion changes numerical representation while preserving the underlying magnitude."

 DISTANCE_SUMMARY =
  "Coordinate distance depends on the selected metric."

 DIMENSION_SUMMARY =
  "Valid geometric formulas must be dimensionally consistent."

 PRECISION_SUMMARY =
  "Exact and approximate measurements must retain their precision and error context."
}

FLOW {
 INPUT
 -> detect_geometric_measurement_context
 -> identify object_quantity_known_parameters_units_metric_angle_convention_and_scale
 -> classify length_distance_angle_perimeter_area_surface_area_volume_or_other measurement
 -> select valid_formula_metric_or_integration pathway
 -> verify dimensional_consistency_and_unit compatibility
 -> compute exact_or_approximate result
 -> apply scale_conversion_or_error propagation when required
 -> route to geometry coordinate plane solid circle triangle polygon polyhedron measurement_units_or_calculus ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Geometric Measurement layer defines the quantitative side of geometry. It supports length, distance, angle, perimeter, circumference, area, surface area, volume, arc length, scale factors, coordinate measurement, units, conversions, dimensional analysis, approximation and error reasoning while connecting geometric formulas to measurement and unit ontologies."
}

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

}

END_FLOW