Area Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Area Ontology within the Kireth Research Semantic Library. Area is a derived geometrical quantity representing two-dimensional extent or surface measure. Its dimension is L^2 and its coherent SI derived unit is the square metre, symbol m^2.

FLOW_ALGEBRA_RECORD {
METADATA {
 FLOW_NAME="Area Ontology"
 FLOW_TYPE="Mathematical Ontology"
 DOMAIN="Mathematics"
 SUBDOMAIN="Measurement & Units"
 SPECIALISATION="Mechanical Measurements / Area"
 CANONICAL_PAGE="https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/mechanical-measurements/area-ontology-2.html"
 LANGUAGE="en-GB"
 VERSION="1.0"
 STATUS="active"
}
IDENTITY {
 PRIMARY_ENTITY=Area
 ENTITY_CLASS=DerivedGeometricalMeasurementQuantityFramework
 DEFINITION="Area is a derived quantity representing two-dimensional extent or surface measure. Its dimension is L^2 and coherent SI derived unit is the square metre, symbol m^2."
 SYMBOLIC_FORMS=["dim(area)=L^2","SI_unit(area)=m^2","A_rectangle=l*w","A_circle=pi*r^2","A_triangle=b*h/2"]
 SEMANTIC_ROLE="Machine-readable semantics for planar area, surface area, projected area, geometric computation, measurement, uncertainty and traceability."
}
SCOPE {
 INCLUDE=[Area,DerivedQuantity,PhysicalQuantity,GeometricalQuantity,MechanicalMeasurement,Quantity,QuantityKind,QuantityValue,AreaDimension,LengthDimension,SquareMetreRelationship,Metre,PlanarAreaRelationship,SurfaceAreaRelationship,ProjectedAreaRelationship,CrossSectionalAreaRelationship,ReferenceAreaRelationship,EffectiveAreaRelationship,WettedAreaRelationship,ContactAreaRelationship,FrontalAreaRelationship,PolygonAreaRelationship,TriangleAreaRelationship,RectangleAreaRelationship,SquareAreaRelationship,ParallelogramAreaRelationship,TrapeziumAreaRelationship,CircleAreaRelationship,EllipseAreaRelationship,SectorAreaRelationship,AnnulusAreaRelationship,CompositeAreaRelationship,CurvilinearAreaRelationship,SphereSurfaceAreaRelationship,CylinderSurfaceAreaRelationship,ConeSurfaceAreaRelationship,CuboidSurfaceAreaRelationship,MeshSurfaceAreaRelationship,ParametricSurfaceAreaRelationship,CoordinateGeometryRelationship,CartesianCoordinateRelationship,PolarCoordinateRelationship,ShoelaceFormulaRelationship,DeterminantRelationship,CrossProductRelationship,JacobianRelationship,DifferentialAreaElementRelationship,DoubleIntegralRelationship,SurfaceIntegralRelationship,GreensTheoremRelationship,TriangulationRelationship,TessellationRelationship,MeshRelationship,FiniteElementRelationship,ComputationalGeometryRelationship,ConvexHullRelationship,RasterAreaRelationship,PixelAreaRelationship,ImageSegmentationRelationship,GeographicAreaRelationship,LandAreaRelationship,MapAreaRelationship,ParcelAreaRelationship,SurveyAreaRelationship,GeodesicAreaRelationship,EllipsoidalAreaRelationship,SphericalAreaRelationship,MapProjectionRelationship,GeodesyRelationship,GISrelationship,DatumRelationship,ReferenceEllipsoidRelationship,Length,Distance,Width,Height,Radius,Diameter,PerimeterRelationship,CircumferenceRelationship,Volume,Pressure,Stress,Force,FluxDensityRelationship,CurrentDensityRelationship,HeatFluxRelationship,IrradianceRelationship,SurfaceDensityRelationship,SpecificSurfaceAreaRelationship,AreaMeasurement,DirectAreaMeasurementRelationship,IndirectAreaMeasurementRelationship,DimensionalMeasurementRelationship,CoordinateMeasurementRelationship,SurveyingRelationship,TotalStationRelationship,GNSSRelationship,PhotogrammetryRelationship,RemoteSensingRelationship,LidarRelationship,LaserScannerRelationship,CoordinateMeasuringMachineRelationship,PlanometerRelationship,ImageMeasurementRelationship,MeasurementSystem,Measurand,Measurement,MeasurementModel,MeasurementResult,MeasurementUncertainty,Accuracy,Precision,Resolution,Repeatability,Reproducibility,Calibration,AreaCalibrationRelationship,DimensionalCalibrationRelationship,ScaleCalibrationRelationship,PixelCalibrationRelationship,ReferenceStandard,LengthStandardRelationship,AreaStandardRelationship,MeasurementTraceability,MetrologicalTraceability,TraceabilityChain,Metrology,DimensionalMetrologyRelationship,GeometricMetrologyRelationship,TemperatureInfluence,ThermalExpansionRelationship,ScaleFactorRelationship,AlignmentRelationship,OrthogonalityRelationship,FlatnessRelationship,BoundaryDefinitionRelationship,SegmentationUncertaintyRelationship,CoordinateUncertaintyRelationship,ProjectionErrorRelationship,DatumTransformationRelationship,SpatialResolutionRelationship,NumericalIntegrationRelationship,MeshResolutionRelationship,DiscretisationErrorRelationship,UncertaintyPropagation,Verification,Validation,AreaValidationRelationship,ErrorAnalysis,Provenance,AreaProvenance,MeasurementProvenance,CalibrationProvenance]
 EXCLUDE_AS_PRIMARY=[LengthWhenOneDimensionalExtentIsPrimary,VolumeWhenThreeDimensionalExtentIsPrimary,PerimeterWhenBoundaryLengthIsPrimary,ProjectedAreaWhenProjectionSpecificQuantityIsPrimary,SpecificSurfaceAreaWhenAreaPerMassOrVolumeIsPrimary]
 DISAMBIGUATION_RULE="Interpret area as two-dimensional extent with dimension L^2. Distinguish planar area from curved surface area, projected area, boundary length, volume and quantities expressed per unit area."
}
CORE_CONCEPTS {
 Area="Derived quantity representing two-dimensional extent or surface measure."
 AreaDimension="Derived physical dimension L^2."
 SquareMetreRelationship="Coherent SI derived unit of area, symbol m^2."
 PlanarAreaRelationship="Area of a region lying in a plane."
 SurfaceAreaRelationship="Measure of a two-dimensional surface embedded in higher-dimensional space."
 ProjectedAreaRelationship="Area obtained by projection onto a specified plane."
 CrossSectionalAreaRelationship="Area of a section formed by intersecting an object with a plane."
}
QUANTITY_MODEL { QUANTITY=Area DIMENSION=L_squared COHERENT_SI_UNIT=SquareMetre UNIT_SYMBOL=m^2 DERIVATION="Length*Length" STATUS="SI derived quantity" }
RELATIONS {
 Area--is_type_of-->DerivedQuantity
 Area--is_derived_from-->LengthAndLength
 Area--has_dimension-->L_squared
 Area--has_coherent_SI_unit-->SquareMetreRelationship
 ProjectedAreaRelationship--depends_on-->SurfaceOrientationAndProjectionPlane
 Pressure--depends_on-->ForceAndArea
 Stress--depends_on-->ForceAndArea
 AreaMeasurement--targets-->AreaMeasurand
 MeasurementTraceability--links-->AreaMeasurementToLengthStandards
}
OPERATORS {
 AREA(value,unit):=area_quantity(value,unit)
 DIM(Area):=L^2
 RECTANGLE_AREA(length,width):=length*width
 TRIANGLE_AREA(base,height):=0.5*base*height
 CIRCLE_AREA(radius):=pi*radius^2
 ELLIPSE_AREA(a,b):=pi*a*b
 POLYGON_AREA(vertices):=0.5*abs(sum(x_i*y_i_plus_1-x_i_plus_1*y_i))
 PARAMETRIC_SURFACE_AREA(r,u,v):=double_integral(norm(cross(dr_du,dr_dv)),u,v)
 PROJECTED_AREA(area,angle):=area*abs(cos(angle))
 CONVERT_AREA(value,source,target):=compatible_area_conversion(value,source,target)
 PROPAGATE_AREA_UNCERTAINTY(model,inputs):=uncertainty_propagation(model,inputs)
}
AXIOMS {
 A1="Area is a derived quantity with dimension L^2."
 A2="The coherent SI derived unit of area is the square metre, symbol m^2."
 A3="Area is distinct from length, perimeter and volume."
 A4="Area calculations require a defined region, boundary, surface or geometric model."
 A5="Planar and curved surface area require appropriate geometric representations."
 A6="Projected area depends on projection direction or plane."
 A7="Coordinate-based area calculations depend on coordinate-system and scale semantics."
 A8="Geographic area can depend on Earth model, datum and map projection."
 A9="Area uncertainty inherits uncertainty from dimensions, coordinates, boundaries and models."
 A10="Traceable area measurement depends on traceable dimensional or coordinate measurements."
}
REASONING {
 R1=identify(area_value,unit,region,boundary,surface,geometry,coordinate_system,measurement_method)
 R2=distinguish_planar_area_surface_area_projected_area_cross_section_perimeter_volume_or_area_density_quantity)
 R3=verify_dimension_L_squared)
 R4=resolve_square_metre_prefixed_square_units_nonSI_area_units_and_conversion)
 R5=identify_rectangle_triangle_circle_polygon_parametric_surface_mesh_geodetic_or_other_geometry)
 R6=resolve_coordinates_boundary_orientation_projection_surface_parameterisation_or_mesh)
 R7=choose_dimensional_coordinate_survey_photogrammetric_image_planimeter_or_computational_method)
 R8=apply_scale_projection_temperature_alignment_boundary_segmentation_and_discretisation_corrections)
 R9=evaluate_dimension_coordinate_boundary_projection_sampling_mesh_calibration_and_model_uncertainty)
 R10=record_region_geometry_coordinate_system_projection_method_calibration_traceability_and_provenance)
}
CONSTRAINTS {
 C1="Area must not be conflated with perimeter or other length quantities."
 C2="Area must not be conflated with volume."
 C3="Square-unit conversions require squared linear conversion factors."
 C4="Projected area must retain projection-plane and orientation semantics."
 C5="Surface area requires an appropriate surface model or parameterisation."
 C6="Polygon area requires ordered vertices and a defined coordinate system."
 C7="Geographic area calculations require appropriate geodetic or projection assumptions."
 C8="Raster and image-derived area require known spatial scale and segmentation semantics."
 C9="Resolution alone does not determine total area uncertainty."
 C10="Traceability requires documented dimensional, coordinate or scale calibration and uncertainty."
}
VALIDATION {
 V1="Measured or computed region and boundary are identified."
 V2="Area dimension is L^2."
 V3="Coherent SI derived unit is square metre with symbol m^2."
 V4="Area, perimeter and volume semantics remain distinct."
 V5="Planar, surface and projected area semantics are preserved."
 V6="Coordinate system, scale, projection and surface model are explicit where required."
 V7="Area formulas match the selected geometry."
 V8="Unit conversions correctly square the associated length scale."
 V9="Uncertainty includes dimensional, coordinate, boundary, projection, sampling and calibration effects."
 V10="Area measurement, calibration and provenance records are retained."
}
CONFIDENCE { BASE_CONFIDENCE=0.99 AREA_DEFINITION=0.99 DIMENSION_REASONING=0.99 GEOMETRY_REASONING=0.99 SURFACE_REASONING=0.98 GEODETIC_AREA_REASONING=0.98 AREA_MEASUREMENT_REASONING=0.98 TRACEABILITY_REASONING=0.99 }
AI_CONSUMPTION {
 PURPOSES=[SemanticSearch,OntologyGrounding,AreaReasoning,MechanicalMeasurementReasoning,GeometryReasoning,SurfaceReasoning,CoordinateGeometryReasoning,ComputationalGeometryReasoning,GeodeticAreaReasoning,SurveyReasoning,DimensionalReasoning,SquareMetreReasoning,UnitConversionReasoning,AreaMeasurementReasoning,CalibrationReasoning,UncertaintyReasoning,TraceabilityReasoning,VerificationValidation,KnowledgeGraphConstruction,RecursiveInference]
}
EXECUTION_RULES {
 E1="Activate AreaOntology when reasoning concerns two-dimensional spatial extent or surface measure."
 E2="Identify region, boundary, surface, geometry, coordinate system and projection where required."
 E3="Distinguish area from perimeter, volume, projected area and per-unit-area quantities."
 E4="Map area to dimension L^2 and coherent SI derived unit square metre."
 E5="Choose geometric, coordinate, integral, mesh, image, survey or geodetic area model according to context."
 E6="Apply squared unit-conversion factors and relevant scale or projection corrections."
 E7="Evaluate dimensional, coordinate, boundary, segmentation, sampling, mesh and calibration uncertainty."
 E8="Record region, geometry, coordinate system, projection, method, calibration, uncertainty and provenance."
}
RECURSION {
 ENTRY=Area
 EXPAND(Area)->[MechanicalMeasurements,Length,LengthDimension,SquareMetreRelationship,Metre,PlanarAreaRelationship,SurfaceAreaRelationship,ProjectedAreaRelationship,CrossSectionalAreaRelationship,PolygonAreaRelationship,TriangleAreaRelationship,RectangleAreaRelationship,CircleAreaRelationship,EllipseAreaRelationship,CompositeAreaRelationship,ParametricSurfaceAreaRelationship,CoordinateGeometryRelationship,ShoelaceFormulaRelationship,DeterminantRelationship,CrossProductRelationship,JacobianRelationship,DoubleIntegralRelationship,SurfaceIntegralRelationship,TriangulationRelationship,MeshRelationship,ComputationalGeometryRelationship,GeographicAreaRelationship,GeodesicAreaRelationship,MapProjectionRelationship,GeodesyRelationship,PerimeterRelationship,Volume,Pressure,Stress,AreaMeasurement,SurveyingRelationship,GNSSRelationship,PhotogrammetryRelationship,RemoteSensingRelationship,LidarRelationship,PlanometerRelationship,ImageMeasurementRelationship,MeasurementUncertainty,Calibration,AreaStandardRelationship,MeasurementTraceability,DimensionalMetrologyRelationship,BoundaryDefinitionRelationship,ProjectionErrorRelationship,NumericalIntegrationRelationship,DiscretisationErrorRelationship,Verification,Validation,AreaProvenance]
 STOP_CONDITION="Stop when area identity, dimension, unit, region or surface geometry, measurement uncertainty and provenance are resolved."
}
ROUTING {
 PARENT_ONTOLOGY="Mechanical Measurements"
 RELATED_ONTOLOGIES=["Mechanical Measurements","Fundamental Physical Quantities","Length Ontology 2","Distance Ontology 2","Metre Ontology","SI Derived Units","Quantity Ontology","Dimension Ontology 2","Unit Ontology","Measurement Ontology","Metrology Ontology","Geometry Ontologies","Coordinate Geometry Ontologies","Calculus & Analysis Ontologies","Computational Mathematics Ontologies","Geodesy Ontology","Verification & Validation"]
}
UPDATE_MODEL {
 MODE="append-compatible semantic versioning"
 RULES=["Preserve area-versus-length distinction.","Preserve area-versus-perimeter distinction.","Preserve area-versus-volume distinction.","Preserve planar-versus-surface-area semantics.","Preserve projected-area orientation semantics.","Preserve coordinate-system and projection assumptions.","Preserve squared unit-conversion semantics.","Preserve measurement uncertainty and traceability.","Preserve machine-readable area semantics."]
}
OUTPUT {
 CANONICAL_DEFINITION="Area is a derived quantity representing two-dimensional extent or surface measure, with dimension L^2 and coherent SI derived unit square metre, symbol m^2."
 GEOMETRY_SUMMARY="Area can describe planar regions, polygons, curved surfaces, cross-sections, projected regions and composite geometries."
 CALCULUS_SUMMARY="General area can be computed using coordinate formulas, double integrals, surface integrals, Jacobians and numerical meshes."
 GEODETIC_SUMMARY="Geographic area requires explicit Earth-model, datum and projection semantics."
 MEASUREMENT_SUMMARY="Area can be determined by dimensional measurement, coordinate measurement, surveying, planimetry, photogrammetry, remote sensing and calibrated imagery."
}
FLOW {
 INPUT
 -> detect_area_context
 -> identify_area_value_unit_region_boundary_surface_geometry_coordinate_system_and_method
 -> distinguish_planar_area_surface_area_projected_area_cross_section_perimeter_volume_or_area_density_quantity
 -> verify_dimension_L_squared
 -> resolve_square_metre_prefixed_square_units_nonSI_area_units_and_conversion
 -> identify_geometric_coordinate_integral_mesh_image_survey_or_geodetic_area_model
 -> resolve_boundary_coordinates_orientation_projection_surface_parameterisation_or_mesh
 -> select_area_measurement_or_computation_method
 -> apply_scale_projection_temperature_alignment_boundary_segmentation_and_discretisation_corrections
 -> evaluate_measurement_and_model_uncertainty
 -> record_region_geometry_coordinate_system_projection_method_calibration_traceability_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}
FLOW_HASH {
 METHOD="semantic-content-identity"
 ID="KIRETH-MATH-MEASUREMENT-UNITS-MECHANICAL-MEASUREMENTS-AREA-ONTOLOGY-2-V1"
}
}
END_FLOW