Surface Area Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of surface area within the Kireth Research Semantic Library. Surface area is the two-dimensional measure of the boundary of a three-dimensional geometric object or the area of a two-dimensional surface embedded in space. This ontology supports total and lateral surface area, prisms, cylinders, cones, spheres, pyramids, polyhedra, parametric surfaces, surface integrals, nets, scaling, units, dimensional analysis, exact and approximate values and relationships with area, volume, solid geometry and differential geometry.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SurfaceArea
 SYMBOLIC_FORMS = [
  "SA",
  "A_surface",
  "double_integral_S dA"
 ]
 ENTITY_CLASS = TwoDimensionalBoundaryMeasure
 PRIMARY_CONTEXT = GeometricMeasurement
 DEFINITION = "Surface area is the nonnegative two-dimensional measure of the boundary surface of a three-dimensional object, or of a surface embedded in space."
 SEMANTIC_ROLE = "Provides the quantitative framework for measuring solid boundaries, lateral and total surface areas, parametric surfaces, polyhedra and curved surfaces."
}

SCOPE {
 INCLUDE = [
  SurfaceArea,
  TotalSurfaceArea,
  LateralSurfaceArea,
  CurvedSurfaceArea,
  PolyhedronSurfaceArea,
  PrismSurfaceArea,
  CuboidSurfaceArea,
  CubeSurfaceArea,
  CylinderSurfaceArea,
  ConeSurfaceArea,
  SphereSurfaceArea,
  HemisphereSurfaceArea,
  PyramidSurfaceArea,
  FrustumSurfaceArea,
  ParametricSurfaceArea,
  SurfaceIntegral,
  SurfaceNet,
  FaceArea,
  ScaleFactor,
  Unit,
  UnitConversion,
  ExactSurfaceArea,
  ApproximateSurfaceArea,
  MeasurementPrecision,
  ErrorBound
 ]

 EXCLUDE_AS_PRIMARY = [
  PlanarAreaWithoutSurfaceContext,
  Volume,
  Perimeter,
  SurfaceTextureAreaUnlessExplicit,
  FractalSurfaceMeasureUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When the task asks for the area of the boundary of a three-dimensional object, the lateral or curved area of a solid, or the area of a parametrised surface, interpret the quantity as surface area."
}

CORE_CONCEPTS {

 SurfaceArea {
  definition = "A two-dimensional measure of a surface or solid boundary."
  dimension = "L^2"
 }

 TotalSurfaceArea {
  definition = "The area of all exterior faces or boundary components of a solid."
 }

 LateralSurfaceArea {
  definition = "The area of side surfaces excluding designated bases."
 }

 CurvedSurfaceArea {
  definition = "The area of curved boundary components."
 }

 PolyhedronSurfaceArea {
  definition = "The sum of the areas of all exterior polygonal faces."
 }

 CubeSurfaceArea {
  relation = "SA=6s^2"
 }

 CuboidSurfaceArea {
  relation = "SA=2(lw+lh+wh)"
 }

 CylinderSurfaceArea {
  total_relation = "SA=2*pi*r^2+2*pi*r*h"
  lateral_relation = "LSA=2*pi*r*h"
 }

 ConeSurfaceArea {
  total_relation = "SA=pi*r^2+pi*r*l"
  lateral_relation = "LSA=pi*r*l"
 }

 SphereSurfaceArea {
  relation = "SA=4*pi*r^2"
 }

 HemisphereSurfaceArea {
  curved_relation = "CSA=2*pi*r^2"
  total_relation = "TSA=3*pi*r^2"
 }

 ParametricSurfaceArea {
  definition = "Surface area computed from the norm of the cross product of surface partial derivatives."
 }
}

RELATIONS {
 SurfaceArea --is_measure_of--> Surface
 SurfaceArea --has_dimension--> AreaDimension
 TotalSurfaceArea --includes--> LateralSurfaceArea
 TotalSurfaceArea --includes--> BaseArea
 PolyhedronSurfaceArea --is_sum_of--> FaceArea
 CubeSurfaceArea --is_special_case_of--> PolyhedronSurfaceArea
 CuboidSurfaceArea --is_special_case_of--> PolyhedronSurfaceArea
 CylinderSurfaceArea --is_special_case_of--> SurfaceArea
 ConeSurfaceArea --is_special_case_of--> SurfaceArea
 SphereSurfaceArea --is_special_case_of--> SurfaceArea
 HemisphereSurfaceArea --is_special_case_of--> SurfaceArea
 ParametricSurfaceArea --is_computed_by--> SurfaceIntegral
 SurfaceNet --represents--> SolidBoundary
 ScaleFactor --transforms--> SurfaceArea
 UnitConversion --preserves--> SurfaceAreaMagnitude
}

OPERATORS {
 CUBE_SURFACE_AREA(s) :=
  6*s^2

 CUBOID_SURFACE_AREA(l,w,h) :=
  2*(l*w+l*h+w*h)

 PRISM_SURFACE_AREA(base_area,base_perimeter,h) :=
  2*base_area + base_perimeter*h

 CYLINDER_LATERAL_AREA(r,h) :=
  2*pi*r*h

 CYLINDER_TOTAL_AREA(r,h) :=
  2*pi*r^2 + 2*pi*r*h

 CONE_LATERAL_AREA(r,l) :=
  pi*r*l

 CONE_TOTAL_AREA(r,l) :=
  pi*r^2 + pi*r*l

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

 HEMISPHERE_CURVED_AREA(r) :=
  2*pi*r^2

 HEMISPHERE_TOTAL_AREA(r) :=
  3*pi*r^2

 PYRAMID_SURFACE_AREA(base_area,face_areas) :=
  base_area + sum(face_areas)

 POLYHEDRON_SURFACE_AREA(faces) :=
  sum(area(face) for face in exterior_faces)

 PARAMETRIC_SURFACE_AREA(X,D) :=
  double_integral_D norm(X_u cross X_v) du dv

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

 CONVERT_SURFACE_AREA(A,unit_from,unit_to) :=
  A*conversion_factor_squared(unit_from,unit_to)
}

DEFINITIONS {
 D1 = "Surface area has dimension L^2."
 D2 = "Surface area is nonnegative."
 D3 = "Total surface area includes all exterior boundary components."
 D4 = "Lateral surface area excludes designated bases."
 D5 = "Curved surface area includes only curved boundary components."
 D6 = "Polyhedron surface area is the sum of exterior face areas."
 D7 = "Parametric surface area is computed by integrating ||X_u cross X_v||."
 D8 = "Similarity scale factor k changes surface area by k^2."
 D9 = "Surface-area units are squared length units."
 D10 = "Surface area and volume must not be conflated."
}

AXIOMS {
 A1_NONNEGATIVITY:
  SurfaceArea >= 0

 A2_BOUNDARY_MEASURE:
  Surface area measures a two-dimensional boundary or embedded surface

 A3_FACE_ADDITIVITY:
  Surface area of a polyhedral boundary is the sum of nonoverlapping exterior face areas

 A4_TOTAL_LATERAL_DISTINCTION:
  Total and lateral surface area differ by the inclusion or exclusion of bases

 A5_SCALE:
  Similarity transformation with scale factor k multiplies surface area by k^2

 A6_UNIT_INVARIANCE:
  Correct unit conversion preserves surface-area magnitude

 A7_DIMENSION:
  Surface area has dimension L^2

 A8_PARAMETRIC_AREA:
  For a regular parameterisation X(u,v), dA=||X_u cross X_v|| du dv
}

REASONING {

 R1_SURFACE_AREA_CLASSIFICATION {
  INPUT = geometric_object
  IDENTIFY = polyhedron_prism_cylinder_cone_sphere_hemisphere_pyramid_frustum_or_parametric_surface
  IDENTIFY = total_lateral_curved_or_partial_goal
  RETURN = surface_area_type
 }

 R2_POLYHEDRON {
  INPUT = face_geometry
  IDENTIFY = exterior_faces
  COMPUTE = each_face_area
  SUM = face_areas
  RETURN = total_surface_area
 }

 R3_PRISM {
  INPUT = base_area_base_perimeter_height
  COMPUTE = 2*base_area + base_perimeter*height
  RETURN = total_surface_area
 }

 R4_CYLINDER {
  INPUT = radius_height_and_goal
  IF lateral
   COMPUTE = 2*pi*r*h
  ELSE IF total
   COMPUTE = 2*pi*r*h + 2*pi*r^2
  RETURN = surface_area
 }

 R5_CONE {
  INPUT = radius_slant_height_and_goal
  IF lateral
   COMPUTE = pi*r*l
  ELSE IF total
   COMPUTE = pi*r*l + pi*r^2
  RETURN = surface_area
 }

 R6_SPHERE_HEMISPHERE {
  INPUT = radius_and_shape
  IF sphere
   COMPUTE = 4*pi*r^2
  ELSE IF hemisphere_curved
   COMPUTE = 2*pi*r^2
  ELSE IF hemisphere_total
   COMPUTE = 3*pi*r^2
  RETURN = surface_area
 }

 R7_PYRAMID {
  INPUT = base_and_lateral_faces
  COMPUTE = base_area
  COMPUTE = lateral_face_areas
  SUM = all_required_exterior_faces
  RETURN = surface_area
 }

 R8_PARAMETRIC_SURFACE {
  INPUT = parameterisation_X_and_domain_D
  VERIFY = regularity
  COMPUTE = X_u
  COMPUTE = X_v
  COMPUTE = norm(X_u cross X_v)
  INTEGRATE = over_D
  RETURN = surface_area
 }

 R9_SCALE_AND_UNITS {
  INPUT = area_scale_or_unit_conversion
  APPLY = k^2_or_squared_conversion_factor
  RETURN = transformed_surface_area
 }
}

CONSTRAINTS {
 C1 = "Surface area and planar area are both dimension L^2 but refer to different geometric contexts."
 C2 = "Total, lateral and curved surface areas must be distinguished."
 C3 = "Internal faces of joined solids must not be counted as exterior surface."
 C4 = "Cone formulas require slant height for lateral area."
 C5 = "Hemisphere total area includes the circular base."
 C6 = "Parametric surface area requires sufficient regularity."
 C7 = "Self-overlapping parameterisations may require multiplicity-aware interpretation."
 C8 = "Scale factors affect surface area quadratically."
 C9 = "Unit conversion must use squared conversion factors."
 C10 = "Approximate dimensions produce approximate surface areas."
}

SEMANTIC_EQUIVALENCE {
 "SurfaceArea" <=> "TwoDimensionalBoundaryMeasure"
 "TotalSurfaceArea" <=> "AreaOfEntireExteriorBoundary"
 "LateralSurfaceArea" <=> "SideSurfaceAreaExcludingBases"
 "CurvedSurfaceArea" <=> "AreaOfCurvedBoundary"
 "PolyhedronSurfaceArea" <=> "SumOfExteriorFaceAreas"
 "ParametricSurfaceArea" <=> "SurfaceIntegralOfAreaElement"
}

VALIDATION {
 V1 = "Result is nonnegative."
 V2 = "Output unit has dimension L^2."
 V3 = "Requested total/lateral/curved interpretation is explicit or correctly inferred."
 V4 = "Only exterior faces are counted."
 V5 = "Cone slant height is used for lateral-area formulas."
 V6 = "Hemisphere base inclusion matches requested surface type."
 V7 = "Parametric area uses norm of cross product."
 V8 = "Scale factor uses second power."
 V9 = "Approximation status is preserved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SURFACE_AREA_DEFINITION = 0.99
 POLYHEDRON_REASONING = 0.99
 CYLINDER_REASONING = 0.99
 CONE_REASONING = 0.99
 SPHERE_REASONING = 0.99
 PARAMETRIC_SURFACE_REASONING = 0.98
 SCALE_REASONING = 0.99
 UNIT_REASONING = 0.99

 POLICY =
  "Reduce confidence when total-versus-lateral intent, hidden/internal faces, slant height, surface parameterisation, scale factor, units or approximation precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SurfaceAreaReasoning,
  GeometricMeasurementReasoning,
  SolidGeometryReasoning,
  PolyhedronReasoning,
  PrismReasoning,
  CylinderReasoning,
  ConeReasoning,
  SphereReasoning,
  ParametricSurfaceReasoning,
  SurfaceIntegralReasoning,
  ScaleReasoning,
  UnitConversionReasoning,
  DimensionalAnalysis,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "surface area" -> SurfaceArea
  "total surface area" -> TotalSurfaceArea
  "lateral surface area" -> LateralSurfaceArea
  "curved surface area" -> CurvedSurfaceArea
  "cube surface area" -> CubeSurfaceArea
  "cuboid surface area" -> CuboidSurfaceArea
  "cylinder surface area" -> CylinderSurfaceArea
  "cone surface area" -> ConeSurfaceArea
  "sphere surface area" -> SphereSurfaceArea
  "hemisphere surface area" -> HemisphereSurfaceArea
 }

 INTERPRETATION_ORDER = [
  detect_surface_area_context,
  identify solid_or_surface_type_dimensions_units_scale_and_precision,
  identify total_lateral_curved_partial_or_parametric goal,
  select valid_formula_face_sum_or_surface_integral,
  compute surface_area,
  verify exterior_boundary_dimension_unit_and_formula assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("surface area")
  OR query_mentions("lateral area")
  OR query_mentions("curved surface area")
  THEN activate(SurfaceAreaOntology)

 E2:
  IF polyhedron
  THEN sum exterior_face_areas

 E3:
  IF prism
  THEN use 2*base_area + base_perimeter*height for total surface area

 E4:
  IF cylinder
  THEN distinguish lateral_from_total formula

 E5:
  IF cone
  THEN require slant_height_for_lateral_formula

 E6:
  IF sphere_or_hemisphere
  THEN distinguish sphere_curved_hemisphere_and_total_hemisphere cases

 E7:
  IF parametric_surface
  THEN integrate norm(X_u cross X_v)

 E8:
  IF similarity_scale_given
  THEN multiply surface_area by k^2

 E9:
  IF unit_conversion_requested
  THEN apply squared_length_conversion
}

RECURSION {
 ENTRY = SurfaceArea

 EXPAND(SurfaceArea) -> [
  TotalSurfaceArea,
  LateralSurfaceArea,
  CurvedSurfaceArea,
  PolyhedronSurfaceArea,
  PrismSurfaceArea,
  CuboidSurfaceArea,
  CubeSurfaceArea,
  CylinderSurfaceArea,
  ConeSurfaceArea,
  SphereSurfaceArea,
  HemisphereSurfaceArea,
  PyramidSurfaceArea,
  FrustumSurfaceArea,
  ParametricSurfaceArea,
  SurfaceIntegral,
  SurfaceNet,
  FaceArea,
  ScaleFactor,
  Unit,
  UnitConversion,
  ErrorBound
 ]

 EXPAND(PolyhedronSurfaceArea) -> [
  FaceArea,
  ExteriorFace,
  InternalFaceExclusion,
  SurfaceNet
 ]

 EXPAND(ParametricSurfaceArea) -> [
  SurfaceParameterisation,
  PartialDerivative,
  CrossProduct,
  AreaElement,
  DoubleIntegral
 ]

 EXPAND(CurvedSurfaceArea) -> [
  CylinderLateralArea,
  ConeLateralArea,
  SphereArea,
  HemisphereCurvedArea
 ]

 STOP_CONDITION =
  "Stop when the requested surface type, total/lateral/curved interpretation, formula, units, scale, exactness and precision status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometric Measurement"

 IMMEDIATE_CONTEXTS = [
  "Geometric Measurement",
  "Area Ontology",
  "Solid Geometry",
  "Surface Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometric Measurement",
  "Area Ontology",
  "Volume Ontology",
  "Solid Geometry",
  "Cube Ontology",
  "Cuboid Ontology",
  "Cylinder Ontology",
  "Cone Ontology",
  "Sphere Ontology",
  "Hemisphere Ontology",
  "Prism Ontology",
  "Pyramid Ontology",
  "Polyhedron Ontology",
  "Surface Ontology",
  "Calculus & Analysis Ontologies",
  "Measurement & Units Ontologies"
 ]

 ROUTE_RULES = [
  "Surface Area -> Geometric Measurement for parent quantitative semantics",
  "Surface Area -> Area for two-dimensional measure semantics",
  "Surface Area -> Solid Geometry for solid-specific formulas",
  "Surface Area -> Surface Ontology for embedded-surface semantics",
  "Surface Area -> Polyhedron for face-sum reasoning",
  "Surface Area -> Calculus for parametric surface integrals",
  "Surface Area -> Measurement & Units for squared-unit conversion",
  "Surface Area -> Volume for related but dimensionally distinct solid measurement"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve two-dimensional boundary-measure semantics.",
  "Preserve total-lateral-curved distinctions.",
  "Preserve exterior-face-only counting.",
  "Preserve parametric area-element semantics.",
  "Preserve squared dimensionality.",
  "Preserve quadratic scaling.",
  "Preserve unit compatibility.",
  "Preserve machine-readable surface-area semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Surface area is the nonnegative two-dimensional measure of the boundary surface of a three-dimensional object or embedded surface."

 DIMENSION_SUMMARY =
  "Surface area has dimension L^2."

 CUBE_SUMMARY =
  "Cube surface area SA=6s^2."

 CUBOID_SUMMARY =
  "Cuboid surface area SA=2(lw+lh+wh)."

 CYLINDER_SUMMARY =
  "Cylinder lateral area=2*pi*r*h and total area=2*pi*r*h+2*pi*r^2."

 CONE_SUMMARY =
  "Cone lateral area=pi*r*l and total area=pi*r*l+pi*r^2."

 SPHERE_SUMMARY =
  "Sphere surface area SA=4*pi*r^2."

 PARAMETRIC_SUMMARY =
  "For X(u,v), surface area=double integral ||X_u cross X_v|| du dv."

 SCALE_SUMMARY =
  "A similarity scale factor k changes surface area by k^2."
}

FLOW {
 INPUT
 -> detect_surface_area_context
 -> identify solid_or_surface_type_dimensions_geometry_units_scale_and_precision
 -> identify total_lateral_curved_partial_or_parametric surface_area goal
 -> select valid_formula_face_decomposition_or_surface_integral
 -> compute surface_area
 -> apply scaling_or_unit_conversion when required
 -> verify exterior_boundary_dimension_regular_surface_unit_and_precision constraints
 -> route to geometric_measurement area volume solid_geometry surface calculus polyhedron_or_units ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Surface Area Ontology defines the two-dimensional measure of solid boundaries and embedded surfaces. It supports total, lateral and curved surface area, polyhedra, prisms, cubes, cuboids, cylinders, cones, spheres, hemispheres, pyramids, frustums, surface nets, parametric surfaces, surface integrals, scaling, units and dimensional validation."
}

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

}

END_FLOW