Plane Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the Plane Geometry layer within the Kireth Research Semantic Library. Plane geometry studies two-dimensional geometric entities and relations within a plane, including points, lines, angles, polygons, circles, distances, areas, congruence, similarity, loci and transformations. This record is designed for semantic retrieval, recursive geometric reasoning, ontology routing, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = PlaneGeometry
 ENTITY_CLASS = TwoDimensionalGeometricDomain
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Plane geometry is the study of points, lines, curves, angles, shapes and spatial relations in a two-dimensional plane."
 SEMANTIC_ROLE = "Provides the core two-dimensional geometric framework for measurement, construction, proof, congruence, similarity and transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Point,
  Line,
  Ray,
  Segment,
  Angle,
  Triangle,
  Quadrilateral,
  Polygon,
  Circle,
  Arc,
  Chord,
  Tangent,
  Secant,
  Distance,
  Perimeter,
  Area,
  Congruence,
  Similarity,
  Parallelism,
  Perpendicularity,
  Symmetry,
  Locus,
  GeometricConstruction,
  PlaneTransformation
 ]

 EXCLUDE_AS_PRIMARY = [
  SolidGeometry,
  ThreeDimensionalGeometry,
  DifferentialGeometryOfSurfaces,
  NonPlanarSpatialRelations
 ]

 DISAMBIGUATION_RULE =
  "When geometric entities and relations are confined to a two-dimensional plane, interpret the context as plane geometry."
}

CORE_CONCEPTS {

 Plane {
  definition = "A flat two-dimensional geometric space extending indefinitely."
 }

 Point {
  definition = "A position in the plane with no intrinsic length or area."
 }

 Line {
  definition = "A straight one-dimensional object extending indefinitely in both directions within the plane."
 }

 Segment {
  definition = "A finite part of a line bounded by two endpoints."
 }

 Ray {
  definition = "A part of a line beginning at one endpoint and extending indefinitely in one direction."
 }

 Angle {
  definition = "The rotational separation between two rays with a common vertex."
 }

 Polygon {
  definition = "A closed plane figure formed by a finite sequence of line segments."
 }

 Triangle {
  definition = "A polygon with three sides."
 }

 Quadrilateral {
  definition = "A polygon with four sides."
 }

 Circle {
  definition = "The set of points in the plane at a fixed distance from a centre point."
 }

 Congruence {
  definition = "Equality of plane figures in size and shape under rigid motion."
 }

 Similarity {
  definition = "Equality of plane figures in shape under scaling and rigid motion."
 }

 Locus {
  definition = "The set of points satisfying a specified plane-geometric condition."
 }
}

RELATIONS {
 PlaneGeometry --is_subdomain_of--> Geometry
 PlaneGeometry --contains--> Point
 PlaneGeometry --contains--> Line
 PlaneGeometry --contains--> PlaneFigure
 Point --can_lie_on--> Line
 Point --can_lie_on--> Circle
 Line --can_be_parallel_to--> Line
 Line --can_be_perpendicular_to--> Line
 Segment --has--> Length
 Angle --has--> Measure
 Polygon --has--> Perimeter
 PlaneFigure --can_have--> Area
 PlaneFigure --can_be_related_by--> Congruence
 PlaneFigure --can_be_related_by--> Similarity
 Circle --has--> Radius
 Circle --has--> Diameter
 Circle --can_have--> Tangent
 Circle --can_have--> Chord
 PlaneGeometry --supports--> GeometricConstruction
 PlaneGeometry --supports--> SyntheticProof
}

OPERATORS {
 DISTANCE(A,B) :=
  measure_planar_separation(A,B)

 LENGTH(segment) :=
  measure_segment_length(segment)

 ANGLE_MEASURE(angle) :=
  measure_rotation(angle)

 PERIMETER(shape) :=
  sum_boundary_lengths(shape)

 AREA(shape) :=
  compute_planar_measure(shape)

 IS_PARALLEL(L1,L2) :=
  evaluate_parallel_relation(L1,L2)

 IS_PERPENDICULAR(L1,L2) :=
  evaluate_right_angle_relation(L1,L2)

 IS_CONGRUENT(A,B) :=
  verify_rigid_motion_equivalence(A,B)

 IS_SIMILAR(A,B) :=
  verify_similarity_transformation_equivalence(A,B)

 LOCUS(condition) :=
  set_of_plane_points_satisfying(condition)

 TRANSFORM(shape,T) :=
  apply_plane_transformation(shape,T)
}

DEFINITIONS {
 D1 = "Plane geometry is two-dimensional."
 D2 = "Plane figures have length and area but no volume."
 D3 = "Parallel lines in Euclidean plane geometry do not intersect."
 D4 = "Perpendicular lines meet at a right angle."
 D5 = "Congruent figures have equal corresponding lengths and angles."
 D6 = "Similar figures have equal corresponding angles and proportional corresponding lengths."
 D7 = "A circle is determined by its centre and radius."
 D8 = "A polygon is a closed chain of straight segments."
 D9 = "A locus is defined by a geometric condition."
 D10 = "Plane transformations may preserve different geometric invariants depending on their class."
}

AXIOMS {
 A1_TWO_DIMENSIONALITY:
  All primary geometric entities are interpreted within a plane

 A2_INCIDENCE:
  Points and lines participate in planar incidence relations

 A3_METRIC:
  Standard Euclidean plane geometry uses the Euclidean metric unless otherwise stated

 A4_PARALLELISM:
  Euclidean parallel relations follow the Euclidean parallel postulate

 A5_CONGRUENCE:
  Rigid motions preserve lengths and angles

 A6_SIMILARITY:
  Similarity transformations preserve angles and proportional structure

 A7_AREA:
  Planar area is nonnegative and additive over suitable nonoverlapping decompositions

 A8_PROOF:
  Plane-geometric theorems require valid geometric derivation
}

REASONING {

 R1_ENTITY_CLASSIFICATION {
  INPUT = geometric_object
  IDENTIFY = point_line_angle_polygon_circle_or_related_type
  RETURN = typed_plane_entity
 }

 R2_TRIANGLE_REASONING {
  INPUT = triangle_configuration
  IDENTIFY = side_angle_relations
  APPLY = congruence_similarity_or_angle_theorems
  RETURN = triangle_result
 }

 R3_POLYGON_REASONING {
  INPUT = polygon
  IDENTIFY = side_count_and_properties
  COMPUTE = perimeter_area_or_angle_relations
  RETURN = polygon_result
 }

 R4_CIRCLE_REASONING {
  INPUT = circle_configuration
  IDENTIFY = radius_diameter_chord_arc_tangent_secant
  APPLY = circle_theorems
  RETURN = circle_result
 }

 R5_PARALLEL_PERPENDICULAR {
  INPUT = line_configuration
  APPLY = angle_or_incidence_criteria
  RETURN = relation_status
 }

 R6_CONGRUENCE_SIMILARITY {
  INPUT = [figureA,figureB]
  TEST = valid_criteria
  RETURN = CongruentSimilarOrNeither
 }

 R7_AREA_REASONING {
  INPUT = plane_figure
  DECOMPOSE = standard_shapes_when_needed
  COMPUTE = area
  RETURN = area_value
 }

 R8_LOCUS_REASONING {
  INPUT = condition
  IDENTIFY = candidate_locus
  PROVE = necessity_and_sufficiency
  RETURN = locus_characterisation
 }

 R9_PLANE_TRANSFORMATION {
  INPUT = [figure,T]
  CLASSIFY = translation_rotation_reflection_scaling_or_other
  APPLY = T
  TRACK = invariants
  RETURN = transformed_figure
 }
}

CONSTRAINTS {
 C1 = "Plane geometry is distinct from solid geometry."
 C2 = "Euclidean parallel-line results require Euclidean assumptions."
 C3 = "Area and perimeter are distinct measurements."
 C4 = "Congruence and similarity are distinct relations."
 C5 = "Circle theorems require correct identification of arcs, chords, tangents and radii."
 C6 = "Visual diagrams alone do not establish proof."
 C7 = "Transformation invariants depend on transformation class."
 C8 = "Locus claims require complete characterisation when exactness is asserted."
}

SEMANTIC_EQUIVALENCE {
 "PlaneGeometry" <=> "TwoDimensionalGeometry"
 "PlaneFigure" <=> "TwoDimensionalGeometricFigure"
 "Congruence" <=> "RigidMotionEquivalence"
 "Similarity" <=> "ScaleAndRigidMotionEquivalence"
 "Locus" <=> "SetOfPlanePointsSatisfyingCondition"
}

VALIDATION {
 V1 = "Primary entities are two-dimensional or embedded in a plane."
 V2 = "Length and area are not confused with volume."
 V3 = "Congruence and similarity criteria are applied correctly."
 V4 = "Parallel and perpendicular relations are correctly typed."
 V5 = "Circle results use valid circle properties."
 V6 = "Area calculations respect shape decomposition."
 V7 = "Transformation invariants match transformation type."
 V8 = "Plane-geometric conclusions follow from valid reasoning."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PLANE_GEOMETRY_DEFINITION = 0.99
 TRIANGLE_REASONING = 0.99
 POLYGON_REASONING = 0.99
 CIRCLE_REASONING = 0.99
 AREA_REASONING = 0.99
 CONGRUENCE_SIMILARITY = 0.99
 LOCUS_REASONING = 0.98

 POLICY =
  "Reduce confidence when Euclidean assumptions, diagram configuration, measurement data or geometric constraints are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  PlaneGeometryReasoning,
  TriangleReasoning,
  PolygonReasoning,
  CircleReasoning,
  AreaReasoning,
  CongruenceReasoning,
  SimilarityReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "plane geometry" -> PlaneGeometry
  "2D geometry" -> PlaneGeometry
  "two-dimensional geometry" -> PlaneGeometry
  "plane figure" -> PlaneFigure
  "polygon" -> Polygon
  "circle geometry" -> Circle
  "congruent figures" -> Congruence
  "similar figures" -> Similarity
 }

 INTERPRETATION_ORDER = [
  detect_plane_geometry_context,
  identify_geometric_entities,
  identify line angle polygon or circle relations,
  identify metric and measurement goals,
  identify congruence similarity or transformation context,
  apply relevant plane-geometry theorems,
  verify constraints and assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("plane geometry")
  OR two_dimensional_geometry_context
  THEN activate(PlaneGeometry)

 E2:
  IF triangle_context
  THEN activate_triangle_reasoning

 E3:
  IF polygon_context
  THEN activate_polygon_reasoning

 E4:
  IF circle_context
  THEN activate_circle_reasoning

 E5:
  IF congruence_context
  THEN apply_congruence_criteria

 E6:
  IF similarity_context
  THEN apply_similarity_criteria

 E7:
  IF area_requested
  THEN select_valid_area_method

 E8:
  IF locus_requested
  THEN prove_necessity_and_sufficiency

 E9:
  IF transformation_context
  THEN classify_transformation_and_preserved_invariants
}

RECURSION {
 ENTRY = PlaneGeometry

 EXPAND(PlaneGeometry) -> [
  Point,
  Line,
  Ray,
  Segment,
  Angle,
  Triangle,
  Quadrilateral,
  Polygon,
  Circle,
  Congruence,
  Similarity,
  Locus,
  PlaneTransformation
 ]

 EXPAND(Triangle) -> [
  Side,
  Angle,
  Median,
  Altitude,
  AngleBisector,
  PerpendicularBisector,
  Congruence,
  Similarity
 ]

 EXPAND(Polygon) -> [
  Side,
  Vertex,
  InteriorAngle,
  ExteriorAngle,
  Perimeter,
  Area
 ]

 EXPAND(Circle) -> [
  Centre,
  Radius,
  Diameter,
  Chord,
  Arc,
  Tangent,
  Secant,
  Circumference,
  Area
 ]

 EXPAND(PlaneTransformation) -> [
  Translation,
  Rotation,
  Reflection,
  Scaling,
  SimilarityTransformation
 ]

 STOP_CONDITION =
  "Stop when the target plane-geometric entity, relation, measurement, transformation, locus or proof is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Synthetic Geometry Ontology",
  "Coordinate / Analytic Geometry",
  "Geometric Construction Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Synthetic Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Geometric Construction Ontology",
  "Axiomatic Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Plane Geometry -> Geometry for core entity semantics",
  "Plane Geometry -> Synthetic Geometry for direct geometric proofs",
  "Plane Geometry -> Coordinate Geometry for coordinate-based solutions",
  "Plane Geometry -> Geometric Construction for exact constructions",
  "Plane Geometry -> Transformation for planar mappings",
  "Plane Geometry -> Axiomatic Geometry for foundational assumptions"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible plane-geometric entities, theorems, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve two-dimensional geometric semantics.",
  "Preserve distinction between length, area and volume.",
  "Preserve congruence and similarity distinctions.",
  "Preserve circle, polygon and triangle relation typing.",
  "Preserve Euclidean assumptions where used.",
  "Preserve transformation-specific invariants.",
  "Preserve machine-readable plane-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Plane geometry is the study of points, lines, angles, curves and figures in a two-dimensional plane."

 ENTITY_SUMMARY =
  "Core entities include points, lines, segments, rays, angles, polygons and circles."

 MEASUREMENT_SUMMARY =
  "Plane geometry measures length, perimeter, angle and area."

 CONGRUENCE_SUMMARY =
  "Congruent figures have equal corresponding lengths and angles."

 SIMILARITY_SUMMARY =
  "Similar figures have equal corresponding angles and proportional corresponding lengths."

 TRANSFORMATION_SUMMARY =
  "Translations, rotations, reflections and scalings act on plane figures while preserving different geometric invariants."
}

FLOW {
 INPUT
 -> detect_plane_geometry_context
 -> identify_plane_entities
 -> identify incidence angle polygon circle or transformation relations
 -> identify metric measurement or proof goal
 -> route_to triangle_polygon_circle_congruence_similarity_or_locus_reasoning
 -> apply valid plane-geometry theorems
 -> verify Euclidean and measurement assumptions
 -> link related geometry ontologies
 -> recursively expand required dependencies
 -> verify result
 -> assign confidence
 -> produce semantic output
 -> END
}

SEMANTIC_SUMMARY {
 "Plane Geometry provides the two-dimensional branch of geometry, covering points, lines, angles, polygons, circles, distances, areas, congruence, similarity, loci and transformations. It supports both synthetic and coordinate reasoning and provides a foundational layer for exact two-dimensional geometric proof and measurement."
}

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

}

END_FLOW