Transformation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of geometric transformation within the Kireth Research Semantic Library. A transformation is a mapping that sends points, figures or geometric structures from a source configuration to an image configuration according to a specified rule. This ontology supports translations, rotations, reflections, glide reflections, dilations, similarities, affine transformations, compositions, inverses, fixed points, invariant sets, coordinate rules, matrix representations and symmetry.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Transformation Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Transformational Geometry / Geometric Mappings"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/transformational-geometry/transformation-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = GeometricTransformation
 SYMBOLIC_FORMS = ["T:X->Y", "P->T(P)", "T(x)=f(x)"]
 ENTITY_CLASS = GeometricMapping
 PRIMARY_CONTEXT = TransformationalGeometry
 DEFINITION = "A geometric transformation is a mapping that assigns each point or geometric entity in a source space an image in a target space according to a defined geometric rule."
 SEMANTIC_ROLE = "Provides the core mapping entity for geometric motion, deformation, congruence, similarity, affine structure, coordinate change, invariance and symmetry reasoning."
}

SCOPE {
 INCLUDE = [
  Transformation,
  GeometricTransformation,
  Image,
  Preimage,
  Domain,
  Codomain,
  Translation,
  Rotation,
  Reflection,
  GlideReflection,
  Dilation,
  Scaling,
  Isometry,
  SimilarityTransformation,
  AffineTransformation,
  Shear,
  Projection,
  IdentityTransformation,
  InverseTransformation,
  Composition,
  FixedPoint,
  FixedSet,
  Invariant,
  MatrixRepresentation
 ]

 EXCLUDE_AS_PRIMARY = [
  ArbitraryFunctionWithoutGeometricInterpretation,
  PureDataTransformation,
  NongeometricCoordinateRelabellingWithoutSpatialMeaning
 ]

 DISAMBIGUATION_RULE =
  "When transformation acts on geometric points, figures, vectors, coordinates, lines, planes or solids and questions concern images, preimages, invariants, congruence, similarity or symmetry, interpret the context as a geometric transformation."
}

CORE_CONCEPTS {

 GeometricTransformation {
  definition = "A mapping between geometric spaces or configurations."
  general_form = "T:X->Y"
 }

 Image {
  definition = "The transformed point or object produced by applying T."
  form = "T(P)"
 }

 Preimage {
  definition = "An original point or object mapped by T to a specified image."
 }

 Domain {
  definition = "The source set on which the transformation is defined."
 }

 Codomain {
  definition = "The target space in which transformed points or objects lie."
 }

 IdentityTransformation {
  definition = "A transformation satisfying I(P)=P for all P in the domain."
 }

 Translation {
  definition = "An isometry moving every point by the same displacement vector."
  form = "T(x)=x+v"
 }

 Rotation {
  definition = "An isometry turning points by a fixed angle about a centre or axis."
 }

 Reflection {
  definition = "An isometry mapping points across a mirror line or plane."
 }

 GlideReflection {
  definition = "Composition of a reflection with a translation parallel to the mirror line or plane."
 }

 Dilation {
  definition = "A transformation scaling distances from a centre by a common factor."
  form = "T(P)=C+k(P-C)"
 }

 Isometry {
  definition = "A transformation preserving all pairwise distances."
 }

 SimilarityTransformation {
  definition = "A transformation preserving angles and multiplying all lengths by one positive scale factor."
 }

 AffineTransformation {
  definition = "A transformation of the form T(x)=A*x+b."
 }

 FixedPoint {
  definition = "A point P satisfying T(P)=P."
 }

 Invariant {
  definition = "A property, set or relation preserved by a transformation."
 }
}

RELATIONS {
 GeometricTransformation --maps--> Domain
 GeometricTransformation --maps_into--> Codomain
 GeometricTransformation --produces--> Image
 Image --has_preimage_under--> GeometricTransformation
 IdentityTransformation --is_special_case_of--> GeometricTransformation
 Translation --is_special_case_of--> Isometry
 Rotation --is_special_case_of--> Isometry
 Reflection --is_special_case_of--> Isometry
 GlideReflection --is_special_case_of--> Isometry
 Isometry --is_special_case_of--> SimilarityTransformation
 SimilarityTransformation --is_special_case_of--> GeometricTransformation
 AffineTransformation --is_special_case_of--> GeometricTransformation
 GeometricTransformation --can_have--> FixedPoint
 GeometricTransformation --can_preserve--> Invariant
 GeometricTransformation --can_be_composed_with--> GeometricTransformation
 BijectiveTransformation --can_have--> InverseTransformation
}

OPERATORS {
 APPLY(T,P) :=
  T(P)

 APPLY_TO_OBJECT(T,S) :=
  {T(P) | P in S}

 COMPOSE(T2,T1) :=
  T2∘T1

 IDENTITY(P) :=
  P

 INVERSE(T) :=
  T^-1 WHEN T is bijective

 FIXED_POINTS(T) :=
  {P | T(P)=P}

 PREIMAGE(T,Q) :=
  {P | T(P)=Q}

 TRANSLATE(P,v) :=
  P+v

 DILATE(P,C,k) :=
  C+k*(P-C)

 AFFINE_MAP(x,A,b) :=
  A*x+b

 CHECK_INVARIANT(property,T,S) :=
  compare(property(S),property(T(S)))
}

DEFINITIONS {
 D1 = "A transformation maps source geometric entities to image entities."
 D2 = "The image of P under T is T(P)."
 D3 = "The identity transformation fixes every point."
 D4 = "A bijective transformation has a unique inverse."
 D5 = "Composition applies transformations sequentially."
 D6 = "An isometry preserves distance."
 D7 = "A similarity preserves angles and scales lengths uniformly."
 D8 = "An affine transformation preserves affine combinations."
 D9 = "A fixed point satisfies T(P)=P."
 D10 = "An invariant is defined relative to a specified transformation or class."
}

AXIOMS {
 A1_MAPPING:
  Each point in the domain has a uniquely defined image under a function-valued transformation

 A2_IDENTITY:
  I(P)=P for every P

 A3_ASSOCIATIVE_COMPOSITION:
  T3∘(T2∘T1)=(T3∘T2)∘T1 whenever compositions are defined

 A4_INVERSE:
  If T is bijective then T^-1∘T=I and T∘T^-1=I

 A5_ISOMETRY:
  If T is an isometry then d(T(P),T(Q))=d(P,Q)

 A6_SIMILARITY:
  If T is a similarity with factor k then d(T(P),T(Q))=k*d(P,Q)

 A7_AFFINE:
  Affine transformations preserve affine combinations

 A8_CONTEXTUAL_INVARIANCE:
  Preserved properties depend on transformation class
}

REASONING {

 R1_TRANSFORMATION_CLASSIFICATION {
  INPUT = transformation_rule
  TEST = identity
  TEST = translation_structure
  TEST = rotation_structure
  TEST = reflection_structure
  TEST = uniform_scaling
  TEST = distance_preservation
  TEST = affine_form
  RETURN = most_specific_valid_transformation_class
 }

 R2_IMAGE_REASONING {
  INPUT = [T,P_or_object]
  APPLY = transformation_rule
  RETURN = image
 }

 R3_PREIMAGE_REASONING {
  INPUT = [T,Q]
  SOLVE = T(P)=Q
  RETURN = preimage_set
 }

 R4_COMPOSITION {
  INPUT = [T1,T2]
  APPLY = T1_first_then_T2
  FORM = T2∘T1
  SIMPLIFY = composite_rule
  RETURN = composite_transformation
 }

 R5_INVERSE {
  INPUT = T
  VERIFY = bijectivity_or_matrix_invertibility
  SOLVE = source_variables_in_terms_of_target_variables
  RETURN = inverse_transformation
 }

 R6_FIXED_POINT {
  INPUT = T
  SOLVE = T(P)=P
  CLASSIFY = no_fixed_points_unique_fixed_point_fixed_line_fixed_plane_or_larger_set
  RETURN = fixed_set
 }

 R7_INVARIANT_REASONING {
  INPUT = [T,properties]
  CLASSIFY = transformation_type
  TEST = distance_angle_collinearity_parallelism_area_orientation_volume_or_other
  RETURN = preserved_and_changed_properties
 }

 R8_MATRIX_REASONING {
  INPUT = coordinate_transformation
  REPRESENT = matrix_or_homogeneous_matrix
  IF composition_requested
   MULTIPLY = matrices_in_correct_order
  IF inverse_requested
   INVERT = matrix_when_nonsingular
  RETURN = geometric_interpretation
 }

 R9_CONGRUENCE_SIMILARITY {
  INPUT = [figureA,figureB]
  SEARCH = isometry_or_similarity mapping A_to_B
  IF isometry_found
   RETURN Congruent
  ELSE IF similarity_found
   RETURN Similar
  ELSE
   RETURN NeitherEstablished
 }
}

CONSTRAINTS {
 C1 = "Transformation composition order must be preserved."
 C2 = "Not every transformation is invertible."
 C3 = "Image and preimage must not be conflated."
 C4 = "Affine transformations do not generally preserve distances or angles."
 C5 = "Similarities preserve angles but may scale lengths."
 C6 = "Reflections reverse orientation."
 C7 = "Matrix representations depend on coordinate basis and convention."
 C8 = "Fixed-point claims require solving T(P)=P."
 C9 = "Invariants must be associated with an explicit transformation class."
}

SEMANTIC_EQUIVALENCE {
 "GeometricTransformation" <=> "GeometryPreservingOrGeometryActingMap"
 "Image" <=> "TransformedEntity"
 "Preimage" <=> "OriginalEntityMappingToImage"
 "RigidTransformation" <=> "Isometry"
 "Dilation" <=> "UniformScalingAboutCentre"
 "AffineTransformation" <=> "LinearTransformationPlusTranslation"
 "FixedPoint" <=> "PointInvariantUnderTransformation"
}

VALIDATION {
 V1 = "Every domain point has a defined image."
 V2 = "Identity fixes every point."
 V3 = "Composition order is correct."
 V4 = "Inverse is used only when transformation is bijective."
 V5 = "Isometry preserves distance."
 V6 = "Similarity scales all lengths by a common factor."
 V7 = "Affine transformations preserve collinearity and parallelism."
 V8 = "Fixed points satisfy T(P)=P."
 V9 = "Matrix form agrees with geometric transformation."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRANSFORMATION_DEFINITION = 0.99
 CLASSIFICATION = 0.99
 IMAGE_PREIMAGE_REASONING = 0.99
 COMPOSITION_REASONING = 0.99
 INVERSE_REASONING = 0.99
 FIXED_POINT_REASONING = 0.99
 INVARIANT_REASONING = 0.98
 MATRIX_REASONING = 0.99

 POLICY =
  "Reduce confidence when source space, target space, coordinate basis, transformation order, centre, axis, scale factor or invertibility are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TransformationReasoning,
  ImageReasoning,
  PreimageReasoning,
  CompositionReasoning,
  InverseReasoning,
  FixedPointReasoning,
  CongruenceReasoning,
  SimilarityReasoning,
  AffineReasoning,
  SymmetryReasoning,
  MatrixReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "transformation" -> GeometricTransformation WHEN geometry_context
  "geometric transformation" -> GeometricTransformation
  "image" -> Image WHEN transformation_context
  "preimage" -> Preimage WHEN transformation_context
  "inverse transformation" -> InverseTransformation
  "identity transformation" -> IdentityTransformation
  "fixed point" -> FixedPoint
  "rigid transformation" -> Isometry
 }

 INTERPRETATION_ORDER = [
  detect_geometric_transformation_context,
  identify domain_codomain_and_geometric_entities,
  parse transformation_rule_and_parameters,
  classify transformation_type,
  identify image_preimage_composition_inverse_fixed_point_invariant_or_matrix goal,
  apply transformation_operator,
  verify mapping_order_basis_and_invertibility_constraints,
  interpret preserved_and_changed_geometry,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("transformation")
  AND geometry_context
  THEN activate(TransformationOntology2)

 E2:
  IF transformation_rule_given
  THEN classify most_specific_valid_type

 E3:
  IF image_requested
  THEN apply T directly

 E4:
  IF preimage_requested
  THEN solve T(P)=Q

 E5:
  IF composition_requested
  THEN apply rightmost transformation first under standard function notation

 E6:
  IF inverse_requested
  THEN verify bijectivity_or_nonsingular_matrix

 E7:
  IF fixed_points_requested
  THEN solve T(P)=P

 E8:
  IF invariants_requested
  THEN classify transformation before testing properties

 E9:
  IF matrix_representation_used
  THEN record coordinate_basis_and_vector_convention
}

RECURSION {
 ENTRY = GeometricTransformation

 EXPAND(GeometricTransformation) -> [
  Image,
  Preimage,
  IdentityTransformation,
  Translation,
  Rotation,
  Reflection,
  GlideReflection,
  Dilation,
  Isometry,
  SimilarityTransformation,
  AffineTransformation,
  Shear,
  Composition,
  InverseTransformation,
  FixedPoint,
  Invariant,
  MatrixRepresentation
 ]

 EXPAND(Isometry) -> [
  Translation,
  Rotation,
  Reflection,
  GlideReflection,
  Congruence,
  DistanceInvariant,
  AngleInvariant
 ]

 EXPAND(SimilarityTransformation) -> [
  Isometry,
  Dilation,
  ScaleFactor,
  Similarity,
  AngleInvariant
 ]

 EXPAND(AffineTransformation) -> [
  LinearPart,
  TranslationPart,
  Shear,
  Scaling,
  CollinearityInvariant,
  ParallelismInvariant
 ]

 EXPAND(FixedPoint) -> [
  FixedSet,
  FixedLine,
  FixedPlane,
  CentreOfRotation,
  ReflectionMirror
 ]

 STOP_CONDITION =
  "Stop when the transformation's type, image, preimage, composition, inverse, fixed set, invariants, matrix form, congruence or similarity relation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Transformational Geometry"

 IMMEDIATE_CONTEXTS = [
  "Transformational Geometry",
  "Linear Transformation Ontology",
  "Affine Transformation Ontology",
  "Vector Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Transformational Geometry",
  "Geometry Ontology",
  "Plane Geometry",
  "Solid Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Vector Geometry Ontology",
  "Linear Transformation Ontology",
  "Affine Transformation Ontology",
  "Matrix Ontology",
  "Isomorphism Ontology",
  "Automorphism Ontology"
 ]

 ROUTE_RULES = [
  "Transformation Ontology 2 -> Transformational Geometry for parent mapping framework",
  "Transformation Ontology 2 -> Linear Transformation for origin-preserving linear maps",
  "Transformation Ontology 2 -> Affine Transformation for Ax+b maps",
  "Transformation Ontology 2 -> Coordinate Geometry for coordinate rules",
  "Transformation Ontology 2 -> Vector Geometry for vector actions and directions",
  "Transformation Ontology 2 -> Matrix for computational representation",
  "Transformation Ontology 2 -> Isomorphism or Automorphism when structure-preserving bijections are the central issue"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve transformation as a mapping.",
  "Preserve image-versus-preimage distinction.",
  "Preserve composition order.",
  "Preserve bijectivity requirements for inverses.",
  "Preserve class-specific invariants.",
  "Preserve coordinate-basis awareness for matrix forms.",
  "Preserve machine-readable transformation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A geometric transformation is a mapping that sends points or geometric entities from a source configuration to an image configuration according to a defined rule."

 IMAGE_SUMMARY =
  "Image of P under T is T(P)."

 COMPOSITION_SUMMARY =
  "(T2∘T1)(P)=T2(T1(P))."

 INVERSE_SUMMARY =
  "T^-1 exists when T is bijective."

 FIXED_POINT_SUMMARY =
  "P is fixed iff T(P)=P."

 ISOMETRY_SUMMARY =
  "An isometry preserves all distances."

 SIMILARITY_SUMMARY =
  "A similarity preserves angles and scales lengths uniformly."

 AFFINE_SUMMARY =
  "An affine transformation has form T(x)=A*x+b and preserves affine structure."
}

FLOW {
 INPUT
 -> detect_geometric_transformation_context
 -> identify source_space_target_space_entities_and_rule
 -> classify transformation_type
 -> identify image_preimage_composition_inverse_fixed_point_invariant_matrix_congruence_or_similarity goal
 -> perform transformation_specific reasoning
 -> verify domain_codomain_order_basis_invertibility_and_parameter constraints
 -> identify preserved_and_changed_geometric_properties
 -> route to transformational_geometry linear_affine_coordinate_vector_matrix_or_structure_preserving ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Transformation Ontology 2 defines geometric transformations as mappings between geometric configurations. It supports image and preimage reasoning, classification into rigid, similarity and affine transformations, composition, inverses, fixed points, invariants, coordinate and matrix representations, congruence and symmetry, while preserving explicit information about mapping order, invertibility and geometric structure."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-TRANSFORMATION-ONTOLOGY-2-V1"
}

}

END_FLOW