Transformational Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the Transformational Geometry layer within the Kireth Research Semantic Library. Transformational geometry studies geometric mappings that move, resize, reflect, rotate, shear or otherwise transform geometric objects while tracking preserved and altered properties. This ontology layer covers rigid motions, similarities, affine transformations, coordinate transformations, compositions, inverses, fixed points, invariants, matrices and symmetry. It is designed for semantic retrieval, recursive geometric reasoning, ontology routing, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = TransformationalGeometry
 ENTITY_CLASS = GeometricMappingFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Transformational geometry is the study of geometric objects through transformations and the properties that are preserved or changed under those mappings."
 SEMANTIC_ROLE = "Provides the mapping-centred framework for translation, rotation, reflection, scaling, similarity, affine transformation, symmetry, composition and invariance reasoning."
}

SCOPE {
 INCLUDE = [
  Transformation,
  GeometricTransformation,
  Translation,
  Rotation,
  Reflection,
  GlideReflection,
  Scaling,
  Dilation,
  SimilarityTransformation,
  RigidTransformation,
  Isometry,
  AffineTransformation,
  Shear,
  Projection,
  CoordinateTransformation,
  Composition,
  InverseTransformation,
  IdentityTransformation,
  FixedPoint,
  Invariant,
  Symmetry,
  MatrixRepresentation
 ]

 EXCLUDE_AS_PRIMARY = [
  ArbitraryFunctionWithoutGeometricMeaning,
  PureTopologicalMapWithoutGeometryContext,
  NoninvertibleDataTransformationWithoutGeometricInterpretation
 ]

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

CORE_CONCEPTS {

 Transformation {
  definition = "A mapping from geometric points or objects in a source space to points or objects in a target space."
  symbolic_form = "T:X->Y"
 }

 Image {
  definition = "The result T(P) or T(S) of applying a transformation to a point or geometric object."
 }

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

 IdentityTransformation {
  definition = "A transformation I satisfying I(P)=P for every point P."
 }

 InverseTransformation {
  definition = "A transformation T^-1 satisfying T^-1(T(P))=P when T is bijective."
 }

 Composition {
  definition = "Sequential application of transformations."
  form = "(T2∘T1)(P)=T2(T1(P))"
 }

 Isometry {
  definition = "A transformation preserving distances."
  examples = [Translation, Rotation, Reflection, GlideReflection]
 }

 SimilarityTransformation {
  definition = "A transformation preserving shape and angles while scaling all lengths by a common positive factor."
 }

 AffineTransformation {
  definition = "A map of the form T(x)=A*x+b preserving lines, parallelism and affine combinations."
 }

 Translation {
  form = "T(x)=x+v"
 }

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

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

 Scaling {
  definition = "A transformation multiplying distances from a centre or origin by a scale factor."
 }

 Shear {
  definition = "An affine transformation displacing points in directions proportional to another coordinate."
 }

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

 Invariant {
  definition = "A property unchanged under a specified transformation."
 }
}

RELATIONS {
 TransformationalGeometry --is_subdomain_of--> Geometry
 Transformation --maps--> Point
 Transformation --maps--> GeometricObject
 Transformation --produces--> Image
 Image --has_preimage_under--> Transformation
 IdentityTransformation --is_special_case_of--> Transformation
 Isometry --is_special_case_of--> Transformation
 SimilarityTransformation --is_special_case_of--> Transformation
 AffineTransformation --is_special_case_of--> Transformation
 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 --preserves--> Angles
 Isometry --preserves--> Distance
 AffineTransformation --preserves--> Collinearity
 AffineTransformation --preserves--> Parallelism
 Transformation --can_have--> FixedPoint
 Transformation --can_be_composed_with--> Transformation
 BijectiveTransformation --can_have--> InverseTransformation
 Symmetry --is_described_by--> Transformation
}

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

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

 COMPOSE(T2,T1) :=
  T2∘T1

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

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

 IS_INVARIANT(property,T) :=
  property(T(S))==property(S)

 TRANSLATE(P,v) :=
  P+v

 ROTATE_2D(P,C,theta) :=
  C + R(theta)*(P-C)

 REFLECT(P,MIRROR) :=
  geometric_reflection(P,MIRROR)

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

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

 TRANSFORM_MATRIX(T) :=
  matrix_or_homogeneous_matrix_representation(T)
}

DEFINITIONS {
 D1 = "A transformation maps geometric entities to geometric entities."
 D2 = "An image is the transformed entity."
 D3 = "An isometry preserves distance."
 D4 = "Rigid motions preserve distance and angle."
 D5 = "A similarity transformation preserves angles and scales lengths uniformly."
 D6 = "An affine transformation preserves lines, collinearity, parallelism and ratios along a line."
 D7 = "Composition applies transformations in sequence."
 D8 = "An invertible transformation has an inverse."
 D9 = "A fixed point is unchanged by the transformation."
 D10 = "A geometric invariant is preserved under the specified transformation class."
}

AXIOMS {
 A1_MAPPING:
  A geometric transformation assigns each source point a target point according to a defined rule

 A2_IDENTITY:
  Identity transformation leaves every point fixed

 A3_COMPOSITION:
  Composition of compatible geometric transformations is associative

 A4_INVERSE:
  A bijective transformation possesses an inverse

 A5_ISOMETRY:
  Isometries preserve pairwise distances

 A6_SIMILARITY:
  Similarity transformations preserve angles and uniform length ratios

 A7_AFFINE:
  Affine transformations preserve affine combinations

 A8_STRUCTURE_AWARENESS:
  Preserved properties depend on transformation class
}

REASONING {

 R1_CLASSIFY_TRANSFORMATION {
  INPUT = mapping_rule
  TEST = distance_preservation
  TEST = angle_preservation
  TEST = affine_form
  IDENTIFY = translation_rotation_reflection_scaling_shear_similarity_or_affine
  RETURN = transformation_type
 }

 R2_IMAGE {
  INPUT = [T,geometric_entity]
  APPLY = transformation_rule
  RETURN = image
 }

 R3_COMPOSITION {
  INPUT = [T1,T2]
  SUBSTITUTE = output_of_T1_into_T2
  SIMPLIFY = composite_rule
  RETURN = T2∘T1
 }

 R4_INVERSE {
  INPUT = T
  TEST = bijectivity_or_invertible_matrix
  SOLVE = original_coordinates_in_terms_of_image
  RETURN = inverse_transformation
 }

 R5_FIXED_POINTS {
  INPUT = T
  SOLVE = T(P)=P
  RETURN = fixed_point_set
 }

 R6_INVARIANTS {
  INPUT = [T,property_set]
  CLASSIFY = transformation_class
  TEST = each_property
  RETURN = preserved_and_changed_properties
 }

 R7_MATRIX_REASONING {
  INPUT = linear_or_affine_transformation
  REPRESENT = matrix_or_homogeneous_matrix
  COMPUTE = determinant_eigenstructure_or_composition_as_required
  RETURN = geometric_interpretation
 }

 R8_SYMMETRY {
  INPUT = geometric_object S
  FIND = transformations T such_that T(S)=S
  CLASSIFY = reflection_rotation_translation_glide_or_other_symmetry
  RETURN = symmetry_group_or_generators
 }

 R9_CONGRUENCE_SIMILARITY {
  INPUT = [figureA,figureB]
  SEARCH = isometry_or_similarity_mapping
  IF isometry_exists
   RETURN Congruent
  ELSE IF similarity_exists
   RETURN Similar
  ELSE
   RETURN NeitherEstablished
 }
}

CONSTRAINTS {
 C1 = "Transformation order matters in composition."
 C2 = "Not every transformation is invertible."
 C3 = "Affine transformations do not generally preserve distances or angles."
 C4 = "Similarity transformations preserve angles but not absolute lengths unless scale factor is one."
 C5 = "Reflections reverse orientation."
 C6 = "Rotations require a centre or axis and an angle."
 C7 = "Scaling requires a centre or origin and scale factor."
 C8 = "Matrix representation depends on coordinate basis."
 C9 = "Invariants must always be stated relative to a transformation class."
}

SEMANTIC_EQUIVALENCE {
 "RigidTransformation" <=> "Isometry"
 "Dilation" <=> "UniformScalingAboutCentre"
 "Image" <=> "TransformedGeometricEntity"
 "FixedPoint" <=> "InvariantPoint"
 "AffineTransformation" <=> "LinearMapPlusTranslation"
 "SymmetryTransformation" <=> "TransformationLeavingObjectInvariant"
}

VALIDATION {
 V1 = "Identity leaves every point unchanged."
 V2 = "Composition order is respected."
 V3 = "Inverse satisfies T^-1∘T=I and T∘T^-1=I where defined."
 V4 = "Isometries preserve distances."
 V5 = "Similarities preserve angles and scale lengths uniformly."
 V6 = "Affine transformations preserve collinearity and parallelism."
 V7 = "Fixed points satisfy T(P)=P."
 V8 = "Matrix and coordinate implementations match geometric definitions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRANSFORMATION_DEFINITION = 0.99
 ISOMETRY_REASONING = 0.99
 SIMILARITY_REASONING = 0.99
 AFFINE_REASONING = 0.99
 COMPOSITION_REASONING = 0.99
 INVERSE_REASONING = 0.99
 SYMMETRY_REASONING = 0.98

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

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TransformationReasoning,
  CongruenceReasoning,
  SimilarityReasoning,
  AffineReasoning,
  SymmetryReasoning,
  MatrixReasoning,
  FixedPointReasoning,
  CoordinateReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "transformational geometry" -> TransformationalGeometry
  "transformation geometry" -> TransformationalGeometry
  "rigid transformation" -> Isometry
  "rigid motion" -> Isometry
  "dilation" -> Scaling
  "similarity transformation" -> SimilarityTransformation
  "affine transformation" -> AffineTransformation
  "image point" -> Image
  "fixed point" -> FixedPoint
 }

 INTERPRETATION_ORDER = [
  detect_transformational_geometry_context,
  identify source_target_space_and_objects,
  identify transformation_rule,
  classify transformation_type,
  identify centre_axis_vector_scale_matrix_or_fixed_data,
  compute image composition inverse or invariant as requested,
  interpret geometric properties preserved_or_changed,
  verify transformation_constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("transformational geometry")
  OR geometric_mapping_context
  THEN activate(TransformationalGeometry)

 E2:
  IF mapping_preserves_distance
  THEN classify(Isometry)

 E3:
  IF mapping_preserves_angles_and_uniform_scale
  THEN classify(SimilarityTransformation)

 E4:
  IF map_has_form_Ax_plus_b
  THEN classify(AffineTransformation)

 E5:
  IF composition_requested
  THEN respect right_to_left function_application_order

 E6:
  IF inverse_requested
  THEN verify_bijectivity_or_matrix_invertibility

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

 E8:
  IF symmetry_requested
  THEN find transformations leaving_object_invariant

 E9:
  IF matrix_representation_used
  THEN track active_coordinate_basis
}

RECURSION {
 ENTRY = TransformationalGeometry

 EXPAND(TransformationalGeometry) -> [
  Transformation,
  Translation,
  Rotation,
  Reflection,
  GlideReflection,
  Scaling,
  SimilarityTransformation,
  Isometry,
  AffineTransformation,
  Shear,
  Composition,
  InverseTransformation,
  FixedPoint,
  Invariant,
  Symmetry,
  MatrixRepresentation
 ]

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

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

 EXPAND(AffineTransformation) -> [
  LinearTransformation,
  Translation,
  Shear,
  Scaling,
  ParallelismInvariant,
  CollinearityInvariant,
  BarycentricInvariant
 ]

 EXPAND(Symmetry) -> [
  ReflectionSymmetry,
  RotationalSymmetry,
  TranslationalSymmetry,
  GlideReflectionSymmetry,
  SymmetryGroup
 ]

 STOP_CONDITION =
  "Stop when the transformation type, image, composition, inverse, fixed-point set, invariant structure, congruence, similarity or symmetry property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

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

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Plane Geometry",
  "Solid Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Vector Geometry Ontology",
  "Transformation Ontology",
  "Linear Transformation Ontology",
  "Affine Transformation Ontology",
  "Matrix Ontology",
  "Symmetry Ontologies"
 ]

 ROUTE_RULES = [
  "Transformational Geometry -> Geometry for core entity semantics",
  "Transformational Geometry -> Plane Geometry for 2D mappings",
  "Transformational Geometry -> Solid Geometry for 3D mappings",
  "Transformational Geometry -> Coordinate Geometry for coordinate rules",
  "Transformational Geometry -> Vector Geometry for direction and matrix actions",
  "Transformational Geometry -> Linear Transformation for origin-preserving linear maps",
  "Transformational Geometry -> Affine Transformation for Ax+b mappings",
  "Transformational Geometry -> Matrix for computational representation",
  "Transformational Geometry -> Symmetry for object-invariant transformation groups"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve mapping-centred geometric semantics.",
  "Preserve transformation-class-specific invariants.",
  "Preserve composition order.",
  "Preserve invertibility requirements for inverse transformations.",
  "Preserve coordinate-basis awareness for matrix forms.",
  "Preserve distinction among isometry, similarity and affine transformation.",
  "Preserve machine-readable transformational-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Transformational geometry studies geometric objects through mappings and the properties preserved or changed by those mappings."

 ISOMETRY_SUMMARY =
  "Translations, rotations, reflections and glide reflections preserve distance."

 SIMILARITY_SUMMARY =
  "Similarity transformations preserve angles and scale all lengths uniformly."

 AFFINE_SUMMARY =
  "Affine transformations have form T(x)=A*x+b and preserve lines, collinearity and parallelism."

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

 FIXED_POINT_SUMMARY =
  "Fixed points satisfy T(P)=P."

 SYMMETRY_SUMMARY =
  "A symmetry is a transformation that maps a geometric object onto itself."
}

FLOW {
 INPUT
 -> detect_transformational_geometry_context
 -> identify source_target_geometry_and_objects
 -> identify transformation_rule_and_parameters
 -> classify translation_rotation_reflection_scaling_isometry_similarity_affine_or_other
 -> identify image_composition_inverse_fixed_point_invariant_symmetry_or_matrix goal
 -> perform transformation_specific reasoning
 -> verify coordinate_basis_invertibility_order_and_invariant constraints
 -> route to plane_solid_coordinate_vector_linear_affine_matrix_or_symmetry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Transformational Geometry provides the mapping-based branch of geometry, describing how points and figures move or change under translations, rotations, reflections, scalings, similarities and affine maps. It supports image and preimage reasoning, composition, inverses, fixed points, invariants, congruence, similarity, symmetry and matrix representations across plane and solid geometry."
}

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

}

END_FLOW