Translation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of geometric translation within the Kireth Research Semantic Library. A translation is an isometry that moves every point by the same displacement vector. This ontology supports translation vectors, coordinate rules, images and preimages, composition, inverses, fixed-point behaviour, invariants, matrix and homogeneous-coordinate representations, symmetry and relationships with isometries, affine transformations and vector geometry.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Translation
 SYMBOLIC_FORMS = [
  "T_v(P)=P+v",
  "(x,y)->(x+a,y+b)",
  "(x,y,z)->(x+a,y+b,z+c)"
 ]
 ENTITY_CLASS = Isometry
 PRIMARY_CONTEXT = TransformationalGeometry
 DEFINITION = "A translation is a geometric transformation that moves every point by the same displacement vector."
 SEMANTIC_ROLE = "Provides the constant-displacement isometry for coordinate, vector, congruence, composition, inverse and symmetry reasoning."
}

SCOPE {
 INCLUDE = [
  Translation,
  TranslationVector,
  DisplacementVector,
  Image,
  Preimage,
  CoordinateTranslation,
  VectorTranslation,
  PlaneTranslation,
  SpatialTranslation,
  IdentityTranslation,
  InverseTranslation,
  CompositionOfTranslations,
  TranslationalSymmetry,
  HomogeneousCoordinateRepresentation,
  Isometry,
  AffineTransformation
 ]

 EXCLUDE_AS_PRIMARY = [
  Rotation,
  Reflection,
  Scaling,
  Shear,
  GeneralAffineMapWithoutPureTranslationForm
 ]

 DISAMBIGUATION_RULE =
  "When a geometric transformation moves all points by the same vector and preserves orientation, distance and angle, interpret it as a translation."
}

CORE_CONCEPTS {

 Translation {
  definition = "An isometry of the form T_v(P)=P+v."
 }

 TranslationVector {
  symbol = "v"
  definition = "The fixed displacement applied to every point."
 }

 Image {
  definition = "For point P, the translated image is P'=P+v."
 }

 Preimage {
  definition = "For image Q under translation vector v, the preimage is Q-v."
 }

 IdentityTranslation {
  definition = "The translation with zero vector."
  condition = "v=0"
 }

 InverseTranslation {
  definition = "The translation by the opposite vector."
  form = "T_v^-1=T_-v"
 }

 CompositionOfTranslations {
  definition = "The composition of translations is another translation."
  form = "T_u∘T_v=T_(u+v)"
 }

 TranslationalSymmetry {
  definition = "A nonzero translation that maps a geometric object or pattern onto itself."
 }
}

RELATIONS {
 Translation --is_special_case_of--> Isometry
 Translation --is_special_case_of--> AffineTransformation
 Translation --uses--> TranslationVector
 Translation --maps--> Point
 Point --maps_to--> Image
 Translation --has_inverse--> InverseTranslation
 Translation --can_compose_with--> Translation
 CompositionOfTranslations --is--> Translation
 IdentityTranslation --is_special_case_of--> Translation
 Translation --preserves--> Distance
 Translation --preserves--> Angle
 Translation --preserves--> Parallelism
 Translation --preserves--> Collinearity
 Translation --preserves--> Orientation
 Translation --preserves--> Area
 Translation --preserves--> Volume
 TranslationalSymmetry --uses--> Translation
}

OPERATORS {
 TRANSLATE(P,v) :=
  P+v

 TRANSLATE_2D((x,y),(a,b)) :=
  (x+a,y+b)

 TRANSLATE_3D((x,y,z),(a,b,c)) :=
  (x+a,y+b,z+c)

 PREIMAGE(Q,v) :=
  Q-v

 INVERSE_TRANSLATION(v) :=
  -v

 COMPOSE_TRANSLATIONS(u,v) :=
  u+v

 IS_IDENTITY_TRANSLATION(v) :=
  v=0

 IS_TRANSLATIONAL_SYMMETRY(S,v) :=
  v!=0 AND TRANSLATE(S,v)=S

 TRANSLATION_DISTANCE(v) :=
  norm(v)
}

DEFINITIONS {
 D1 = "A translation moves every point by the same vector."
 D2 = "Translation preserves all distances."
 D3 = "Translation preserves all angles."
 D4 = "Translation preserves orientation."
 D5 = "Translation preserves parallelism and collinearity."
 D6 = "The inverse of translation by v is translation by -v."
 D7 = "The composition of translations by u and v is translation by u+v."
 D8 = "A translation has no fixed points unless it is the identity translation."
 D9 = "Translation by the zero vector is the identity transformation."
 D10 = "Translations form an abelian group under composition."
}

AXIOMS {
 A1_CONSTANT_DISPLACEMENT:
  For every point P, T_v(P)-P=v

 A2_DISTANCE_PRESERVATION:
  d(T_v(P),T_v(Q))=d(P,Q)

 A3_ANGLE_PRESERVATION:
  Translation preserves angle measure

 A4_ORIENTATION_PRESERVATION:
  Translation preserves orientation

 A5_INVERSE:
  T_-v∘T_v=I

 A6_COMPOSITION:
  T_u∘T_v=T_(u+v)

 A7_COMMUTATIVITY:
  T_u∘T_v=T_v∘T_u

 A8_FIXED_POINT:
  If v!=0 then T_v has no fixed points in Euclidean space
}

REASONING {

 R1_POINT_IMAGE {
  INPUT = [P,v]
  COMPUTE = P+v
  RETURN = image_point
 }

 R2_OBJECT_IMAGE {
  INPUT = [object S,v]
  APPLY = translation_to_every_point_or_vertex
  RETURN = translated_object
 }

 R3_PREIMAGE {
  INPUT = [Q,v]
  COMPUTE = Q-v
  RETURN = preimage_point
 }

 R4_COMPOSITION {
  INPUT = [T_v,T_u]
  ADD = vectors
  RETURN = T_(u+v)
 }

 R5_INVERSE {
  INPUT = T_v
  NEGATE = v
  RETURN = T_-v
 }

 R6_FIXED_POINTS {
  INPUT = v
  IF v==0
   RETURN = AllPointsFixed
  ELSE
   RETURN = NoFixedPoints
 }

 R7_INVARIANTS {
  INPUT = translated_object
  RETURN = [
   distances_preserved,
   angles_preserved,
   parallelism_preserved,
   collinearity_preserved,
   orientation_preserved,
   area_preserved,
   volume_preserved
  ]
 }

 R8_TRANSLATIONAL_SYMMETRY {
  INPUT = [object_or_pattern S,v]
  REQUIRE = v!=0
  TEST = T_v(S)=S
  RETURN = SymmetryOrNot
 }

 R9_COORDINATE_RULE {
  INPUT = vector components
  BUILD = coordinate_addition_rule
  RETURN = translation_formula
 }
}

CONSTRAINTS {
 C1 = "All points must receive the same displacement vector."
 C2 = "A nonzero translation has no fixed points in ordinary Euclidean space."
 C3 = "Translation must not be confused with rotation or reflection."
 C4 = "Composition order is harmless among translations because translation composition is commutative, but standard function notation should still be respected."
 C5 = "Coordinate components depend on the active basis."
 C6 = "Translational symmetry requires a nonzero translation vector."
 C7 = "Translation distance equals the magnitude of the translation vector."
 C8 = "Pure translation has linear part equal to the identity matrix."
}

SEMANTIC_EQUIVALENCE {
 "Translation" <=> "ConstantDisplacementIsometry"
 "TranslationVector" <=> "CommonDisplacementVector"
 "IdentityTranslation" <=> "TranslationByZeroVector"
 "InverseTranslation" <=> "TranslationByNegativeVector"
 "CompositionOfTranslations" <=> "VectorAdditionOfDisplacements"
}

VALIDATION {
 V1 = "Every point displacement equals v."
 V2 = "Pairwise distances are preserved."
 V3 = "Angles are preserved."
 V4 = "Orientation is preserved."
 V5 = "Inverse vector is -v."
 V6 = "Composition vector is u+v."
 V7 = "Nonzero translation has no fixed points."
 V8 = "Coordinate implementation matches vector addition."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRANSLATION_DEFINITION = 0.99
 IMAGE_REASONING = 0.99
 COMPOSITION_REASONING = 0.99
 INVERSE_REASONING = 0.99
 INVARIANT_REASONING = 0.99
 FIXED_POINT_REASONING = 0.99
 SYMMETRY_REASONING = 0.98

 POLICY =
  "Reduce confidence when translation vector, coordinate basis, dimension or symmetry domain are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TranslationReasoning,
  IsometryReasoning,
  CoordinateReasoning,
  VectorReasoning,
  CompositionReasoning,
  InverseReasoning,
  FixedPointReasoning,
  SymmetryReasoning,
  CongruenceReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "translation" -> Translation WHEN geometry_context
  "geometric translation" -> Translation
  "translation vector" -> TranslationVector
  "shift" -> Translation WHEN geometry_context
  "translational symmetry" -> TranslationalSymmetry
  "inverse translation" -> InverseTranslation
 }

 INTERPRETATION_ORDER = [
  detect_translation_context,
  identify source_point_or_object,
  identify translation_vector,
  identify dimension_and_coordinate_basis,
  compute image_preimage_composition_inverse_or_symmetry as requested,
  verify constant_displacement_and_invariants,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("translation")
  AND geometry_context
  THEN activate(TranslationOntology)

 E2:
  IF point_and_vector_given
  THEN compute point_plus_vector

 E3:
  IF preimage_requested
  THEN subtract translation_vector

 E4:
  IF inverse_requested
  THEN negate translation_vector

 E5:
  IF composition_of_translations_requested
  THEN add translation_vectors

 E6:
  IF fixed_points_requested
  THEN return all_points_if_zero_vector_else_none

 E7:
  IF translational_symmetry_requested
  THEN require nonzero_vector
  AND test object_invariance

 E8:
  IF matrix_form_requested
  THEN use homogeneous_affine_representation_when_translation_requires_matrix_form
}

RECURSION {
 ENTRY = Translation

 EXPAND(Translation) -> [
  TranslationVector,
  Image,
  Preimage,
  IdentityTranslation,
  InverseTranslation,
  CompositionOfTranslations,
  TranslationalSymmetry,
  CoordinateTranslation,
  VectorTranslation,
  Isometry,
  AffineTransformation
 ]

 EXPAND(TranslationVector) -> [
  Magnitude,
  Direction,
  CoordinateComponents,
  DisplacementVector
 ]

 EXPAND(CompositionOfTranslations) -> [
  VectorAddition,
  Commutativity,
  Identity,
  Inverse
 ]

 EXPAND(TranslationalSymmetry) -> [
  PeriodicPattern,
  Lattice,
  FundamentalTranslation,
  SymmetryGroup
 ]

 STOP_CONDITION =
  "Stop when the translation's vector, image, preimage, inverse, composition, invariant structure, fixed-point behaviour or symmetry property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Transformational Geometry"

 IMMEDIATE_CONTEXTS = [
  "Transformation Ontology 2",
  "Vector Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Affine Transformation Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Transformational Geometry",
  "Transformation Ontology 2",
  "Vector Geometry Ontology",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Affine Transformation Ontology",
  "Linear Transformation Ontology",
  "Matrix Ontology",
  "Automorphism Ontology"
 ]

 ROUTE_RULES = [
  "Translation -> Transformation Ontology 2 for general mapping semantics",
  "Translation -> Vector Geometry for displacement vectors",
  "Translation -> Coordinate Geometry for coordinate update rules",
  "Translation -> Affine Transformation for T(x)=x+v representation",
  "Translation -> Matrix for homogeneous-coordinate computation",
  "Translation -> Automorphism when translation acts as structure-preserving self-map"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve constant-displacement semantics.",
  "Preserve distance and angle invariance.",
  "Preserve orientation preservation.",
  "Preserve zero-vector identity case.",
  "Preserve inverse-by-negation relation.",
  "Preserve composition-by-vector-addition relation.",
  "Preserve machine-readable translation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A translation is an isometry that moves every point by the same displacement vector."

 VECTOR_FORM_SUMMARY =
  "T_v(P)=P+v"

 COORDINATE_2D_SUMMARY =
  "(x,y)->(x+a,y+b)"

 COORDINATE_3D_SUMMARY =
  "(x,y,z)->(x+a,y+b,z+c)"

 INVERSE_SUMMARY =
  "T_v^-1=T_-v"

 COMPOSITION_SUMMARY =
  "T_u∘T_v=T_(u+v)"

 FIXED_POINT_SUMMARY =
  "A nonzero Euclidean translation has no fixed points."

 INVARIANT_SUMMARY =
  "Translations preserve distance, angle, orientation, parallelism, collinearity, area and volume."
}

FLOW {
 INPUT
 -> detect_translation_context
 -> identify point_object_and_translation_vector
 -> identify coordinate_dimension_and_basis
 -> compute image_preimage_inverse_composition_or_symmetry
 -> verify same_displacement_for_all_points
 -> verify isometry_orientation_and_affine_invariants
 -> route to transformation vector coordinate affine matrix or symmetry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Translation Ontology defines the constant-displacement isometry T_v(P)=P+v. It supports coordinate and vector representations, images and preimages, inverses, composition through vector addition, fixed-point behaviour, translational symmetry and invariant reasoning while linking directly to affine, coordinate and vector geometry."
}

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

}

END_FLOW