Translation Transformation Ontology

FLOW ALGEBRA
TITLE: Translation Transformation Ontology
VERSION: 1.0
DOMAIN: Mathematical Entities / Transformation Entities
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/transformation-entities/translation-transformation-ontology.html
STATUS: Proposed formalisation; source-page wording not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. ENTITY DEFINITION
entity TranslationTransformation : AffineTransformation
  parameters: translation_vector t in V
  domain: affine_space A modelled on vector_space V
  codomain: A
  mapping: T_t(p) = p + t
  interpretation: every point is displaced by the same vector

2. TYPE SYSTEM
  AffineSpace(A,V)
  Point(p,A)
  Vector(t,V)
  Translation(T_t,A,V)
  ImagePoint(q,A)
  Composition(T_s,T_t)
  Inverse(T_t,T_-t)
  Displacement(p,q,t)

3. AXIOMS AND INVARIANTS
  A1: forall p in A, T_0(p) = p.
  A2: forall p in A, T_t(p) = p + t.
  A3: forall s,t in V, T_s(T_t(p)) = T_(s+t)(p).
  A4: T_t is bijective, with inverse T_(-t).
  A5: forall p,q in A, T_t(q)-T_t(p) = q-p.
  A6: In a Euclidean affine space, distances and angles are preserved.
  A7: Translation preserves collinearity, parallelism and affine ratios.
  A8: Nonzero translations have no fixed points in an ordinary affine space.
  A9: Translation maps lines to parallel lines and preserves orientation.

4. COORDINATE REPRESENTATIONS
  Cartesian R^n: x' = x + t.
  R^2: (x,y) -> (x+a,y+b).
  R^3: (x,y,z) -> (x+a,y+b,z+c).
  Homogeneous coordinates: [x';1] = [[I,t],[0,1]] [x;1].
  Note: A translation is affine, not generally linear when t != 0.

5. TYPED RELATIONSHIPS
  TranslationTransformation --IS_A--> AffineTransformation
  TranslationTransformation --HAS_PARAMETER--> TranslationVector
  TranslationTransformation --MAPS--> Point
  TranslationTransformation --PRESERVES--> DisplacementVector
  TranslationTransformation --PRESERVES--> EuclideanDistance [Euclidean context]
  TranslationTransformation --COMPOSES_WITH--> TranslationTransformation
  TranslationTransformation --HAS_INVERSE--> InverseTranslation
  TranslationTransformation --REPRESENTED_BY--> HomogeneousTransformationMatrix
  TranslationTransformation --DIFFERS_FROM--> RotationTransformation
  TranslationTransformation --DIFFERS_FROM--> ScalingTransformation

6. EXECUTABLE FLOW RULES (PSEUDOCODE)
  flow ApplyTranslation(p:Point[A], t:Vector[V]) -> Point[A]:
    require compatible(A,V,p,t)
    return p + t

  flow ComposeTranslations(t:Vector[V], s:Vector[V]) -> Vector[V]:
    return t + s

  flow InvertTranslation(t:Vector[V]) -> Vector[V]:
    return -t

  flow ValidateTranslation(points:list[Point[A]], images:list[Point[A]]) -> bool:
    require len(points) == len(images) and len(points) > 0
    d = images[0] - points[0]
    return all(images[i] - points[i] == d for i in indices(points))

7. INFERENCE RULES
  R1: Translation(T_t) AND Point(p) => Image(T_t(p),p+t).
  R2: Translation(T_s) AND Translation(T_t) => CompositeTranslation(T_(s+t)).
  R3: Translation(T_t) => InverseTranslation(T_(-t)).
  R4: Translation(T_t) AND EuclideanMetric => Isometry(T_t).
  R5: Translation(T_t) AND t != 0 => NoFixedPoints(T_t).
  R6: EqualDisplacementAcrossAllPoints => CandidateTranslation [verify domain and mapping].

8. SEMANTIC EQUIVALENCE AND DISTINCTIONS
  SEMANTIC_EQUIVALENCE:
    translation transformation ~ geometric translation ~ displacement mapping [in affine geometry]
  NOT_EQUIVALENT:
    translation != rotation
    translation != scaling
    translation != arbitrary affine transformation
    translation != language translation
  CONTEXT: Interpret 'translation' as spatial/mathematical unless otherwise typed.

9. VALIDATION
  V1: Confirm points and translation vectors share compatible affine/vector spaces.
  V2: Confirm all image-minus-source displacements equal t.
  V3: Check inverse composition yields identity.
  V4: Check composition law for representative inputs.
  V5: Do not assert distance preservation without an appropriate metric.
  V6: Use exact arithmetic where possible; otherwise record numeric tolerance.

10. CONFIDENCE AND PROVENANCE
  CONFIDENCE: 0.98 for standard mathematical identities; not a source-fidelity score.
  PROVENANCE: Standard affine geometry; source URL identifies ontology topic.
  SOURCE_VERIFICATION: Not independently verified against complete source page.

11. FLOW_HASH AND UPDATE_MODEL
  FLOW_HASH: SHA-256 of canonical UTF-8 file content excluding the FLOW_HASH field itself; compute on publication.
  UPDATE_MODEL:
    version: 1.0
    change_policy: increment version on altered definitions, axioms, rules or semantics
    dependency_review: affine-space, vector, matrix, isometry and transformation ontologies

12. AI_CONSUMPTION
  entity_type: TranslationTransformation
  preferred_relation_format: typed subject-predicate-object triples
  required_context: affine space, vector space, displacement vector
  reasoning_mode: deterministic symbolic inference with type checking
  validation_mode: axiom and invariant verification
  avoid: conflating affine translation with a linear map on vectors

END FLOW ALGEBRA