Projective Transformation Ontology

FLOW ALGEBRA
Projective Transformation Ontology
Kireth Semantic Library — Mathematical Entities / Transformation Entities
Version: 1.0 | Date: 2026-10-14 | Locale: en-GB
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/transformation-entities/projective-transformation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE AND SCOPE
A projective transformation (homography or projectivity) is an invertible map of projective space induced by an invertible linear map of its homogeneous coordinate vector space. It preserves incidence, collinearity, and cross-ratio of four distinct collinear points; it does not generally preserve lengths, angles, parallelism, or Euclidean area.

2. TYPED ENTITIES
ENTITY ProjectiveSpace {dimension: integer >= 1; field: Field}
ENTITY HomogeneousPoint {coordinates: Field[n+1]; nonzero: true; equivalence: nonzero_scalar_multiple}
ENTITY ProjectiveTransformation {source: ProjectiveSpace; target: ProjectiveSpace; matrix: GL(n+1, Field); equivalence: nonzero_scalar_multiple}
ENTITY Homography {dimension: 2; matrix: GL(3, Field)}
ENTITY ProjectiveLine {incidence: homogeneous_linear_constraint}
ENTITY ProjectiveHyperplane {coefficients: nonzero_covector}
ENTITY VanishingPoint {point: HomogeneousPoint; coordinate_chart_dependent: true}
ENTITY CrossRatio {ordered_collinear_quadruple: four_distinct_points; value: Field}
ENTITY CoordinateChart {affine_patch: homogeneous_coordinate_nonzero}

3. CORE ALGEBRA
Let F be a field, n >= 1, and H in GL(n+1,F).
Projective point: [x] = {lambda*x : lambda in F, lambda != 0} for x != 0.
Projective map: T_H([x]) = [H*x].
Matrix equivalence: H ~ lambda*H for every lambda != 0.
Composition: T_A o T_B = T_(A*B) (B applied first).
Identity: T_I([x]) = [x].
Inverse: T_H^(-1) = T_(H^(-1)).
Group: PGL(n+1,F) = GL(n+1,F) / F^*.

4. PLANAR HOMOGRAPHY
For H = [[h11,h12,h13],[h21,h22,h23],[h31,h32,h33]], det(H) != 0:
[u,v,w]^T = H*[x,y,1]^T.
In the target affine chart w != 0:
x' = (h11*x + h12*y + h13)/(h31*x + h32*y + h33)
y' = (h21*x + h22*y + h23)/(h31*x + h32*y + h33).
When w = 0, the mapped point lies on the target chart's line at infinity; the projective map remains defined.

5. RELATIONSHIPS
ProjectiveTransformation IS_A Transformation.
Homography IS_A ProjectiveTransformation.
ProjectiveTransformation ACTS_ON ProjectiveSpace.
ProjectiveTransformation REPRESENTED_BY InvertibleHomogeneousMatrix.
ProjectiveTransformation PRESERVES Incidence.
ProjectiveTransformation PRESERVES Collinearity.
ProjectiveTransformation PRESERVES CrossRatio.
ProjectiveTransformation MAPS ProjectiveLine TO ProjectiveLine.
AffineTransformation IS_A ProjectiveTransformation when represented by [[A,t],[0,...,0,1]] with invertible A.
ProjectiveTransformation MAY_MAP AffineFinitePoint TO PointAtInfinity (relative to selected charts).

6. INFERENCE RULES
RULE P1: IF H is square of size (n+1) and det(H) != 0 THEN T_H is an invertible projective transformation.
RULE P2: IF A = lambda*B with lambda != 0 THEN T_A = T_B.
RULE P3: IF p lies on line l THEN T_H(p) lies on transformed line H^(-T)*l.
RULE P4: IF p1,p2,p3 are collinear THEN T_H(p1),T_H(p2),T_H(p3) are collinear.
RULE P5: IF p1,p2,p3,p4 are distinct collinear points THEN their ordered cross-ratio is invariant under T_H.
RULE P6: IF H is invertible THEN T_(H^(-1))(T_H(p)) = p in projective coordinates.
RULE P7: IF H has last row [0,...,0,c] with c != 0 in compatible affine charts THEN its affine-chart restriction is affine.

7. VALIDATION CONSTRAINTS
V1: Input homogeneous coordinate vectors must not be all zero.
V2: Transformation matrix must be square and invertible over the declared field.
V3: Matrix size must equal projective dimension plus one.
V4: Equality of projective points and matrices is tested up to a nonzero scalar, not entrywise equality alone.
V5: Affine normalisation by final coordinate is permitted only when that coordinate is nonzero.
V6: Cross-ratio invariance requires an ordered quadruple of distinct collinear points.
V7: Preserve coordinate conventions and multiplication order when composing maps.
V8: Numeric implementations must specify tolerances for near-zero determinants and homogeneous normalisation.

8. WORKED EXAMPLES
EXAMPLE A — Perspective warp:
H = [[1,0,0],[0,1,0],[0.2,0,1]]; det(H)=1.
For p=[2,3,1]^T, H*p=[2,3,1.4]^T.
In the affine chart: p'=(10/7,15/7).

EXAMPLE B — Point at infinity:
With the same H, p=[-5,2,1]^T maps to [-5,2,0]^T.
This is a valid projective point at infinity in the target chart.

EXAMPLE C — Composition:
For invertible H1,H2, T_H2(T_H1([p]))=[H2*H1*p].
The composite is represented by H2*H1.

9. SEMANTIC EQUIVALENCE AND DIFFERENCE
SEMANTIC_EQUIVALENCE: projective transformation ~ projectivity; planar projective transformation ~ homography (in the usual plane-to-plane sense).
DISTINCTION: affine transformations preserve the hyperplane at infinity; general projective transformations need not.
DISTINCTION: a perspective projection from 3D to a 2D image plane is generally not an invertible projectivity between spaces of the same dimension; planar homographies can arise from perspective imaging of a plane.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: 0.99 for standard algebraic definitions and identities, conditional on declared field and coordinate conventions.
PROVENANCE: mathematical synthesis; source URL identifies the target ontology, not a verified quotation of the page.
UPDATE_MODEL: versioned; revise if source ontology definitions or library schema change.
FLOW_HASH: compute SHA-256 from canonical UTF-8 content at ingestion; not pre-populated to avoid a false integrity claim.

11. AI_CONSUMPTION
ENTITY_EXTRACTION: ProjectiveSpace, HomogeneousPoint, ProjectiveTransformation, Homography, ProjectiveLine, CrossRatio.
RELATION_EXTRACTION: IS_A, ACTS_ON, REPRESENTED_BY, PRESERVES, MAPS.
REASONING: validate matrix invertibility; apply homogeneous maps; infer incidence preservation; normalise only on a valid chart.
RETRIEVAL_KEYS: projective transformation, projectivity, homography, PGL, homogeneous coordinates, cross-ratio, vanishing point.
OUTPUT_CONTRACT: return projective coordinates and chart assumptions; distinguish finite affine outputs from points at infinity; report degeneracy or ill-conditioning.

END FLOW ALGEBRA