Plane Ontology

FLOW ALGEBRA
ONTOLOGY: Plane Ontology
VERSION: 4.0
DOMAIN: Mathematics / Entity Graph / Geometric Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/plane-ontology.html
SOURCE_ALIGNMENT: Independent mathematical formalisation; source page not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SITE: https://kirethresearch.org
DATE: 2026-10-13

[01] PURPOSE
Represent planes as typed geometric entities with explicit assumptions about ambient space, incidence, dimension, metrics, coordinate frames and inference validity. Separate Euclidean affine planes from abstract projective planes and from bounded planar regions.

[02] ENTITY MODEL
ENTITY Plane: geometric two-dimensional flat affine subset of a specified ambient affine space.
ENTITY AmbientSpace: affine space A over field F with translation vector space V.
ENTITY Point: element of an affine space.
ENTITY DirectionSubspace: two-dimensional linear subspace W of V.
ENTITY PlaneOrigin: point p0 in A.
ENTITY NormalVector: nonzero vector n perpendicular to plane, for Euclidean 3-space.
ENTITY PlaneEquation: affine linear equation with nonzero coefficient vector.
ENTITY CoordinateFrame: origin and ordered independent direction vectors.
ENTITY Line: one-dimensional affine subspace.
ENTITY LineSegment: bounded portion of a line.
ENTITY HalfSpace: one side of a plane in Euclidean 3-space.
ENTITY Intersection: common points of geometric entities.
ENTITY Distance: nonnegative metric quantity, defined only with a metric.
ENTITY Angle: metric-dependent measure between lines or planes.
ENTITY PlanarRegion: subset of a plane, not synonymous with the plane.
ENTITY ProjectivePlane: distinct incidence-geometric structure; not assumed affine or metric.

[03] TYPES AND SIGNATURES
Field F; AffineSpace A(F,V); VectorSpace V(F).
Plane(A) = {p0 + w | w in W}, where p0 in A and W <= V with dim(W)=2.
Point(A) = p in A.
Line(A) = {p0 + t*u | t in F, u != 0}.
contains: Plane x Point -> Boolean.
parallel: Plane x Plane -> Boolean, relative to a common affine ambient space.
intersects: Plane x Plane -> Boolean.
normal: Plane(Euclidean3) -> Vector3 \ {0}, defined up to nonzero scale.
distance: Point x Plane -> RealNonnegative, when Euclidean metric available.

[04] CORE AXIOMS
AX-01: A plane is an affine translate of a two-dimensional vector subspace.
AX-02: Any plane has infinitely many points when its base field is infinite.
AX-03: If p,q lie on plane P, the affine line through distinct p,q lies in P.
AX-04: Three non-collinear points determine exactly one plane in Euclidean 3-space.
AX-05: Two distinct intersecting lines determine exactly one plane.
AX-06: Two distinct parallel lines determine exactly one plane in Euclidean 3-space.
AX-07: An affine plane is unbounded over R; a planar region may be bounded.
AX-08: A Euclidean 3-space plane has codimension one.
AX-09: A plane has no unique normal orientation unless an orientation is specified.
AX-10: Coincident planes represent the same geometric point set, despite different equations.

[05] REPRESENTATIONS
REP-01 Parametric: P = {p0 + s*u + t*v | s,t in R}; u,v linearly independent.
REP-02 Cartesian: ax+by+cz=d; (a,b,c) != (0,0,0).
REP-03 Point-normal: n dot (x-p0)=0; n != 0.
REP-04 Implicit: f(x,y,z)=0 with f affine linear and nonzero linear part.
REP-05 Coordinate: z=0 is a special case, not the definition of every plane.
REP-06 General affine: P=p0+W with dim(W)=2, applicable beyond R^3.
REP-07 Homogeneous coordinates: ax+by+cz-dw=0 for affine plane's projective closure, with w != 0 for affine points.

[06] RELATIONSHIPS
REL-01 Plane IS_A GeometricEntity.
REL-02 Plane IS_A AffineSubspace.
REL-03 Plane HAS_DIMENSION 2.
REL-04 Plane IS_EMBEDDED_IN AmbientSpace.
REL-05 Plane CONTAINS Point.
REL-06 Plane CONTAINS Line when all line points lie in plane.
REL-07 Plane HAS_DIRECTION_SUBSPACE W.
REL-08 Plane HAS_PARAMETRIC_REPRESENTATION PlaneEquation.
REL-09 Plane HAS_CARTESIAN_REPRESENTATION PlaneEquation in R^3.
REL-10 Plane MAY_HAVE NormalVector when a Euclidean inner product and codimension one are available.
REL-11 Plane MAY_INTERSECT Plane.
REL-12 Plane MAY_BE_PARALLEL_TO Plane.
REL-13 Plane MAY_BE_PERPENDICULAR_TO Plane when metric is defined.
REL-14 Plane MAY_BOUND HalfSpace in Euclidean 3-space.
REL-15 PlanarRegion IS_SUBSET_OF Plane.
REL-16 ProjectivePlane RELATED_TO Plane but NOT_SEMANTICALLY_IDENTICAL_TO EuclideanAffinePlane.

[07] FORMAL INFERENCE RULES
RULE-01 IF u,v independent THEN span(u,v) has dimension 2.
RULE-02 IF P=p0+span(u,v) AND u,v independent THEN P is a plane.
RULE-03 IF p in P AND w in direction(P) THEN p+w in P.
RULE-04 IF p,q in P AND p != q THEN line(p,q) subset P.
RULE-05 IF p,q,r are non-collinear THEN exists unique plane containing all three.
RULE-06 IF p,q,r are collinear THEN they do not determine a unique plane in R^3.
RULE-07 IF n != 0 THEN n dot (x-p0)=0 defines a plane in R^3.
RULE-08 IF a=b=c=0 THEN ax+by+cz=d is not a plane equation.
RULE-09 IF n1,n2 proportional AND equations consistent THEN planes coincide.
RULE-10 IF n1,n2 proportional AND equations inconsistent THEN planes are distinct and parallel.
RULE-11 IF n1,n2 nonproportional THEN two planes in R^3 intersect in a line.
RULE-12 IF P,Q are distinct parallel planes THEN P intersect Q = empty.
RULE-13 IF P=Q THEN P intersect Q=P.
RULE-14 IF P,Q distinct and intersecting in R^3 THEN intersection dimension=1.
RULE-15 IF n1 dot n2=0 AND both nonzero THEN planes are perpendicular.
RULE-16 IF n1 cross n2 != 0 THEN intersection-line direction is parallel to n1 cross n2.
RULE-17 IF n dot (p-p0)=0 THEN p lies on plane with normal n through p0.
RULE-18 IF n dot (p-p0)!=0 THEN p lies off that plane.
RULE-19 IF P defined by ax+by+cz=d THEN point-plane distance = abs(ax0+by0+cz0-d)/sqrt(a^2+b^2+c^2).
RULE-20 IF n is multiplied by nonzero scalar THEN its plane equation describes same plane when d scaled identically.
RULE-21 IF p0 replaced by another point of P THEN point-normal equation defines same P.
RULE-22 IF u,v replaced by an invertible linear combination THEN parametric representation defines same P.
RULE-23 IF an isometry T acts on P THEN T(P) is a plane.
RULE-24 IF an invertible affine transformation T acts on P THEN T(P) is a plane.
RULE-25 IF an affine map is non-injective THEN image of a plane may have dimension 0,1,or 2.
RULE-26 IF a planar subset is bounded THEN it is not the entire real affine plane.
RULE-27 IF a line has direction u in direction(P) AND one of its points is in P THEN line subset P.
RULE-28 IF line direction u has n dot u != 0 THEN line intersects plane in exactly one point.
RULE-29 IF line direction u has n dot u=0 AND a point lies on plane THEN line lies in plane.
RULE-30 IF line direction u has n dot u=0 AND a point lies off plane THEN line and plane are disjoint.
RULE-31 IF P is plane in R^3 THEN P is closed and connected in Euclidean topology.
RULE-32 IF P is plane in R^3 THEN P has zero 3-dimensional Lebesgue measure.
RULE-33 IF P is Euclidean plane THEN P is intrinsically flat with zero Gaussian curvature.
RULE-34 IF a,b are distinct points of P THEN their midpoint lies in P.
RULE-35 IF p,q in P THEN all affine combinations (1-t)p+tq lie in P.
RULE-36 IF two nonzero normals are proportional THEN their planes are parallel or coincident.

[08] VALIDATION CONSTRAINTS
CHECK-01: Ambient space and field must be declared.
CHECK-02: A plane requires affine dimension exactly two.
CHECK-03: Parametric direction vectors must be independent.
CHECK-04: Cartesian normal must be nonzero.
CHECK-05: Distance and angle rules require an inner product or metric.
CHECK-06: Equation comparison must be invariant under nonzero scalar multiplication.
CHECK-07: Avoid conflating geometric plane with drawing surface, bounded region or projective plane.
CHECK-08: Three-point construction requires non-collinearity.
CHECK-09: Parallel relation must distinguish strict parallelism from coincidence when required.
CHECK-10: Check floating-point tolerances separately from exact symbolic equality.
CHECK-11: Normal-vector claims in higher-dimensional ambient spaces require codimension-one qualification.
CHECK-12: Projective incidence axioms must not be imported automatically into affine Euclidean geometry.

[09] MATHEMATICAL EXAMPLES
EX-01: z=0 is xy-plane with normal (0,0,1).
EX-02: x=0 is yz-plane with normal (1,0,0).
EX-03: y=0 is xz-plane with normal (0,1,0).
EX-04: x+y+z=1 is plane through (1,0,0),(0,1,0),(0,0,1).
EX-05: x+y+z=1 and 2x+2y+2z=2 are coincident.
EX-06: x+y+z=1 and x+y+z=2 are distinct parallel planes.
EX-07: x=0 and y=0 intersect in z-axis.
EX-08: z=0 and x=0 are perpendicular planes.
EX-09: P=(1,2,3)+s(1,0,0)+t(0,1,0) is z=3.
EX-10: Three points (0,0,0),(1,0,0),(2,0,0) do not determine a unique plane.
EX-11: Distance from (0,0,2) to z=0 equals 2.
EX-12: The set x^2+y^2<=1,z=0 is a planar disk, not an entire plane.
EX-13: The zero set x^2+y^2+z^2=1 is a sphere, not a plane.
EX-14: The image of z=0 under translation (0,0,5) is z=5.
EX-15: The image of z=0 under invertible rotation is a plane.
EX-16: In R^4, {(s,t,0,0):s,t in R} is a plane with codimension two.

[10] SEMANTIC EQUIVALENCE
EQUIVALENT: affine plane; two-dimensional affine subspace (when ambient space specified).
CONTEXTUAL_ALIAS: Euclidean plane (when Euclidean metric and ambient setting specified).
REPRESENTATIONAL_EQUIVALENCE: point-normal, implicit linear, and parametric forms when nondegenerate and describing same point set.
NOT_EQUIVALENT: half-plane; bounded planar region; plane curve; projective plane; arbitrary two-dimensional manifold; hyperplane in ambient dimension >3.
SEMANTIC_EQUIVALENCE_RULE: Same ambient space and same set of points, regardless of valid equation scaling or coordinate change.

[11] KNOWLEDGE GRAPH TRIPLES
(Plane, rdf:type, GeometricEntity)
(Plane, rdf:type, AffineSubspace)
(Plane, hasAffineDimension, 2)
(Plane, definedBy, PointAndTwoIndependentDirections)
(Plane, contains, Point)
(Plane, contains, Line)
(Plane, mayIntersect, Plane)
(Plane, mayBeParallelTo, Plane)
(Plane, mayHaveNormal, NormalVector)
(Plane, bounds, HalfSpace)
(PlanarRegion, subsetOf, Plane)
(ProjectivePlane, distinctFrom, EuclideanAffinePlane)

[12] CONFIDENCE
MATHEMATICAL_DEFINITIONS: 0.99 (standard, subject to stated assumptions)
FORMAL_RULES: 0.98 (assumption-sensitive)
SOURCE_PAGE_ALIGNMENT: unverified
INFERENCE_POLICY: confidence is not a substitute for mathematical proof; validate premises and algebra.

[13] FLOW_HASH
ALGORITHM: SHA-256
CANONICALISATION: UTF-8; Unicode NFC; LF line endings; hash canonical body excluding the FLOW_HASH value itself.
FLOW_HASH_VALUE: COMPUTE_ON_PUBLICATION
VERSION_ID: plane-ontology-flow-algebra-v4.0

[14] UPDATE_MODEL
TRIGGER: source ontology changes; corrections to mathematics; new relationships; schema migration.
PROCESS: fetch source; compare semantic definitions; verify premises; revise affected rules; rerun examples; increment version; regenerate hash; publish changelog.
CONFLICT_POLICY: preserve mathematical truth; mark source discrepancies for review rather than silently propagating them.

[15] AI_CONSUMPTION
INPUT: user query, entity type, ambient space, coordinate representation, optional metric.
NORMALISE: identify affine versus projective interpretation; canonicalise equations and variable names.
RESOLVE: map entity to plane, point, line, normal, coordinate frame or planar region.
VALIDATE: test nondegeneracy, field, dimension and metric prerequisites.
INFER: apply only rules with satisfied premises; record inference chain.
OUTPUT: structured entities, relationships, derived facts, confidence and source provenance.
FAIL_SAFE: if assumptions are absent or ambiguous, request clarification or label conclusions conditional.

[16] PROVENANCE AND LICENCE
PROJECT: Kireth Semantic Library
AUTHORS: Mark Higgins — Sky (Recursive AI)
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/plane-ontology.html
STATUS: Independently prepared; source text not retrieved.
END FLOW ALGEBRA