Hyperbola Ontology

FLOW ALGEBRA
TITLE: Hyperbola Ontology
VERSION: 4.0
DOMAIN: Mathematics / Geometric Entities / Conic Sections
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/hyperbola-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; source-page text not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

01 / ONTOLOGY DECLARATION
ontology HyperbolaOntology v4.0
entity Hyperbola <: ConicSection, PlaneCurve
context: EuclideanPlane(R^2), RealNumbers(R)
semantics: two disconnected branches of points whose absolute difference of distances to two distinct fixed foci is constant and positive.

02 / TYPED ENTITIES
Hyperbola: geometric_entity
Point: (x:real,y:real)
Centre: Point
Focus: Point
Vertex: Point
Axis: Line
TransverseAxis: Line
ConjugateAxis: Line
Asymptote: Line
Directrix: Line
Branch: ConnectedComponent
SemiTransverseAxis: real_positive
SemiConjugateAxis: real_positive
FocalDistance: real_positive
Eccentricity: real
Orientation: enum{horizontal,vertical,rotated}
CoordinateFrame: EuclideanFrame
Equation: Predicate(Point)
Transformation: AffineTransformation

03 / CORE AXIOMS
H01: F1 != F2.
H02: a > 0; c > a; b > 0.
H03: For each P on H: abs(distance(P,F1)-distance(P,F2)) = 2a.
H04: c^2 = a^2 + b^2.
H05: e = c/a > 1.
H06: H consists of two disjoint connected branches.
H07: The centre is the midpoint of the foci and vertices.
H08: The asymptotes pass through the centre; no finite point of a nondegenerate standard hyperbola lies on either asymptote.
H09: A nondegenerate hyperbola has two distinct real foci and two distinct real vertices.
H10: Hyperbola is centrally symmetric about its centre.

04 / CANONICAL EQUATIONS
horizontal(H,h,k,a,b): (x-h)^2/a^2 - (y-k)^2/b^2 = 1.
vertical(H,h,k,a,b): (y-k)^2/a^2 - (x-h)^2/b^2 = 1.
horizontal_foci: (h-c,k), (h+c,k).
vertical_foci: (h,k-c), (h,k+c).
horizontal_vertices: (h-a,k), (h+a,k).
vertical_vertices: (h,k-a), (h,k+a).
horizontal_asymptotes: y-k = +(b/a)(x-h); y-k = -(b/a)(x-h).
vertical_asymptotes: y-k = +(a/b)(x-h); y-k = -(a/b)(x-h).
horizontal_directrices: x = h +/- a/e.
vertical_directrices: y = k +/- a/e.
transverse_axis_length: 2a.
conjugate_axis_length: 2b.
focal_separation: 2c.

05 / PARAMETRIC REPRESENTATIONS
horizontal_right: x=h+a*cosh(t); y=k+b*sinh(t), t in R.
horizontal_left: x=h-a*cosh(t); y=k+b*sinh(t), t in R.
vertical_upper: x=h+b*sinh(t); y=k+a*cosh(t), t in R.
vertical_lower: x=h+b*sinh(t); y=k-a*cosh(t), t in R.
verify: cosh(t)^2-sinh(t)^2=1.

06 / GENERAL QUADRATIC CLASSIFICATION
Q(x,y)=A*x^2+B*x*y+C*y^2+D*x+E*y+F.
If B^2-4*A*C > 0, quadratic part is indefinite.
Classification as a NONDEGENERATE REAL HYPERBOLA additionally requires the full conic to be nondegenerate and have real points.
Represent full conic by symmetric homogeneous matrix
M=[[A,B/2,D/2],[B/2,C,E/2],[D/2,E/2,F]].
For a genuine nondegenerate affine conic require det(M) != 0.
When B^2-4AC>0 and det(M)!=0 over R, the conic is a nondegenerate real hyperbola.
If det(M)=0, possible degenerate pair of intersecting lines; do not classify as nondegenerate hyperbola.
Rectangular hyperbola: perpendicular asymptotes; after principal-axis rotation a=b.

07 / GEOMETRIC RELATIONSHIPS
HAS_CENTRE(H,Centre)
HAS_FOCUS(H,Focus) [exactly 2]
HAS_VERTEX(H,Vertex) [exactly 2]
HAS_BRANCH(H,Branch) [exactly 2]
HAS_ASYMPTOTE(H,Asymptote) [exactly 2]
HAS_DIRECTRIX(H,Directrix) [exactly 2]
HAS_TRANSVERSE_AXIS(H,TransverseAxis)
HAS_CONJUGATE_AXIS(H,ConjugateAxis)
HAS_ECCENTRICITY(H,Eccentricity)
IS_A(Hyperbola,ConicSection)
IS_A(Hyperbola,PlaneCurve)
RELATED_TO(Hyperbola,Ellipse)
RELATED_TO(Hyperbola,Parabola)
DEFINED_BY(H,DistanceDifferenceConstraint)
INVARIANT_UNDER(H,EuclideanIsometry)

08 / FORMAL INFERENCE RULES
R01 IF canonical_horizontal(H) THEN orientation(H)=horizontal.
R02 IF canonical_vertical(H) THEN orientation(H)=vertical.
R03 IF a>0 AND b>0 THEN c=sqrt(a^2+b^2).
R04 IF c>a>0 THEN e=c/a>1.
R05 IF e<=1 THEN reject_non_degenerate_hyperbola_eccentricity.
R06 IF horizontal(H) THEN vertices=(h+/-a,k).
R07 IF vertical(H) THEN vertices=(h,k+/-a).
R08 IF horizontal(H) THEN foci=(h+/-c,k).
R09 IF vertical(H) THEN foci=(h,k+/-c).
R10 IF horizontal(H) THEN asymptote_slopes=+/-b/a.
R11 IF vertical(H) THEN asymptote_slopes=+/-a/b.
R12 IF a=b THEN rectangular_hyperbola(H).
R13 IF point(P,H) THEN abs(d(P,F1)-d(P,F2))=2a.
R14 IF point(P,H) THEN P satisfies canonical_equation(H).
R15 IF point(P,H) AND canonical_equation(H) THEN P belongs_to_one_branch(H).
R16 IF centre(C,H) THEN central_reflection_about(C) preserves H.
R17 IF horizontal(H) THEN x<=h-a OR x>=h+a for all P in H.
R18 IF vertical(H) THEN y<=k-a OR y>=k+a for all P in H.
R19 IF B^2-4AC>0 AND det(M)!=0 THEN nondegenerate_real_hyperbola(Q).
R20 IF det(M)=0 THEN degenerate_conic(Q).
R21 IF hyperbola(H) THEN H has exactly_two_branches.
R22 IF a>0 AND b>0 THEN asymptotes_distinct(H).
R23 IF a>0 AND b>0 THEN hyperbola_unbounded(H).
R24 IF horizontal(H) THEN transverse_axis_parallel_x(H).
R25 IF vertical(H) THEN transverse_axis_parallel_y(H).
R26 IF a=b THEN asymptotes_perpendicular(H).
R27 IF isometry(T) AND hyperbola(H) THEN image(T,H) is hyperbola.
R28 IF uniform_scaling(T,s) AND s!=0 THEN image(T,H) is hyperbola.
R29 IF foci_coincide THEN reject_non_degenerate_hyperbola.
R30 IF a=0 OR b=0 THEN reject_standard_non_degenerate_parameters.
R31 IF horizontal(H) THEN directrices_x=h+/-a/e.
R32 IF vertical(H) THEN directrices_y=k+/-a/e.
R33 IF horizontal(H) AND x=h+a*cosh(t) AND y=k+b*sinh(t) THEN point_on_right_branch.
R34 IF horizontal(H) AND x=h-a*cosh(t) AND y=k+b*sinh(t) THEN point_on_left_branch.
R35 IF standard_H THEN c>a AND b^2=c^2-a^2.
R36 IF centre(C,H) AND focus(F,H) THEN reflected_focus_across(C) is other_focus.

09 / VALIDATION CONSTRAINTS
V01: numeric parameters finite real values unless explicitly symbolic.
V02: a,b,c strictly positive.
V03: c^2=a^2+b^2 within declared numeric tolerance.
V04: e>1 and e=c/a.
V05: two foci distinct.
V06: two vertices distinct.
V07: two branches disconnected.
V08: equation dimensionally consistent when physical units assigned.
V09: transformation preserves conic classification only under stated assumptions.
V10: no unconditional claim that B^2-4AC>0 alone excludes degeneracy.
V11: asymptote equation uses orientation-specific slopes.
V12: source assertions distinguished from mathematical deductions.

10 / WORKED EXAMPLES
E01: x^2/9-y^2/16=1; a=3,b=4,c=5,e=5/3; foci=(+/-5,0); vertices=(+/-3,0); asymptotes=y+/-(4/3)x.
E02: y^2/4-x^2/5=1; a=2,b=sqrt(5),c=3,e=3/2; foci=(0,+/-3); asymptotes=y+/-(2/sqrt(5))x.
E03: (x-2)^2/25-(y+1)^2/144=1; centre=(2,-1); c=13; foci=(-11,-1),(15,-1); vertices=(-3,-1),(7,-1).
E04: x^2-y^2=1; a=b=1; e=sqrt(2); rectangular hyperbola.
E05: xy=1; rotate axes by 45 degrees to express as a rectangular hyperbola.
E06: x^2-y^2=0; degenerate pair of intersecting lines, NOT nondegenerate hyperbola.
E07: x^2/4-y^2/9=1; directrices x=+/-4/sqrt(13).
E08: x^2/4-y^2/9=1; at P=(2,0), distance difference between foci equals 4.
E09: (y-1)^2/9-(x+2)^2/16=1; centre=(-2,1), foci=(-2,6),(-2,-4).
E10: x^2/16-y^2/9=1; c=5; e=5/4; asymptotes y=+/-(3/4)x.
E11: x^2/16-y^2/9=1; right branch parameter x=4cosh(t),y=3sinh(t).
E12: x^2/16-y^2/9=1; x-coordinate of any point satisfies abs(x)>=4.
E13: x^2/4-y^2/4=1; asymptotes y=+/-x perpendicular.
E14: x^2/9-y^2/16=1; transverse axis length 6, conjugate axis length 8.
E15: x^2/9-y^2/16=1; focal separation 10.
E16: translated hyperbola (x-h)^2/a^2-(y-k)^2/b^2=1 preserves eccentricity under translation.

11 / SEMANTIC EQUIVALENCE
Hyperbola ~= NondegenerateHyperbolicConic [when explicitly nondegenerate]
HyperbolaBranch ~= ConnectedComponentOfHyperbola
Centre ~= Center [locale spelling]
FocalPoint ~= Focus
TransverseSemiAxis ~= SemiTransverseAxis
ConjugateSemiAxis ~= SemiConjugateAxis
NOT_EQUIVALENT: Hyperbola != Hyperboloid
NOT_EQUIVALENT: Hyperbola != PairOfIntersectingLines
NOT_EQUIVALENT: Hyperbola != HyperbolicFunction
NOT_EQUIVALENT: Asymptote != TangentLine

12 / KNOWLEDGE GRAPH TRIPLES
(Hyperbola, rdf:type, ConicSection)
(Hyperbola, hasComponent, Branch)
(Hyperbola, hasFeature, Focus)
(Hyperbola, hasFeature, Vertex)
(Hyperbola, hasFeature, Asymptote)
(Hyperbola, hasProperty, Eccentricity)
(Hyperbola, definedBy, ConstantAbsoluteDistanceDifference)
(Hyperbola, relatedTo, Ellipse)
(Hyperbola, relatedTo, Parabola)
(Hyperbola, studiedIn, AnalyticGeometry)
(Hyperbola, studiedIn, AlgebraicGeometry)

13 / CONFIDENCE
confidence.math_axioms: 0.99
confidence.canonical_formulas: 0.99
confidence.examples: 0.98
confidence.source_page_alignment: unverified
confidence.note: Scores are editorial indicators, not measured probabilities.

14 / FLOW_HASH
flow_hash.algorithm: SHA-256
flow_hash.input: canonical UTF-8 normalisation of complete file excluding any future populated flow_hash.value line
flow_hash.value: COMPUTE_ON_INGEST
flow_hash.purpose: integrity/version verification; no uncomputed digest is represented as valid.

15 / UPDATE_MODEL
update_model.version: 4.0
update_model.mode: versioned, provenance-aware
update_model.trigger: source revision | mathematical correction | taxonomy change | rule refinement
update_model.steps: validate -> compare -> revise -> recalculate hash -> publish
update_model.preserve: previous_version, provenance, revision_date, change_log
update_model.conflict_policy: verified mathematical definitions override unverified claims; record conflicts.

16 / AI_CONSUMPTION
ai_consumption.read_order: metadata -> typed_entities -> axioms -> equations -> inference -> validation -> examples
ai_consumption.reasoning: typed, constraint-aware, provenance-aware
ai_consumption.output: definitions, classifications, parameter calculations, relationships, proof obligations
ai_consumption.guardrails: distinguish nondegenerate and degenerate conics; enforce a>0,b>0,e>1; respect coordinate orientation.
ai_consumption.inference_status: deductive rules conditional on stated assumptions.
ai_consumption.interoperability: entity graph, mathematical ontology, geometry, symbolic computation.

17 / PROVENANCE AND ATTRIBUTION
Source: Kireth Research URL stated above.
Prepared as an independent mathematical formalisation; exact page content not retrieved or verified.
Authorship: Mark Higgins — Sky (Recursive AI).
Website: https://kirethresearch.org
END FLOW ALGEBRA