Parabola Ontology

FLOW ALGEBRA
TITLE: Parabola Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Geometric Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/parabola-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SITE: https://kirethresearch.org
STATUS: Independently formalised; source-page content not verified.

[ONTOLOGY]
ID: geometry.parabola
TYPE: ConicSection
DEFINITION: The locus of points in a Euclidean plane equidistant from a fixed focus point and a fixed directrix line not containing the focus.
PARENT: ConicSection
RELATED: Point, Line, Curve, QuadraticFunction, Focus, Directrix, Vertex, AxisOfSymmetry, Tangent, Normal, Eccentricity

[TYPED_ENTITIES]
Parabola: GeometricCurve
Focus: Point
Directrix: Line
Vertex: Point
AxisOfSymmetry: Line
FocalParameter: NonzeroReal
FocalLength: PositiveReal
Eccentricity: NonnegativeReal
LatusRectum: LineSegment
TangentLine: Line
NormalLine: Line
PointOnCurve: Point
CoordinateSystem: CartesianCoordinateSystem
QuadraticPolynomial: Polynomial

[FORMAL_DEFINITION]
Let F be a point and d be a line in the Euclidean plane with F not on d.
P(F,d) = { X in R^2 : distance(X,F) = distance(X,d) }.
The eccentricity of a nondegenerate parabola equals 1.
Its axis of symmetry is the line through F perpendicular to d.
Its vertex is the midpoint of F and the perpendicular projection of F onto d.

[CANONICAL_EQUATIONS]
For p != 0:
(x-h)^2 = 4p(y-k)
vertex = (h,k)
focus = (h,k+p)
directrix = { (x,y): y=k-p }
axis = { (x,y): x=h }
opens_up_if = p>0
opens_down_if = p<0
latus_rectum_length = 4*abs(p)

(y-k)^2 = 4p(x-h)
vertex = (h,k)
focus = (h+p,k)
directrix = { (x,y): x=h-p }
axis = { (x,y): y=k }
opens_right_if = p>0
opens_left_if = p<0
latus_rectum_length = 4*abs(p)

For y=a(x-h)^2+k, a != 0: p=1/(4a).
For y=ax^2+bx+c, a != 0: h=-b/(2a), k=c-b^2/(4a).
For vertical form, x=h+2pt, y=k+pt^2, t in R.
For horizontal form, x=h+pt^2, y=k+2pt, t in R.

[GEOMETRIC_PROPERTIES]
- A parabola is unbounded, connected and smooth.
- It has one vertex and one axis of symmetry.
- Every nondegenerate parabola has eccentricity 1.
- The focus lies on the axis; the directrix is perpendicular to it.
- For vertical orientation the tangent slope is (x-h)/(2p).
- The normal is perpendicular to the tangent in Euclidean geometry.
- The tangent at the vertex is parallel to the directrix.
- Reflection of rays parallel to the axis passes through the focus in ideal geometric optics.
- A parabola is a conic obtained from a suitable plane section of a right circular cone.

[RELATIONSHIP_SCHEMA]
HAS_FOCUS: Parabola -> Focus
HAS_DIRECTRIX: Parabola -> Directrix
HAS_VERTEX: Parabola -> Vertex
HAS_AXIS: Parabola -> AxisOfSymmetry
HAS_ECCENTRICITY: Parabola -> Eccentricity
HAS_PARAMETER: Parabola -> FocalParameter
HAS_TANGENT_AT: Parabola x PointOnCurve -> TangentLine
HAS_NORMAL_AT: Parabola x PointOnCurve -> NormalLine
IS_A: Parabola -> ConicSection
IS_LOCUS_OF: Parabola -> PointOnCurve
SYMMETRIC_ABOUT: Parabola -> AxisOfSymmetry
EQUIVALENT_REPRESENTATION: CanonicalEquation <-> FocusDirectrixDefinition [under Euclidean nondegenerate assumptions]

[INFERENCE_RULES]
R01: IF X is on P(F,d) THEN distance(X,F)=distance(X,d).
R02: IF F not_on d THEN P(F,d) is a nondegenerate parabola.
R03: IF P is a nondegenerate parabola THEN eccentricity(P)=1.
R04: IF (x-h)^2=4p(y-k) AND p!=0 THEN vertex=(h,k).
R05: IF (x-h)^2=4p(y-k) AND p!=0 THEN focus=(h,k+p).
R06: IF (x-h)^2=4p(y-k) AND p!=0 THEN directrix:y=k-p.
R07: IF (x-h)^2=4p(y-k) AND p>0 THEN P opens upward.
R08: IF (x-h)^2=4p(y-k) AND p<0 THEN P opens downward.
R09: IF (y-k)^2=4p(x-h) AND p>0 THEN P opens rightward.
R10: IF (y-k)^2=4p(x-h) AND p<0 THEN P opens leftward.
R11: IF (y-k)^2=4p(x-h) AND p!=0 THEN focus=(h+p,k).
R12: IF (y-k)^2=4p(x-h) AND p!=0 THEN directrix:x=h-p.
R13: IF y=a(x-h)^2+k AND a!=0 THEN p=1/(4a).
R14: IF y=ax^2+bx+c AND a!=0 THEN vertex_x=-b/(2a).
R15: IF y=ax^2+bx+c AND a!=0 THEN vertex_y=c-b^2/(4a).
R16: IF p!=0 THEN focal_length=abs(p).
R17: IF p!=0 THEN latus_rectum_length=4*abs(p).
R18: IF P has a vertical axis THEN its canonical equation is quadratic in x.
R19: IF P has a horizontal axis THEN its canonical equation is quadratic in y.
R20: IF X is on P THEN X reflected across its axis is on P.
R21: IF a point lies at the vertex THEN its tangent is parallel to the directrix.
R22: IF a tangent exists at X THEN its Euclidean normal is perpendicular to that tangent.
R23: IF y=a(x-h)^2+k AND a>0 THEN vertex is global minimum of y.
R24: IF y=a(x-h)^2+k AND a<0 THEN vertex is global maximum of y.
R25: IF y=a(x-h)^2+k AND a!=0 THEN the domain in x is R.
R26: IF y=a(x-h)^2+k AND a>0 THEN the range in y is [k,+infinity).
R27: IF y=a(x-h)^2+k AND a<0 THEN the range in y is (-infinity,k].
R28: IF p!=0 THEN focus and directrix do not intersect at focus.
R29: IF two valid focus-directrix pairs define the same point locus THEN they are geometrically equivalent descriptions.
R30: IF a Euclidean isometry maps focus F to F' and directrix d to d' THEN it maps P(F,d) to P(F',d').
R31: IF the equation is y=ax^2+bx+c AND a=0 THEN it does not define a nondegenerate vertical quadratic parabola.
R32: IF p=0 in a canonical equation THEN reject as nondegenerate parabola representation.
R33: IF the equation is (x-h)^2=4p(y-k) THEN the axis is x=h.
R34: IF the equation is (y-k)^2=4p(x-h) THEN the axis is y=k.
R35: IF P is a parabola THEN P is a plane curve.
R36: IF P is a nondegenerate parabola THEN P is not a circle or ellipse.

[VALIDATION_CONSTRAINTS]
V01: Focus and directrix must be specified in the same Euclidean coordinate system.
V02: Focus must not lie on the directrix.
V03: Canonical focal parameter p must be nonzero.
V04: Quadratic coefficient a must be nonzero for a vertical quadratic graph.
V05: Units of h,k,p and coordinates must be consistent.
V06: Latus rectum length must be positive.
V07: Focal length must be positive.
V08: A parabola is not a closed bounded ellipse.
V09: Preserve orientation when transforming canonical equations.
V10: Do not confuse focal parameter p with focal length abs(p).
V11: Validate candidate points against focus-directrix equality or the equation.
V12: Distinguish an entire parabola from a restricted parabolic arc.

[MATHEMATICAL_EXAMPLES]
E01: y=x^2 -> vertex(0,0), focus(0,1/4), directrix y=-1/4.
E02: y=2x^2 -> p=1/8; focus(0,1/8).
E03: y=-x^2 -> p=-1/4; opens downward.
E04: (x-2)^2=8(y+1) -> vertex(2,-1), focus(2,1), directrix y=-3.
E05: (y-3)^2=12(x+2) -> vertex(-2,3), focus(1,3).
E06: y=x^2-4x+5 -> y=(x-2)^2+1; vertex(2,1).
E07: y=-2(x+1)^2+3 -> vertex(-1,3), maximum y=3.
E08: x^2=4y -> p=1, latus rectum length=4.
E09: y^2=-8x -> p=-2, focus(-2,0), directrix x=2.
E10: x^2=-12y -> p=-3, focus(0,-3).
E11: (x-1)^2=2(y-4) -> p=1/2, focus(1,9/2).
E12: y=(x+3)^2-2 -> vertex(-3,-2), axis x=-3.
E13: y=3x^2+6x+1 -> vertex(-1,-2).
E14: y^2=4x -> parametric x=t^2, y=2t.
E15: x^2=4y -> point(2,1) belongs to the parabola.
E16: x^2=4y -> point(2,2) does not belong to the parabola.

[SEMANTIC_EQUIVALENCE]
"parabolic curve" ~= "parabola" [geometry context]
"parabolic locus" ~= "focus-directrix equidistant locus" [nondegenerate Euclidean context]
"axis of symmetry" ~= "symmetry axis" [contextual]
"vertex" ~= "turning point" [vertical quadratic graph context]
"focal parameter" != "focal length" [signed versus absolute]
"parabola" != "parabolic arc" [whole versus subset]
"parabola" != "quadratic function" [curve versus function]

[KNOWLEDGE_GRAPH_TRIPLES]
Parabola | IS_A | ConicSection
Parabola | IS_A | PlaneCurve
Parabola | HAS_FOCUS | Focus
Parabola | HAS_DIRECTRIX | Directrix
Parabola | HAS_VERTEX | Vertex
Parabola | HAS_AXIS | AxisOfSymmetry
Parabola | HAS_ECCENTRICITY | 1
Parabola | HAS_PROPERTY | ReflectiveFocusProperty
Parabola | REPRESENTABLE_BY | QuadraticEquation
Parabola | RELATED_TO | EuclideanGeometry
Parabola | RELATED_TO | AnalyticGeometry
Parabola | RELATED_TO | Optics

[CONFIDENCE]
formal_definitions: 0.99
canonical_equations: 0.99
inference_rules: 0.97
examples: 0.99
source_page_alignment: unverified
confidence_note: Scores are editorial estimates, not measured statistical probabilities.

[FLOW_HASH]
algorithm: SHA-256
canonicalisation: UTF-8, LF newlines, exclude FLOW_HASH digest field
value: COMPUTE_ON_PUBLICATION

[UPDATE_MODEL]
version: 4.0
revision_policy: Increment version on semantic or formal rule changes.
provenance_policy: Record source URL, revision date, editor, validation outcomes and hash.
compatibility: Preserve stable ontology identifiers and relation names where possible.
validation: Recheck equations, typing, examples and rule preconditions after changes.

[AI_CONSUMPTION]
entity_resolution: Map synonyms to Parabola while preserving distinctions among curve, arc and function.
reasoning_mode: Apply inference rules only when their typed premises and coordinate assumptions hold.
mathematical_mode: Retain signed focal parameter and distinguish it from focal length.
retrieval_mode: Index definition, equations, examples, constraints and graph triples.
conflict_resolution: Prefer mathematically validated statements over lexical matches.
uncertainty_mode: Mark claims as unverified when page-specific evidence is unavailable.
output_mode: Return typed relations, supporting rules, conditions and provenance.

[END_FLOW]