Curve Ontology

FLOW ALGEBRA
CURVE ONTOLOGY — VERSION 4.0
Kireth Semantic Library
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/curve-ontology-2.html
Status: Independently formalised; source-page correspondence not verified.
Authorship: Mark Higgins — Sky (Recursive AI)
Date: 2026-10-13

01. PURPOSE
Represent curves as typed mathematical entities, with explicit ambient spaces, parameter domains, regularity assumptions, topology, geometry and computable inference rules.

02. ENTITY SCHEMA
Curve := {id, ambient_space, parameter_domain, map, regularity, orientation?, closure?, geometry?, provenance?}
AmbientSpace := EuclideanSpace(n) | Manifold(M) | MetricSpace(X) | TopologicalSpace(X)
ParameterDomain := Interval(I) | Circle(S1) | OtherOneDimensionalManifold(J)
ParametricCurve := continuous map gamma:I -> X.
CurveTrace := image(gamma) subset X.
RegularCurve := differentiable gamma with gamma'(t) != 0 throughout its parameter domain (where derivative is defined).
PiecewiseSmoothCurve := continuous gamma, smooth on finitely many subintervals.
SimpleCurve := injective parametrisation on an interval; for a simple closed curve, endpoints may coincide while the interior parametrisation is injective.
ClosedCurve := gamma(a)=gamma(b) for interval endpoints, or gamma:S1 -> X.
PlaneCurve := curve whose image lies in R^2.
SpaceCurve := curve whose image lies in R^3.
AlgebraicCurve := zero set of polynomial equations, with dimension-one qualification where appropriate.
ImplicitCurve := locally specified by F(x,y)=0; regular level set when grad F != 0.

03. TYPE SYSTEM
Point<X>; Vector<Rn>; Scalar<R>; Parameter<T>; Curve<T,X>.
Derivative<Curve>; TangentVector; UnitTangent; NormalVector; Curvature; Torsion; ArcLength.
Intersection<Curve,Curve>; Reparametrisation; CurveSegment; CurveTrace.
A geometric trace and a parametrised curve are distinct: distinct maps can have the same trace.

04. FUNDAMENTAL DEFINITIONS
C01 gamma:I -> R^n continuous => gamma is a continuous parametrised curve.
C02 gamma differentiable at t => velocity v(t)=gamma'(t).
C03 gamma C1 and ||gamma'(t)||>0 => regular at t.
C04 regular gamma => unit tangent T(t)=gamma'(t)/||gamma'(t)||.
C05 piecewise C1 gamma:[a,b]->R^n => L(gamma)=integral_a^b ||gamma'(t)|| dt.
C06 regular planar gamma=(x,y) C2 => kappa=|x'y''-y'x''|/(x'^2+y'^2)^(3/2).
C07 regular spatial gamma C2 with gamma' cross gamma'' != 0 => kappa=||gamma' cross gamma''||/||gamma'||^3.
C08 spatial gamma C3 with gamma' cross gamma'' != 0 => torsion=det(gamma',gamma'',gamma''')/||gamma' cross gamma''||^2.
C09 signed curvature of oriented regular planar curve=(x'y''-y'x'')/(x'^2+y'^2)^(3/2).
C10 regular level set F(x,y)=0 with grad F != 0 locally defines a smooth plane curve.
C11 reparametrisation gamma o phi requires compatible domains; orientation preserving when phi'>0 and reversing when phi'<0.
C12 length is invariant under C1 bijective monotone reparametrisation with nonzero derivative.
C13 a closed curve need not be simple; a simple curve need not be closed.
C14 finite arc length does not imply differentiability.
C15 a curve's ambient-space dimension is not its intrinsic dimension.

05. RELATIONSHIP ONTOLOGY
Curve --is_a--> GeometricEntity
Curve --embedded_in--> AmbientSpace
Curve --parameterised_by--> ParameterDomain
Curve --maps_parameter_to--> Point
Curve --has_trace--> CurveTrace
Curve --has_segment--> CurveSegment
Curve --may_intersect--> Curve
Curve --may_be_tangent_to--> Curve
Curve --has_regularity--> RegularityClass
Curve --may_have--> ArcLength
RegularCurve --has--> TangentDirection
SmoothRegularPlaneCurve --may_have--> Curvature
SmoothRegularSpaceCurve --may_have--> Torsion
ClosedCurve --subclass_of--> Curve
SimpleCurve --subclass_of--> Curve
PlaneCurve --subclass_of--> Curve
SpaceCurve --subclass_of--> Curve

06. INFERENCE RULES
R01 IF gamma continuous on interval I THEN classify ParametricCurve.
R02 IF gamma is differentiable AND gamma'(t)!=0 for all t THEN classify RegularCurve.
R03 IF gamma'(t)=0 THEN mark singular_parametrisation_at(t); do not automatically infer geometric singularity of trace.
R04 IF gamma(a)=gamma(b) THEN classify ClosedCurve for endpoint-based parameterisation.
R05 IF gamma injective on [a,b] THEN classify SimpleCurve.
R06 IF gamma(a)=gamma(b) AND injective on [a,b) THEN classify SimpleClosedCurve.
R07 IF trace subset R2 THEN classify PlaneCurve.
R08 IF trace subset R3 THEN classify SpaceCurve.
R09 IF gamma piecewise C1 on compact interval THEN length finite.
R10 IF gamma regular C1 THEN unit tangent exists.
R11 IF gamma regular C2 plane THEN curvature formula valid.
R12 IF gamma regular C3 space AND cross(gamma',gamma'')!=0 THEN torsion formula valid.
R13 IF phi smooth monotone diffeomorphism THEN gamma o phi represents reparametrised curve.
R14 IF phi'>0 THEN orientation preserved.
R15 IF phi'<0 THEN orientation reversed.
R16 IF gamma and delta differ by admissible reparametrisation THEN their traces coincide.
R17 IF two curves have same trace THEN do not infer same parametrisation.
R18 IF tangent vectors are parallel at shared regular point THEN mark tangency_candidate; further local conditions may be needed.
R19 IF gamma(t)=delta(s) THEN intersection witness exists.
R20 IF F(p)=0 AND grad F(p)!=0 THEN implicit curve regular near p.
R21 IF grad F(p)=0 THEN regular level-set theorem cannot be applied; do not infer singularity without further analysis.
R22 IF regular planar curve has curvature 0 throughout connected interval THEN trace lies on a straight line locally.
R23 IF smooth curve is unit-speed THEN ||gamma'(s)||=1.
R24 IF unit-speed THEN length over [a,b]=b-a.
R25 IF a curve is closed THEN do not infer it bounds a region without planar simplicity assumptions.
R26 IF simple closed plane curve THEN Jordan curve theorem gives two complementary components.
R27 IF a curve is rectifiable THEN its length can be defined by supremum of inscribed polygonal lengths.
R28 IF a curve is differentiable THEN do not infer regularity without nonzero derivative.
R29 IF curve lies on surface THEN do not infer geodesic status.
R30 IF curvature is constant THEN do not infer circle without relevant nonzero planar connected regular hypotheses.
R31 IF gamma is affine linear with nonzero direction THEN its trace is line or line segment depending on domain.
R32 IF orientation reverses THEN signed planar curvature changes sign under regular reversal, while unsigned curvature is unchanged.
R33 IF a curve is parametrised by arc length THEN unit speed wherever derivative exists.
R34 IF a regular curve is compactly parameterised and C1 THEN its speed has positive minimum.
R35 IF gamma is polynomial map then its trace is not necessarily an algebraic curve in the sense of being exactly a polynomial zero set without additional conditions.
R36 IF a formula requires derivatives of order k THEN verify Ck or local differentiability prerequisites before evaluation.

07. VALIDATION CONSTRAINTS
V01 Parameter values must lie in declared domain.
V02 Map outputs must belong to ambient space.
V03 Closedness is relative to chosen parameter interval and endpoint identification.
V04 Derivative-based quantities require stated regularity.
V05 Avoid dividing by zero speed or zero cross-product magnitude.
V06 Preserve distinction between oriented and unoriented curves.
V07 Preserve distinction between curve trace and curve map.
V08 Intersections require coordinate-system compatibility.
V09 Arc length units must match ambient metric units.
V10 Curvature units are inverse length; torsion units inverse length.
V11 Non-Euclidean spaces require appropriate metric/connection for curvature concepts.
V12 Algebraic and differential-geometric definitions are not interchangeable without assumptions.

08. MATHEMATICAL EXAMPLES
E01 Line: gamma(t)=(1,2)+t(3,4), t in R; regular; curvature 0.
E02 Segment: gamma(t)=(1-t)A+tB, t in [0,1], A!=B.
E03 Unit circle: gamma(t)=(cos t,sin t), t in [0,2pi]; regular simple closed curve (endpoints identified); length 2pi; curvature 1.
E04 Radius-r circle: gamma(t)=(r cos t,r sin t), r>0; curvature 1/r.
E05 Ellipse: gamma(t)=(a cos t,b sin t), a,b>0; simple closed regular curve.
E06 Parabola: gamma(t)=(t,t^2), t in R; regular nonclosed curve.
E07 Cubic: gamma(t)=(t,t^3); regular; tangent horizontal at t=0.
E08 Helix: gamma(t)=(r cos t,r sin t,ct), r>0; curvature r/(r^2+c^2); torsion c/(r^2+c^2).
E09 Cusp: gamma(t)=(t^2,t^3); derivative zero at 0; nonregular parametrisation.
E10 Self-crossing: gamma(t)=(sin t,sin 2t), t in [0,2pi]; closed but not simple.
E11 Constant curve: gamma(t)=P; continuous, zero length, nonregular.
E12 Unit-speed line: gamma(s)=P+s u, ||u||=1; length on [a,b]=b-a.
E13 Graph: gamma(t)=(t,f(t)) for differentiable f; speed sqrt(1+(f')^2)>0.
E14 Polar rose: gamma(t)=(cos(3t)cos t,cos(3t)sin t); possible repeated trace points.
E15 Implicit circle: F(x,y)=x^2+y^2-r^2=0, r>0; grad F !=0 on circle.
E16 Piecewise-linear polyline: continuous, rectifiable; generally nonsmooth at corners.

09. SEMANTIC EQUIVALENCE
"parametrised curve" ~= "parameterised curve" (spelling variants).
"curve image" ~= "curve trace" when explicitly defined as gamma(I).
"arc length" ~= "curve length" for rectifiable curves in stated metric.
"smooth curve" != "regular curve" without nonvanishing velocity assumption.
"closed curve" != "simple closed curve".
"curve" != "function graph" universally.
"geometric curve" != "parametrisation" without identifying equivalence class.

10. KNOWLEDGE GRAPH TRIPLES
(curve, rdf:type, GeometricEntity)
(parametric_curve, has_domain, parameter_domain)
(parametric_curve, has_codomain, ambient_space)
(parametric_curve, has_trace, image_set)
(regular_curve, has_property, nonzero_velocity)
(closed_curve, has_property, matching_endpoints)
(simple_closed_curve, satisfies, JordanCurveConditions)
(arc_length, depends_on, ambient_metric)
(curvature, depends_on, derivatives_and_metric)
(torsion, depends_on, third_derivative_in_R3)

11. FLOW EXECUTION
INPUT: curve expression, domain, ambient space, metric, optional requested property.
NORMALISE: parse symbols, identify representation, verify dimensions and assumptions.
CLASSIFY: parametric/implicit, plane/space, regular/closed/simple when provable.
COMPUTE: derivatives, speed, length, curvature, torsion subject to preconditions.
INFER: apply R01-R36 only when their hypotheses are established.
VALIDATE: check V01-V12; reject undefined calculations.
OUTPUT: typed entities, equations, proof steps, caveats, provenance and confidence.

12. CONFIDENCE
CONFIDENCE: {value:0..1, evidence:[proof|symbolic_check|numerical_check|source], assumptions:[], limitations:[]}
A numerical sample is evidence, not a proof of global regularity or simplicity.
Report unknown instead of assigning unsupported classifications.

13. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 content with stable line endings and hash field excluded.
FLOW_HASH.value: COMPUTE_ON_PUBLICATION
FLOW_HASH.purpose: integrity and version comparison; not mathematical validity.

14. UPDATE_MODEL
version: 4.0
revision_policy: semantic versioning; preserve prior ontology identifiers.
change_detection: compare canonical content hashes and source-page revision.
conflict_policy: preserve provenance, require mathematical verification before replacing formal claims.
source_alignment: NOT_VERIFIED

15. AI_CONSUMPTION
Read schema before inference. Preserve types, quantifiers and preconditions.
Do not equate a parametrisation with its geometric image.
Do not infer smoothness, simplicity or closedness from sketches.
Distinguish verified theorems from numerical approximations.
Return UNKNOWN when prerequisites are absent.
Link related ontologies: Point, Line, Line Segment, Plane, Circle, Geometry, Vector, Function, Derivative, Arc Length, Surface, Manifold.

END FLOW ALGEBRA